Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours

TL;DR

This paper introduces a novel method for improving Christofides algorithm on specific TSP instances by gluing tours over 3-edge cuts, reducing edge usage and achieving better approximation ratios.

Contribution

It develops a new approach for gluing tours over 3-edge cuts in TSP, leading to improved bounds and a new approximation algorithm for certain problem instances.

Findings

Reduced usage of edges with x-value 1 from 3/2 to 3/2 - θ/10.

Constructed convex combinations of tours for specific fractional points.

Achieved a 17/12-approximation for TSP under certain conditions.

Abstract

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a $\theta$-cyclic point: $x_e \in \{0,\theta, 1-\theta, 1\}$, where the support graph is subcubic and each vertex is incident to at least one edge with $x$-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with $x$-value 1 from the $\frac{3}{2}$ of Christofides algorithm to $\frac{3}{2}-\frac{\theta}{10}$ while keeping the usage of edges with fractional $x$-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a $\frac{2}{3}$-uniform point: $x_e \in \{0, \frac{2}{3}\}$, we give a $\frac{17}{12}$-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of $\frac{3}{2}$ of Christofides algorithm and $\frac{4}{3}$ implied by the famous "four-thirds conjecture".

Equations

$$\min \{cx:\sum _{j\in V_n\setminus \{i\}}{x}_{ij}=2\text{ for }i\in V_n,\sum _{i\in U,j\notin U}{x}_{ij}\ge 2\text{ for }\varnothing \subset U\subset V_n,x\in [0,1{]}^{E_n}\}.$$

$$g(\mathrm{TSP})=\underset{n\ge 3,c\in {\mathbb{R}}_{\ge 0}^n}{\max }\frac{{\min }_{x\in \mathrm{TSP}(K_n)}cx}{{\min }_{x\in \mathrm{SEP}(K_n)}cx}.$$

$$g(\mathrm{TSP})=\min \{\alpha :\alpha \cdot x\in \mathrm{TSP}(K_n)\text{for}n\ge 3,x\in \mathrm{SEP}(K_n)\}.$$

$$v\text{textsc}-\mathrm{Tree}(G)=$$

$$\displaystyle\;x(E[U])\leq|U|-1\mbox{ for all $\emptyset\subset U\subseteq V\setminus\{v\}$},\;x(E)=|V|\}.$$

$$O\text{textsc}-\mathrm{Join}(G)=\{z\in [0,1{]}^E:$$

$$\text{ for }U\subseteq V,A\subseteq \delta (U),|U\cap O|+|A|\text{ odd}\}.$$

$$\sum _{T\in \mathcal{T}}{\lambda }_T\sum _{J\in {\mathcal{J}}^T}{\theta }_J{\chi }^{T+J}$$

$${\mathbb{P}}_u=\{\{2e_u\},\{e_u,f_u\},\{e_u,g_u\},\{2e_u,2f_u\},\{2e_u,2g_u\},\{2e_u,f_u,g_u\},\{e_u,2f_u,g_u\},\{e_u,f_u,2g_u\}\}.$$

$$\sum _{p\in {\mathbb{P}}_u:{\chi }_e^p=2}\varphi (p)$$

$$\sum _{p\in {\mathbb{P}}_u:{\chi }_e^p=1}\varphi (p)$$

$$\sum _{p\in {\mathbb{P}}_u}\varphi (p)$$

$$\varphi (\{2e_u\})$$

$$\sum _{i=1}^h\alpha (x_e+z_e^i)=\sum _{\ell =1}^h\alpha \cdot \frac{3}{2}-\alpha \cdot \frac{\theta }{2}=\frac{3}{2}-\frac{\alpha \theta }{2}.$$

$$\mathrm{Pr}[e_u\text{ is doubled in F }]=\mathrm{Pr}[e_u\in J\text{leavevmode}\text{and}\text{leavevmode}{e}_u\in T]=\mathrm{Pr}[e_u\in T]\cdot \mathrm{Pr}[e_u\in J].$$

$$\mathrm{Pr}[e_u\text{ is doubled in F }]$$

$$=\mathrm{Pr}[e_u\in J|e_u\in M]\cdot \mathrm{Pr}[e_u\in M]+\mathrm{Pr}[e_u\in J|e_u\notin M]\cdot \mathrm{Pr}[e_u\notin M]$$

$$=\frac{1-\theta }{2}\cdot \alpha +\frac{1}{2}\cdot (1-\alpha )$$

$$=\frac{1}{2}-\frac{\alpha \theta }{2}.$$

$${\mathcal{T}}_f=\{T\in \mathcal{T}:f\in T\text{ and }g\notin T\}\text{ and }{\mathcal{T}}_g=\{T\in \mathcal{T}:g\in T\text{ and }f\notin T\},$$

$$\sum _{T\in \mathcal{T}}{\lambda }_T{\chi }_f^T$$

$$=0+(\theta -\Lambda )+\Lambda +0=x_f.$$

$$\sum _{T\in \mathcal{T}}{\lambda }_T{\chi }_g^T$$

$$=\Lambda +0+0+(1-\theta -\Lambda )=x_g.$$

$$|U|=\sum _{C\in \mathcal{C}:V(C)\subseteq U}|V(C)\cap U|+\sum _{C\in \mathcal{C}:V(C)\cap U\ne \varnothing }|V(C)\cap U|.$$

$$\hat{z}=\sum _{\mathcal{C}\in \mathcal{C}}{\lambda }_{\mathcal{C}}{y}^{\mathcal{C}}\in \mathrm{TSP}(K_n).$$

$${\hat{z}}_e=$$

$$=$$

$$\mathrm{Pr}[e\in J|e\in M]$$

$$=\frac{1}{6}\cdot \frac{1}{7}+\frac{1}{2}\cdot \frac{6}{7}=\frac{19}{42}.$$

$$\mathrm{Pr}[e\in J]$$

$$=\frac{19}{42}\cdot \frac{1}{4}+\frac{1}{6}\cdot \frac{3}{4}=\frac{5}{21}.$$

$$\mathbb{E}[{\chi }_e^F]$$

$$=2\cdot \frac{1}{2}\cdot \frac{5}{21}+\frac{1}{2}\cdot \frac{16}{21}+\frac{1}{2}\cdot \frac{5}{21}=\frac{3}{4}-\frac{1}{84}.$$

Full text

1 Introduction

In the Traveling Salesperson Problem (TSP) we are given an integer $n\geq 3$ as the number of vertices and a non-negative cost vector $c$ defined on the edges of the complete graph $K_{n}=(V_{n}=\{1,\ldots,n\},E_{n}={\{1,\ldots,n\}\choose 2})$. We wish to find the minimum cost Hamilton cycle in the graph $K_{n}$ with respect to costs $c$. This problem is NP-hard and it is even NP-hard to approximate within any constant factor [WS11]. A natural assumption is that the cost vector $c$ is metric: $c_{ij}+c_{jk}\geq c_{ik}$ for $i,j,k\in V_{n}$. This special case of TSP is called metric TSP. Metric TSP is NP-hard [GJ90]. In fact, metric TSP is APX-hard and NP-hard to approximate with a ratio better than 220/219 [PV06]. Since we never deal with non-metric TSP in this paper, we henceforth refer to metric TSP by TSP.

The following linear programming relaxation for the TSP is known as the subtour elimination relaxation.

[TABLE]

We let $\operatorname{{SEP}}(K_{n})$ denote the feasible region of this linear programming relaxation. Since vector $c$ is metric, any spanning, connected Eulerian multi-subgraph of $K_{n}$ (henceforth a tour of $K_{n}$) can be used to find a Hamilton cycle of $K_{n}$ of no greater cost.222For a graph $G=(V_{n},E)$, we define a tour of $G$ to be a tour of $K_{n}$ that uses only edges in $E$. Notice that the incidence vector for such a tour lives in $\mathbb{R}^{E}$. We define $\operatorname{{{TSP}}}(K_{n})$ to be the convex hull of incidence vectors of tours of $K_{n}$. The integrality gap of the subtour elimination relaxation for the TSP is333 We use $\mathbb{R}^{p}_{\geq 0}$ to denote $\{x\in\mathbb{R}^{p},x_{i}\geq 0\text{ for }i\in\{1,\dots,p\},x\neq 0\}$.

[TABLE]

By the characterization of the integrality gap by Goemans [Goe95] (see also [CV04]), $g(\operatorname{{{TSP}}})$ can also be defined as

[TABLE]

It is well-known that $g(\operatorname{{{TSP}}})\geq\frac{4}{3}$. Based on the definition of $g(\operatorname{{{TSP}}})$ in (1.2), we can interpret this lower bound as follows: for any $\epsilon>0$, there is a point $x$ such that $x\in\operatorname{{SEP}}(K_{n})$ and $(\frac{4}{3}-\epsilon)x\notin\operatorname{{{TSP}}}(K_{n})$. As for upper bounds, a polyhedral analysis of the classical algorithm of Christofides proves $g(\operatorname{{{TSP}}})\leq\frac{3}{2}$, as well as providing a $\frac{3}{2}$-approximation algorithm for the TSP [Chr76, Wol80].

Theorem 1.1** **(Polyhedral proof of Christofides

algorithm [Chr76, Wol80]).

If $x\in\operatorname{{SEP}}(K_{n})$, then $\frac{3}{2}x\in\operatorname{{{TSP}}}(K_{n})$.

For more than four decades, there was no result that showed for all $x\in\operatorname{{SEP}}(K_{n})$, the vector $(\frac{3}{2}-\epsilon)x\in\operatorname{{{TSP}}}(K_{n})$ for some constant $\epsilon>0$. After we submitted this journal paper, this was resolved for a tiny value of $\epsilon$ [KKO21a, KKO21b]. Motivated by the lower bound of $\frac{4}{3}$ on $g(\operatorname{{{TSP}}})$, the following has been conjectured and is wide open.

Conjecture 1** (The four-thirds conjecture).**

If $x\in\operatorname{{SEP}}(K_{n})$, then $\frac{4}{3}x\in\operatorname{{{TSP}}}(K_{n})$.

Despite the lack of progress towards resolution of Conjecture 1, there has been great success in providing new bounds on $g(\operatorname{{{TSP}}})$ for special cases in the past decade [OSS11, MS16, SV14]. Next we present two equivalent formulations of Conjecture 1 that are relevant for our results.

1.1 Fundamental Classes for TSP

One approach to the four-thirds conjecture is to consider fundamental classes for TSP. Fundamental classes of points were introduced by Carr and Ravi [CR98] and further developed by Boyd and Carr [BC11] and Carr and Vempala [CV04]. A set of vectors $\mathcal{X}$ is a fundamental class for TSP if (i) for every $x\in\mathcal{X}$ we have $x\in\operatorname{{SEP}}(K_{n})$ and (ii) proving $\alpha\cdot x\in\operatorname{{{TSP}}}(K_{n})$ for all $x\in\mathcal{X}$ implies $g(\operatorname{{{TSP}}})\leq\alpha$.

1.1.1 Cyclic Points

For $x\in\operatorname{{SEP}}(K_{n})$, we define $G_{x}=(V_{n},E_{x})$ to be the subgraph of $K_{n}$ whose edge set corresponds to the support of $x$ (i.e., $E_{x}=\{e\;:\;x_{e}>0\}$).444We sometimes abuse notation and treat $x$ as a vector in $\mathbb{R}^{E_{x}}$. The set of cyclic points form a fundamental class for TSP with a very simple structure [CV04].

Definition 1**.**

A point $x$ is called a $\theta$-cyclic point for some $0<\theta\leq\frac{1}{2}$ if:

• •

Vector $x$ is in $\operatorname{{SEP}}(K_{n})\cap\{0,\theta,1-\theta,1\}^{E_{n}}$.

• •

The support graph of $x$, $G_{x}=(V_{n},E_{x})$, is subcubic.

• •

For each $v\in V_{n}$ there is at least one edge $e\in\delta(v)$ with $x_{e}=1$.

Observe that for a $\theta$-cyclic point $x$ we have: (i) the set of 1-edges in $G_{x}$, $W_{x}=\{e:x_{e}=1\}$, forms vertex-disjoint paths of $G_{x}$, (ii) the fractional edges in $G_{x}$, $H_{x}=\{e:x_{e}<1\}$, form vertex-disjoint cycles of $G_{x}$. It is easy to see that if $\theta<\frac{1}{2}$ all the cycles in $H_{x}$ have even length since $x\in\operatorname{{SEP}}(K_{n})$. Conjecture 1 can be restated as follows.

Conjecture 2**.**

For any $\theta\in(0,1)$ and any $\theta$-cyclic point $x\in\mathbb{R}^{E_{n}}$, we have $\frac{4}{3}x\in\operatorname{{{TSP}}}(K_{n})$.

Similarly, Theorem 1.1 can also be restated as follows.

Theorem 1.2**.**

For any $\theta\in(0,1)$ and any $\theta$-cyclic point $x\in\mathbb{R}^{E_{n}}$, we have $\frac{3}{2}x\in\operatorname{{{TSP}}}(K_{n})$.

The main result of this paper is to show that we can save on the 1-edges of $\theta$-cyclic points.

Theorem 1.3**.**

Let $x\in\mathbb{R}^{E_{n}}$ be a $\theta$-cyclic point. Define vector $y$ as follows: $y_{e}=\frac{3}{2}-\frac{\theta}{10}$ for $e\in W_{x}$, $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$ and $y_{e}=0$ for $e\notin E_{x}$. Then $y\in\operatorname{{{TSP}}}(K_{n})$.

In fact a bound on $g(\operatorname{{{TSP}}})$ restricted to $\frac{1}{2}$-cyclic points would also provide a bound on $g(\operatorname{{{TSP}}})$ when restricted to all half-integral points of the subtour elimination relaxation [CV04], a special case that has received some attention [CR98, BS21, KKO20], and was highlighted in a recent conjecture of Schalekamp, Williamson and van Zuylen stating that the maximum on $g(\operatorname{{{TSP}}})$ is achieved for half-integral points of the subtour elimination polytope [SWvZ13]. For $\frac{1}{2}$-cyclic points Theorem 1.3 implies the following.

Corollary 1.4**.**

Let $x$ be a $\frac{1}{2}$-cyclic point. Define vector $y$ as follows: $y_{e}=\frac{3}{2}-\frac{1}{20}$ for $e\in W_{x}$, $y_{e}=\frac{3}{4}$ for $e\in H_{x}$ and $y_{e}=0$ for $e\notin E_{x}$. Then $y\in\operatorname{{{TSP}}}(K_{n})$.

Theorem 1.3 gives an improved bound for recently studied special case of uniform points, which form another fundamental class for TSP.

1.1.2 Uniform Points

A point $x\in\operatorname{{SEP}}(K_{n})$ is a $\frac{2}{k}$-uniform point for some $k\in\mathbb{Z}_{\geq 3}$ if for each edge $e\in E_{n}$, either $x_{e}$ is a multiple of $\frac{2}{k}$ or $x_{e}=0$. It is clear that $g(\operatorname{{{TSP}}})\leq\alpha$ if and only if for all $k\in\mathbb{Z}_{\geq 3}$ and all $\frac{2}{k}$-uniform points $x$ we have $\alpha\cdot x\in\operatorname{{{TSP}}}(K_{n})$. Therefore, $\frac{2}{k}$-uniform points form a fundamental class for TSP. We can restate the four-thirds conjecture as follows.

Conjecture 3**.**

For any integer $k\geq 3$ and $\frac{2}{k}$-uniform point $x\in\mathbb{R}^{E_{n}}$ we have $\frac{4}{3}x\in\operatorname{{{TSP}}}(K_{n})$.

Sebő et al. considered a weakening of Conjecture 3 in the case when $k=3$ [SBS14]. For a $\frac{2}{3}$-uniform point $x\in\mathbb{R}^{E_{n}}$, we have $\frac{3}{2}x\in\operatorname{{{TSP}}}(K_{n})$ by Theorem 1.1. Thus, they asked if there is constant $\epsilon>0$ such that the vector $(\frac{3}{2}-\epsilon)\cdot x\in\operatorname{{{TSP}}}(K_{n})$. Of course, the four-thirds conjecture itself implies that the value of $\epsilon$ is at least $\frac{1}{6}$.

Conjecture 4**.**

If $x\in\mathbb{R}^{E_{n}}$ is a $\frac{2}{3}$-uniform point, then $\frac{4}{3}x\in\operatorname{{{TSP}}}(K_{n})$.

One application of Theorem 1.3 is to show that the value of $\epsilon$ is at least $\frac{1}{12}$, which brings us “halfway” towards resolving Conjecture 4. The proof of Theorem 1.5 can be found in Section 5.1.

Theorem 1.5**.**

Let $x\in\mathbb{R}^{E_{n}}$ be a $\frac{2}{3}$-uniform point. Then $\frac{17}{12}x\in\operatorname{{{TSP}}}(K_{n})$. If $G_{x}$ is Hamiltonian, then $\frac{87}{68}x\in\operatorname{{{TSP}}}(K_{n})$.

The following observation was first made by Carr and Vempala [CV04].

Proposition 1.6**.**

Let $k\in\mathbb{Z}_{\geq 3}$. We have $\alpha\cdot x\in\operatorname{{{TSP}}}(K_{n})$ for all $\frac{2}{k}$-uniform points $x$ if and only if for all $k$-edge-connected $k$-regular multigraphs $G=(V,E)$, the point $\alpha\cdot(\frac{2}{k}\cdot\chi^{E})$ dominates a convex combination of tours of $G$.

Focusing on uniform points (i.e., proving Conjecture 3) is a possible path towards the four-thirds conjecture. One way to attack this conjecture is to decompose a uniform point into a convex combination of different points with a useful structure. For example, Theorem 1.5 uses this approach for $\frac{2}{3}$-uniform points by decomposing them into convex combinations of $\frac{1}{2}$-cyclic points and applying our main result (Theorem 1.3).

The next step on this path is to study $\frac{2}{4}$-uniform points. TSP on $\frac{2}{4}$-uniform points is equivalent to so-called “half-integral” TSP: if for any 4-edge-connected 4-regular graph $G$ the vector $\alpha\cdot(\frac{1}{2}\cdot\chi^{E(G)})$ dominates a convex combination of incidence vectors of tours of $G$, then $g(\operatorname{{{TSP}}})$ restricted to half-integral instances is at most $\alpha$.

For graph $G=(V,E)$, a cut $U\subset V$ is proper if $|U|\geq 2$ and $|V\setminus{U}|\geq 2$. If we assume that the 4-regular 4-edge-connected graph $G$ does not contain proper 4-edge cuts (i.e., it is essentially 6-edge-connected), then the following theorem, whose proof appears in Section 5.2, is relevant. Its proof uses essentially the same techniques we use to prove Theorems 1.3 and 1.5, but is simpler (i.e., does not require gluing) because there are no tight proper cuts containing 1-edges.

Theorem 1.7**.**

Let $G=(V,E)$ be a 4-edge-connected 4-regular graph with an even number of vertices and no proper 4-edge cuts. Then the vector $(\frac{3}{2}-\frac{1}{42})\cdot(\frac{1}{2}\cdot\chi^{E})$ dominates a convex combination of incidence vectors of tours of $G$.

Theorem 1.7 could serve as the base case if we could prove an analogous theorem when $G$ has an odd number of vertices and if we could glue over proper 4-edge cuts of $G$. However, the gluing arguments we present for $\theta$-cyclic points can not easily be extended to this case due to the increased complexity of the distribution of patterns over 4-edge cuts.555We remark that [GLL*+*21] did extend Theorem 1.7 to the case when $G$ has an odd number of vertices. This is discussed further in Section 1.2.

1.2 Related Work on Fundamental Points

Fundamental points were introduced in a series of papers for the TSP [CR98] and for the minimum cost 2-edge-connected multigraph problem (2ECM) [CR98, BC11, CV04]. Let $\operatorname{{{2ECM}}}ec(K_{n})$ be the convex hull of incidence vectors of 2-edge-connected multigraphs of $K_{n}$. Clearly, $\operatorname{{{TSP}}}(K_{n})\subseteq\operatorname{{{2ECM}}}ec(K_{n})$.

Consider a $\frac{1}{2}$-cyclic point $x\in\mathbb{R}^{E_{n}}$. If $H_{x}$ is a collection of 3-cycles, then Boyd and Carr showed $\frac{4}{3}x\in\operatorname{{{TSP}}}(K_{n})$ [BC11] and Boyd and Legault showed $\frac{6}{5}x\in\operatorname{{{2ECM}}}ec(K_{n})$ [BL15]. Boyd and Sebő presented a polynomial time algorithm proving that if $H_{x}$ is a collection of 4-cycles, then $\frac{10}{7}x\in\operatorname{{{TSP}}}(K_{n})$ [BS21]. For the same class, we gave an efficient algorithm proving $\frac{9}{7}x\in\operatorname{{{2ECM}}}ec(K_{n})$ [HN21]. The interest in $\frac{1}{2}$-cyclic points stems in part from the aforementioned conjecture that the maximum value of $g(\operatorname{{{TSP}}})$ is achieved for instances of the TSP where the optimal solution to $\min\{cx\;:\;x\in\operatorname{{SEP}}(K_{n})\}$ is $\frac{1}{2}$-cyclic [SWvZ13]. In fact, in each of the classes above there is a family of instances that achieves the largest known lower bound on $g(\operatorname{{{TSP}}})$ and the integrality gap of the subtour elimination relaxation for 2ECM [CR98, ABE06, BC11, BS21].

Now we review the results on uniform points. Let $x\in\mathbb{R}^{E_{n}}$ be a $\frac{2}{k}$-uniform point. Carr and Ravi showed if $k=4$, then $\frac{4}{3}x\in\operatorname{{{2ECM}}}ec(K_{n})$ [CR98]. Boyd and Legault showed that if $k=3$, then $\frac{6}{5}x\in\operatorname{{{2ECM}}}ec(K_{n})$ [BL15]. Legault later improved the factor $\frac{6}{5}$ to $\frac{7}{6}$ [Leg17]. Haddadan, Newman and Ravi proved that for $k=3$, we have $\frac{27}{19}x\in\operatorname{{{TSP}}}(K_{n})$ [HNR21]. Boyd and Sebő showed that if $G_{x}$ is additionally Hamiltonian, then $\frac{9}{7}x\in\operatorname{{{TSP}}}(K_{n})$ [BS21].

Recent Results:

Since the conference version [HN19] of this paper was announced, there have been many developments. First, Karlin, Klein and Oveis Gharan showed that for any $\frac{1}{2}$-cyclic point $x\in\mathbb{R}^{E_{n}}$, we have $(\frac{3}{2}-\epsilon)x\in\operatorname{{{TSP}}}(K_{n})$ for some constant $\epsilon>0$ [KKO20]. Subsequently, Gupta et al. [GLL*+*21] combined our approach with that of [KKO20] resulting in a further improved approximation factor for this case. Essentially, they extend Theorem 1.7 to the odd case and are then able to substitute this combinatorial construction for the max-entropy approach used by [KKO20] in the base case. They then use other methods (from [KKO20]) in lieu of gluing. With respect to uniform points, if $x\in\mathbb{R}^{E_{n}}$ is a $\frac{2}{k}$-uniform point, then these results also imply that when $k=4$, $(\frac{3}{2}-\epsilon)x\in\operatorname{{{TSP}}}(K_{n})$ for some constant $\epsilon>0$ [KKO20, GLL*+*21]. Finally, another recent result showed that $(\frac{3}{2}-\epsilon)x\in\operatorname{{{TSP}}}(K_{n})$ for all $k$ for some tiny value of $\epsilon$ [KKO21a, KKO21b].

1.3 Gluing Convex Combinations Over 3-edge Cuts

A key part of our proof of Theorem 1.3 is gluing solutions over certain 3-edge cuts, thereby reducing to instances without such cuts, which are easier to solve. This approach of gluing solutions over 3-edge cuts, and thereby reducing to a problem on graphs without proper 3-edge cuts was first introduced by Cornuéjols, Naddef and Pulleyblank [CNP85]. For a graph $G=(V,E)$, let $U\subset V$ and denote by $G^{U}$ the graph obtained by contracting $U$ (i.e., identifying all vertices in $U$ to a single vertex and removing the resulting loops). Cornuéjols et al. defined a class of 3-edge-connected graphs $\mathcal{A}$ as fully reducible if

• •

If $G\in\mathcal{A}$ has a proper 3-edge cut $U$, then both $G^{U}$ and $G^{\overline{U}}$ are in $\mathcal{A}$.

• •

The minimum cost Hamilton cycle of $G$ can be found in polynomial time for the graphs in $\mathcal{A}$ that do not have a proper 3-edge cut.

Cornuéjols showed that TSP can be solved in polynomial time for fully reducible graphs [CNP85]. An example of such a fully reducible class are the Halin graphs [CNP83].

For many fully reducible classes $\mathcal{A}$, they showed that for $G\in\mathcal{A}$, if $G$ does not have a proper 3-edge cut, then the convex hull of incidence vectors of Hamilton cycles of $G$ coincides with a system of linear inequalities with polynomial separation [CNP85]. For example, they show if $G=(V,E)$ does not contain any disjoint cycles, then $P_{G}=\{x\in[0,1]^{E}\;:\;x(\delta(v))=2\text{ for }v\in V\}$ is the convex hull of incidence vectors of Hamilton cycles of $G$. Let us describe this result in more detail. Suppose for any graph $G$ with no proper 3-edge cuts that does not contain any disjoint cycles, we have $\operatorname{{{TSP}}}(G)=P_{G}$. Now consider a graph $G=(V,E)$ with no disjoint cycles that has a 3-edge cut $U$. In the graph $G^{U}$, we say the vertex corresponding to the contracted set $U$ is a pseudovertex. Suppose that the graphs $G^{U}$ and $G^{\overline{U}}$ contain no proper 3-edge cuts and suppose we can write $y$ restricted to the edge set of each graph as a convex combination of Hamilton cycles of the respective graph. Let us consider the patterns around the pseudovertices; if the edges adjacent to the pseudovertices are $\{a,b,c\}$ then each vertex can be adjacent to two edges in a Hamilton cycle and therefore, there are only three possible patterns around a vertex: $\{\{a,b\},\{a,c\},\{b,c\}\}$. Moreover, since each pattern appears the same percentage of time (in the respective convex combinations) for each pseudovertex, tours with corresponding patterns can be glued over the 3-edge cut. In this case, the gluing procedure is quite straightforward. This reduction has also proven quite useful for the minimum cost 2-edge-connected subgraph problem (2EC) [CR98, BL15, Leg17].

In contrast, it appears such a reduction is not known for TSP tours. Indeed, gluing proofs cannot be easily extended to tours for several reasons: (1) As just shown, they are often used for gluing subgraphs (no doubled edges). In TSP, we must allow edges to be doubled, so there are too many possible patterns around a vertex. For example, if we allow each possible pattern corresponding to an even degree, there are 13 possible patterns. (2) Gluing tours over a 3-edge cut might result in disconnected Eulerian multigraphs. Finally, (3) many of the algorithms based on gluing are not proven to run in polynomial time [CR98, BL15, Leg17].

1.4 Our Approach

The main technical contribution of this paper is to show that for a carefully chosen set of tours, we can design a gluing procedure over critical cuts, which, roughly speaking are proper 3-edge cuts that are tight: the $x$-values of the three edges crossing the cut sum to 2. As we will see, a tour generated via a polyhedral version of Christofides algorithm on a $\theta$-cyclic point can have eight (rather than 13) possible patterns around a vertex. While this is still a lot, we identify certain conditions for the convex combination, under which controlling the frequency of a single one of these patterns allows us to control the frequency of the other seven. Moreover, we show the frequency of one of the patterns (around an arbitrary vertex $v$) depends on the fraction of times in the convex combination of connectors that vertex $v$ is a leaf. Thus, we can fix a critical cut $U\subset V_{n}$ in $G_{x}$ and find a convex combination of tours for $G_{x}^{U}$. Then we can find a set of tours for $G_{x}^{\overline{U}}$ such that the distribution of patterns around the pseudovertex corresponding to $U$ matches that of the pseudovertex corresponding to $\overline{U}$ in $G_{x}^{U}$, which enables us to glue over critical cuts.

Applying this gluing procedure, we can reduce an instance of TSP on a $\theta$-cyclic point to base-case instances, which contain no critical cuts. On a high level, our proof of Theorem 1.3 for such instances is based on Christofides algorithm: We show that a $\theta$-cyclic point $x$ can be written as a convex combination of connected, spanning subgraphs of $G_{x}$ with no doubled edges (henceforth a connector of $G_{x}$) with certain properties and then we show that the vector $z$, where $z_{e}=\frac{x_{e}}{2}$ for $e\in H_{x}$ and $z_{e}=\frac{1}{2}-\frac{\theta}{10}$ for $e\in W_{x}$, can be written as a convex combination of subgraphs, each of which can be used for parity correction of a connector (henceforth a parity corrector). Tight cuts are generally difficult to handle using an approach based on Christofides algorithm, since $(\frac{1}{2}-\epsilon)x$ is insufficient for parity correction of a tight cut if it is crossed by an odd number of edges in the connector. However, in our base-case instances, there are only two types of tight 3-edge cuts. The first type of cut is a degenerate tight cut. These cuts are easy to handle and we defer their formal definition to Section 2.1. The second type of cut is a vertex cut, which we show are also easy to handle. In particular, the parity of vertex cuts can be addressed with a key tool used by Boyd and Sebő [BS21] called rainbow $v$-trees (see Theorem 2.5). Using this in combination with a decomposition of the 1-edges into few induced matchings, which have some additional required properties, we can prove Theorem 1.3 for the base case.

1.5 Organization

In Section 2, we introduce notation and some definitions relevant to cut structure in cyclic points. We also review some well-known polyhedral tools. In Section 3, we present our main ideas and tools for gluing tours over critical cuts of $G_{x}$, thereby reducing to TSP on base cases that only contain certain types of tight cuts. We then show in Section 4 that we are able to handle these remaining tight cuts via an approach similar—on a high-level—to Christofides algorithm. For our gluing approach to work, we need to choose the connectors to have certain properties and to save on the 1-edges, we need to show that a vector with less than half on each 1-edge belongs to the $O$-join polytope. Both of these key technical ingredients can be found in Section 4. Sections 3 and 4 contain a complete proof of Theorem 1.3. In Section 5, we present an application of Theorem 1.3 to approximating TSP on $\frac{2}{3}$-uniform points. We also give an algorithm for TSP on $\frac{2}{4}$-uniform points under certain assumptions. Finally, in Section 6, we make some concluding remarks and present some problems for future research.

2 Notation and Tools

Let $G=(V,E)$ be a graph. For a subset $U\subset V$ of vertices, let $\delta_{G}(U)=\{uv\in E:u\in U,v\notin U\}$. (We use $\delta(U)=\delta_{G}(U)$ when the graph $G$ is clear from the context.) Let $E[U]=\{uv\in E:u\in U,v\in U\}$.

A multi-subset (henceforth multiset for brevity) of edges of $E$ is a set that can contain multiple copies of edges in $E$. A multi-subgraph (henceforth multigraph for brevity) of $G$ is the graph on vertex set $V$ whose edge set is a multiset of $E$ (i.e., a multigraph can contain multiple copies of an edge). We sometimes consider a multigraph $F$ of $G$ to be a multiset of edges of $G$.

The incidence vector of multigraph $F$ of $G$, denoted by $\chi^{F}$ is a vector in $\mathbb{R}^{E}$ where $\chi^{F}_{e}$ is the multiplicity of $e$ in $F$. Let $F$ and $F^{\prime}$ be two multigraphs of $G$, then $F+F^{\prime}$ is the multigraph that contains $\chi^{F}_{e}+\chi^{F^{\prime}}_{e}$ copies of edge $e$ for $e\in E$. If $e$ is an edge in $F$, then $F-e$ is the multigraph with incidence vector $\chi^{F-e}=\chi^{F}-\chi^{\{e\}}$. For a vector $x\in\mathbb{R}^{E}$ and a multigraph $F$ of $G$, we denote $\sum_{e\in F}x_{e}\cdot\chi^{F}_{e}$ by $x(F)$. We use $\delta_{F}(U)$ to refer to the multiset of edges in $F$ that have exactly one endpoint in $U$. The degree of a vertex $v\in V$ in $F$ is the number of edges in $F$ that are incident on $v$.

2.1 Cuts in Cyclic Points

Let $x$ be a $\theta$-cyclic point with support graph $G_{x}=(V_{n},E_{x})$. The graph $G_{x}$ contains three types of tight cuts, by which we mean a cut $U\subset V_{n}$ such that $x(\delta(U))=2$. A vertex cut is a cut $U=\{u\}$. Notice that for all $u\in V_{n}$, $x(\delta(u))=2$. The second type of cut is a critical cut.

Definition 2**.**

A proper cut $U\subset V_{n}$ in $G_{x}$ is called a critical cut if $|\delta(U)|=3$ and $\delta(U)$ contains exactly one edge $e$ with $x_{e}=1$. Moreover, for each pair of edges in $\delta(U)$, their endpoints in $U$ (and in $V\setminus{U}$) are distinct.

We refer to the third type of cut as a degenerate tight cut.

Definition 3**.**

A proper cut $U\subset V_{n}$ in $G_{x}$ is called a degenerate tight cut if $|\delta(U)|=3$, $|U|>3$ and $|V\setminus{U}|>3$ and the two fractional edges in $\delta(U)$ share an endpoint in either $U$ or $V\setminus{U}$.

For a degenerate tight cut $U$, let $\delta(U)=\{e,f,g\}$, such that $f$ and $g$ are the fractional edges that share an endpoint $v$. Let $e_{v}$ be the unique 1-edge incident on $v$. Observe that $\{e,e_{v}\}$ forms a 2-edge cut in $G_{x}$.

Let $x$ be a $\theta$-cyclic point and $U\subset V_{n}$ be a cut in $G_{x}$. Also, let $\overline{U}=V_{n}\setminus U$. We can obtain $\theta$-cyclic point $x^{U}$ by contracting set $\overline{U}$ in $G_{x}$ to a single vertex. (Respectively, we can obtain a $\theta$-cyclic point $x^{\overline{U}}$ by contracting set $U$ in $G_{x}$ to a single vertex.) We let $v_{\overline{U}}$ denote the vertex corresponding to set $\overline{U}$ in ${G_{x}^{U}}$ (respectively, $v_{U}$ corresponds to $U$ in ${G_{x}^{\overline{U}}}$).

Observation 2.1**.**

Let $U\subseteq V_{n}$ be a minimal critical cut in $G_{x}$ (i.e., for $S\subset U$, the cut defined by $S$ is not critical in $G_{x}$). Then, ${G_{x}^{U}}$ does not contain any critical cuts.

Proof.

Suppose for contradiction that there is $S\subset V({G_{x}^{U}})$ that is a critical cut of ${G_{x}^{U}}$. We can assume that $v_{\overline{U}}\notin S$. Moreover, $S\subsetneq U$. This is a contradiction to minimality of $U$ since $S$ constitutes a critical cut in $G_{x}$ as well. ∎

Observation 2.2**.**

Suppose that $G_{x}$ has $k$ critical cuts. Let $U$ be a critical cut of $G_{x}$. The number of critical cuts in ${G_{x}^{U}}$ is at most $k-1$.

Proof.

Clearly, $U$ is not a critical cut of ${G_{x}^{U}}$. We show that there is correspondence between the critical cuts of ${G_{x}^{U}}$ and $G_{x}$. This implies that ${G_{x}^{U}}$ can have at most $k-1$ critical cuts. If $S$ is a critical cut of ${G_{x}^{U}}$ we can assume without loss of generality that $v_{\overline{U}}\notin S$. Hence, $S$ is also a critical cut of $G_{x}$. ∎

2.2 Polyhedral Basics

Let $G=(V,E)$ and let $x$ be a vector in $\mathbb{R}^{E}$. Consider a collection of multigraphs $\mathcal{F}$ of $G$. We say $\lambda=\{\lambda_{F}\}_{F\in\mathcal{F}}$ are convex multipliers for $\mathcal{F}$ if $\sum_{F\in\mathcal{F}}\lambda_{F}=1$ and $\lambda_{F}\geq 0$ for $F\in\mathcal{F}$. We say $\{\lambda,\mathcal{F}\}$ is a convex combination for $x$ if $\lambda=\{\lambda_{F}\}_{F\in\mathcal{F}}$ are the convex multipliers for $\mathcal{F}$ and $x=\sum_{F\in\mathcal{F}}\lambda_{F}$. We say $x$ can be written as a convex combination of multigraphs in $\mathcal{F}$ if we can find such $\mathcal{F}$ and $\lambda$ in polynomial time in the size of $x$. Here by the size of $x$ we refer to $|E|$ (i.e., the number of edges in the support of $x$).

2.2.1 The $v$-Tree Polytope and Rainbow $v$-Trees

Let $G=(V,E)$ be a graph. For a vertex $v\in V$, a $v$-tree is a subgraph $F$ of $G$ such that $|F\cap\delta(v)|=2$ and $F\setminus\delta(v)$ induces a spanning tree of $V\setminus\{v\}$. Denote by $v\mathrm{\textsc{-Tree}}(G)$ the convex hull of incidence vectors of $v$-trees of $G$. The $v\mathrm{\textsc{-Tree}}(G)$ is characterized by the following linear inequalities.

[TABLE]

Observation 2.3**.**

We have $\operatorname{{SEP}}(K_{n})\subseteq v\mathrm{\textsc{-Tree}}(K_{n})$ for all $v\in K_{n}$.

Observation 2.4**.**

Let $x\in\operatorname{{SEP}}(K_{n})$ be such that $G_{x}$ is 3-edge-connected and cubic. Let $\mathcal{C}$ be any 2-factor in $G_{x}$. Define vector $y$ to have $y_{e}=\frac{1}{2}$ for $e\in\mathcal{C}$, $y_{e}=1$ for $e\in E_{x}\setminus\{C\}$ and $y_{e}=0$ otherwise. Then $y\in\operatorname{{SEP}}(K_{n})\subseteq v\mathrm{\textsc{-Tree}}(K_{n})$.

Proof.

Take $\emptyset\subset U\subset V_{n}$. If $U=\{v\}$ for some $v\in V_{n}$, then $x(\delta(U)=2$ as $\mathcal{C}\cap\delta(v)=2$.

If $|\delta(U)|\geq 4$, then clearly $x(\delta(U))\geq 2$. Otherwise, $|\delta(U)|=3$. Since at most two edges in $\delta(U)$ belong to $\mathcal{C}$, there is at least one edge $e\in\delta(U)$ with $x_{e}=1$. Hence, $x(\delta(U))\geq 2$. Therefore, $x\in\operatorname{{SEP}}(K_{n})$. We have $x\in v\mathrm{\textsc{-Tree}}(K_{n})$ by Observation 2.3. ∎

It can be deduced from the discussion above that a vector $x$ in the subtour elimination relaxation can be written as a convex combination of $v$-trees for any vertex $v$ in $G_{x}$. In fact, the $v$-trees in this convex combination can satisfy some additional properties.

Definition 4**.**

Let $G=(V,E)$ and $v$ be a vertex of $G$. Let $\mathcal{P}$ be a collection of disjoint subsets of $E$. A $\mathcal{P}$-rainbow $v$-tree, namely $T$, is a $v$-tree of $G$ such that $|T\cap P|=1$ for $P\in\mathcal{P}$.

The following theorem can be proved via the matroid intersection theorem [Edm03] and Observation 2.3.

Theorem 2.5** ([BL95],[BS21]).**

Let $x\in\operatorname{{SEP}}(K_{n})$ and $\mathcal{P}$ be a collection of disjoint subsets of $E_{x}$ such that $x(P)=1$ for $P\in\mathcal{P}$. Then $x$ can be written as a convex combination of $\mathcal{P}$-rainbow $v$-trees of $K_{n}$ for any $v\in V_{n}$.

Grötchel and Padberg [GP85] observed that $v$-trees of a connected graph $G=(V,E)$ satisfy the basis axioms of a matroid. For $x\in\operatorname{{SEP}}(K_{n})$ we have $x\in v\mathrm{\textsc{-Tree}}(K_{n})$ by Observation 2.3. Also, $\mathcal{P}$ defines a partition matroid where each base intersect each part of $\mathcal{P}$ exactly once. Therefore, vector $x$ is in the convex hull of incidence vector of common basis of the partition matroid defined by $\mathcal{P}$ and the matroid whose basis are the $v$-trees of $K_{n}$.

2.2.2 The $O$-join Polytope

Let $G=(V,E)$ be a graph and $O\subseteq V$ where $|O|$ is even. An $O$-join of $G$ is a subgraph $J$ of $G$ where a vertex $v\in V$ has odd degree in $J$ if and only if $v\in O$. Let $O\mathrm{\textsc{-{Join}}}(G)$ be the convex hull of incidence vectors of $O$-joins of $G$. Edmonds and Johnson [EJ73] showed the following description for the $O\mathrm{\textsc{-{Join}}}(G)$.

[TABLE]

Observation 2.6**.**

If $x\in\operatorname{{SEP}}(K_{n})$, then $\frac{x}{2}\in O\mathrm{\textsc{-{Join}}}(K_{n})$ for any $O\subseteq V_{n}$ with $|O|$ even.

Proof.

Let $z=\frac{x}{2}$. For $U\subseteq V_{n}$, we have $z(\delta(U))\geq 1$. Moreover, for $e\in E_{n}$, we have $z_{e}\leq\frac{1}{2}$ since $x_{e}\leq 1$. Therefore, for $A\subseteq\delta(U)$ we have $z(A)\leq\frac{|A|}{2}$. This implies $z(\delta(U))-2z(A)\geq 1-|A|$. ∎

The following observation states a useful property of $O$-joins that can be obtained via (2.2).

Observation 2.7**.**

Let $G=(V,E)$ be a graph, and let $O\subseteq V$ be a subset of vertices such that $|O|$ is even. Let $z\in O\mathrm{\textsc{-{Join}}}(G)$, and $z(\delta(u))\leq 1$ for all $u\in V$. Then $z$ can be written as convex combination of $O$-joins of $G$ denoted by $\{\psi,\cal{J}\}$ such that for $u\in O$ we have $|J\cap\delta(u)|=1$ for $J\in\cal{J}$.

Proof.

By [EJ73], if $z\in O\mathrm{\textsc{-{Join}}}(G)$, then $z$ can be written as a convex combination of $O$-joins of $G$ denoted by $\{\psi,\cal{J}\}$. Let $u$ be a vertex in $O$. We have $z(\delta(u))\leq 1$. On the other hand, for every $J\in\mathcal{J}$ we have $|J\cap\delta(u)|\geq 1$. Therefore, $|J\cap\delta(u)|=1$. ∎

2.2.3 Proof of Theorem 1.1: Polyhedral Analysis of Christofides

Now, we are ready to prove Theorem 1.1. See 1.1

Proof.

Let $x\in\operatorname{{SEP}}(K_{n})$. Then by Observation 2.3, $x\in v\mathrm{\textsc{-Tree}}(K_{n})$ for some $v\in V_{n}$. Hence, we can find $v$-trees $\mathcal{T}$ and convex multipliers $\lambda$ for $\mathcal{T}$ such that $x=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{T}$. For each $T\in\mathcal{T}$, let $O_{T}$ be the set of odd degree vertices of $T$. Notice that $\frac{x}{2}\in O_{T}\mathrm{\textsc{-{Join}}}(K_{n})$ for all $T\in\mathcal{T}$. This implies that $\frac{x}{2}$ can be written as a convex combination of $O_{T}$-joins $\mathcal{J}^{T}$ of $K_{n}$ with convex multipliers $\theta=\{\theta_{J}\}_{J\in\mathcal{J}^{T}}$ (i.e., $\frac{x}{2}=\sum_{J\in\mathcal{J}^{T}}\theta_{J}\chi^{J}$). Notice that for $T\in\mathcal{T}$ and $J\in\mathcal{J}^{T}$, multigraph $T+J$ is a tour of $K_{n}$. Hence, $\sum_{T\in\mathcal{T}}\lambda_{T}\sum_{J\in\mathcal{J}^{T}}\theta_{J}\chi^{T+J}\in\operatorname{{{TSP}}}(K_{n})$. Moreover,

[TABLE]

Therefore, $\frac{3}{2}x\in\operatorname{{{TSP}}}(K_{n})$. ∎

3 Matching Patterns: Gluing Tours Over Critical Cuts

Let $x$ be a $\theta$-cyclic point and $G_{x}=(V_{n},E_{x})$ be the support of $x$. In this section, we present an approach for gluing tours over critical cuts of $G_{x}$. One property of the tours we construct, which is crucial to enable this gluing procedure, is that every tour contains at least one copy of each 1-edge. This allows us to assume that $x$ belongs to a subclass of $\theta$-cyclic points in which (i) $G_{x}$ is cubic, (ii) $W_{x}$, the 1-edges of $G_{x}$, form a perfect matching, and (iii) $H_{x}$, the fractional edges of $G_{x}$, form a 2-factor. We can make this assumption, because we can contract a path of 1-edges to a single 1-edge; the tour of the new cubic graph yields a tour for the original subcubic graph. We work under this assumption throughout this section and in Section 4.

For a vertex $u\in V$, denote by $e_{u}$ the unique 1-edge in $G_{x}$ that is incident on $u$. Let $\delta(u)=\{e_{u},{f_{u}},{g_{u}}\}$ where ${f_{u}}$ and ${g_{u}}$ are the two fractional edges incident on $u$ and $x_{{f_{u}}}=\theta$ and $x_{{g_{u}}}=1-\theta$. In each tour of $G_{x}$, a multiset of edges from $\delta(u)$ belongs to the tour. We call this multiset the pattern around $u$. Denote by ${\mathbb{P}}_{u}$ the set of possible patterns around a vertex $u$ in a tour of $G_{x}$ that contain at least one copy of the 1-edge $e_{u}$ and in which $u$ has degree either two or four (see Figure 3.1).

[TABLE]

In every tour we construct, the pattern around each vertex $u\in V$ will be some pattern from $\mathbb{P}_{u}$. There are other multisets of $\delta(u)$ that can be valid patterns around $u$ in a tour. For example, the pattern $\{{f_{u}},{g_{u}}\}$ could be the pattern around $u$ in some tour. However, in our construction this pattern will never be the pattern around $u$ as we always include at least one copy of $e_{u}$ in a tour. Formally, we write $\delta_{F}(u)=p$ if the pattern around $u$ in the tour $F$ is $p\in\mathbb{P}_{u}$.

Definition 5**.**

A tour $F$ of $G_{x}=(V_{n},E_{x})$ is a handpicked tour of $G_{x}$ if for all $u\in V_{n}$, the pattern around $u$ in $F$ belongs to $\mathbb{P}_{u}$ (i.e., if $\delta_{F}(u)\in\mathbb{P}_{u}$ for all $u\in V_{n}$).

If $\mathscr{F}$ is a set of tours and $\phi=\{\phi_{F}\}_{F\in\mathscr{F}}$ is a set of convex multipliers, then we use $\phi(p)=\sum_{F\in\mathscr{F}:\delta_{F}(u)=p}\phi_{F}$ to denote the pattern frequency of $p\in\mathbb{P}_{u}$ in the convex combination $\{\phi,\mathscr{F}\}$. Notice that $\phi(p)\in[0,1]$. Moreover, if $\mathscr{F}$ is a set of handpicked tour of $G_{x}$, then for all $u\in V_{n}$, we have $\sum_{p\in\mathbb{P}_{u}}\phi(p)=1$. For each $u\in V_{n}$, we define the pattern profile of vertex $u$ in the convex combination $\{\phi,\mathscr{F}\}$ to be the eight values $\{\phi(p)\}$ for all $p\in\mathbb{P}_{u}$.

Another key parameter of a convex combination is the frequency of doubled edges. For the convex combination $\{\phi,\mathscr{F}\}$, define $\phi_{2}(e)=\sum_{F\in\mathscr{F}:\chi^{F}_{e}=2}\phi_{F}$ for all $e\in E_{x}$. The pattern profile of a vertex $u$ turns out to be directly related to the occurence of doubled edges from $\delta(u)$. This dependence is formalized in the next observation, which states that for a convex combination of handpicked tours, if the parameters $\phi_{2}(e)$ are fixed for $e\in\delta(u)$, then the pattern profile for each vertex $u$ depends only on the pattern frequency of the pattern $\{2e_{u}\}$.

Observation 3.1**.**

Let $y$ and $q$ be vectors in $\mathbb{R}^{E_{x}}_{\geq 0}$. Suppose $y$ can be written as a convex combination of tours of $G_{x}$ denoted by $\{\phi,\mathscr{F}\}$ such that for all $e\in E_{x}$, we have $\phi_{2}(e)=q_{e}$.

Then for each vertex $u\in V_{n}$ and for each pattern $p\in\mathbb{P}_{u}$, the frequency of pattern $p$, $\phi(p)$, in this convex combination is uniquely determined by the frequency of pattern $\{2e_{u}\}$.

Proof.

Suppose $\phi(\{2e_{u}\})=\zeta_{u}$ for some $\zeta_{u}\in[0,1]$. Then for each $u\in V_{n}$, the following identities hold with respect to the convex combination $\{\phi,\mathscr{F}\}$.

[TABLE]

Since the above system of equations has full rank (i.e., the variables are $\phi(p)$ for $p\in\mathbb{P}_{u}$), it has a unique solution. Therefore, $\phi(p)$ is a function of $\zeta_{u}$ for all $p\in\mathbb{P}_{u}$. ∎

We apply Observation 3.1 to control the pattern profile of a pseudovertex $u$ by constructing tours in which the pattern frequency of $\{2e_{u}\}$ can be set arbitrarily. This enables us to prove Theorem 1.3 with an inductive (gluing) approach. For such an approach to work, we need to prove a stronger statement. Let $\alpha$ be a constant in $(0,1]$ that we will fix later.

Proposition 3.2**.**

Define $y\in\mathbb{R}^{E_{x}}_{\geq 0}$ as follows: $y_{e}=\frac{3}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x}$ and $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$. Then $y$ can be written as a convex combination of handpicked tours of $G_{x}$ denoted by $\{\phi,\mathscr{F}\}$ such that

1. (i)

$\phi_{2}(e)=\frac{1}{2}-\frac{\alpha\theta}{2}$, for $e\in W_{x}$, and 2. (ii)

$\phi_{2}(e)=\frac{x_{e}^{2}}{2}$* for all $e\in H_{x}$.*

Observe that Proposition 3.2 implies Theorem 1.3. As mentioned previously, it is in fact stronger; we construct tours for the base cases ($\theta$-cyclic points whose support graphs have no critical cuts) and the additional properties in Proposition 3.2 enable us to “glue” these tours together over the critical cuts of $G_{x}$. Hence, our induction is on the number of critical cuts in $G_{x}$.

Observation 3.3 gives sufficient conditions under which we can glue tours of $G_{x}^{U}$ and $G_{x}^{\overline{U}}$ together over the critical cut to obtain tours for $G_{x}$ that preserve key properties.

Observation 3.3**.**

Let $U\subset V_{n}$ be a critical cut of $G_{x}$. Suppose $x^{U}$ can be written as a convex combination of handpicked tours of ${G_{x}^{U}}$ denoted by $\{\phi^{U},\mathscr{F}^{U}\}$ with the following properties. (And suppose the same holds for $x^{\overline{U}}$, ${G_{x}^{\overline{U}}}$ $\{\phi^{\overline{U}},\mathscr{F}^{\overline{U}}\}$, respectively.)

1. (i)

The pattern profiles of vertices $v_{U}$ and $v_{\overline{U}}$ are the same in their respective convex combinations. 2. (ii)

For every tour $F\in\mathscr{F}^{U}$, $F\setminus{\delta(v_{\overline{U}})}$ induces a connected multigraph on $U$.

Then $x$ can be written as a combination of handpicked tour of $G_{x}$ denoted by $\{\phi,\mathscr{F}\}$ such that

1. (a)

$\phi_{2}(e)=\phi^{U}_{2}(e)$* for $e\in E({G_{x}^{U}})$,* 2. (b)

$\phi_{2}(e)=\phi^{\overline{U}}_{2}(e)$* for $e\in E({G_{x}^{\overline{U}}})$, and* 3. (c)

$|\mathscr{F}|\leq|\mathscr{F}^{U}|+|\mathscr{F}^{\overline{U}}|$.

Proof.

First, we prove the following simple claim.

Claim 1**.**

Consider a graph $G=(V,E)$ and nonempty $U\subset V$ such that $U$ is a 3-edge-cut in $G=(V,E)$. Let $F_{U}$ be a tour in $G^{U}$ and let $F_{\overline{U}}$ be a tour in $G^{\overline{U}}$ such that $\chi^{F_{U}}_{e}=\chi^{F_{\overline{U}}}_{e}$ for $e\in\delta(U)$. Moreover, assume that $F_{U}\setminus{\delta(v_{\overline{U}})}$ induces a connected multigraph on $U$. Then the multiset of edges $F$ defined as $\chi^{F}_{e}=\chi^{F_{U}}_{e}$ for $e\in E(G^{U})$ and $\chi^{F}_{e}=\chi^{F_{\overline{U}}}_{e}$ for $e\in E(G^{\overline{U}})$ is a tour of $G$.

Proof..

It is clear that $F$ induces an Eulerian spanning multigraph on $G$, but we need to ensure that $F$ is connected. For example, the tour induced on $F_{\overline{U}}\setminus{\delta(v_{U})}$ might not be connected. However, since the subgraph of $F_{U}$ induced on the vertex set $U$ is connected, the tour $F$ is connected: each vertex in $\overline{U}$ is connected to some vertex in $U$. $\Diamond$

We observe that if the pattern profiles of $v_{U}$ and $v_{\overline{U}}$ with respect to the convex combinations $\{\phi^{U},\mathscr{F}^{U}\}$ and $\{\phi^{\overline{U}},\mathscr{F}^{\overline{U}}\}$, respectively, are the same, then we can always find two tours $F_{U}\in\mathscr{F}$ and $F_{\overline{U}}\in\mathscr{F}^{\overline{U}}$ such that the pattern around $v_{\overline{U}}$ in $F_{U}$ is the same as the pattern around $v_{U}$ in $F_{\overline{U}}$. We can apply Claim 1 to obtain a new tour $F$, to which we assign convex multiplier $\phi_{F}=\min\{\phi^{U}_{F_{U}},\phi^{\overline{U}}_{F_{\overline{U}}}\}$ and add to set $\mathscr{F}$. Then we subtract $\phi_{F}$ from each of these convex multipliers, remove tours with convex multipliers zero from $\mathscr{F}^{U}$ and $\mathscr{F}^{\overline{U}}$, and repeat. Observe the total number of tours in $\mathscr{F}$ is at most $|\mathscr{F}^{U}|+|\mathscr{F}^{\overline{U}}|$.

We need to show that each tour $F\in\mathscr{F}$ is handpicked. This follows from the fact that the pattern around each vertex in $U$ in $F$ is the same as the pattern around it in $F_{U}$. Moreover, each edge $e\in E_{x}\cap\delta(U)$ is doubled in a tour $F$ of $G_{x}$ iff it is doubled in both $F_{U}$ and in $F_{\overline{U}}$. For $e\in E({G_{x}^{U}})\setminus{\delta(U)}$, edge $e$ is doubled iff it is doubled in $F_{U}$. Analogously, each vertex in $\overline{U}$ has the same pattern in $F$ as it has in $F_{\overline{U}}$, and each edge $e\in E({G_{x}^{\overline{U}}})\setminus{\delta(U)}$ is doubled iff it is doubled in $F_{\overline{U}}$. Thus, properties (a) and (b) hold for the convex combination $\{\phi,\mathscr{F}\}$. ∎

In the base case (where graphs $G_{x}=(V_{n},E_{x})$ have no critical cuts), each tour in the convex combination that we construct consists of a connector plus a parity correction (as we will describe in Section 4). For each $u\in V_{n}$, the previously introduced (and yet to be fixed) parameter $\alpha$ is a lower bound on the fraction of connectors in this convex combination in which $u$ has degree two. Let the parameter $\eta_{u}$ denote the (exact) fraction of connectors in which vertex $u$ has degree one. (Note that $\eta_{u}$ can be different for every vertex.) In our construction of a convex combination of tours $\{\phi,\mathscr{F}\}$ for the base case, it will be the case that the frequency of pattern $\{2e_{u}\}$ will be equal to $\frac{\eta_{u}}{2}$. Thus, controlling the value of $\eta_{u}$ allows us to control $\phi(\{2e_{u}\})$. A key technical tool is that when we construct a convex combination of tours $\{\phi,\mathscr{F}\}$ for graph $G_{x}$ with no critical cuts (i.e., a base case graph), we can always ensure that for one arbitrarily chosen vertex $v$, the value of $\eta_{v}$ (and hence $\phi(\{2e_{v}\})$) can be chosen arbitrarily. This allows us to ensure that the pattern frequency of $\{2e_{v_{\overline{U}}}\})$ equals the pattern frequency of $\{2e_{v_{U}}\})$ in the given convex combination for ${G_{x}^{\overline{U}}}$ (i.e., Condition (i) in Observation 3.3) when ${G_{x}^{U}}$ is a base case.

Lemma 3.4**.**

Suppose $G_{x}$ contains no critical cuts. Fix any vertex $v\in V$ and fix constant $\zeta$ with $0\leq\zeta\leq\frac{(1-\alpha)\theta}{2}$. Define $y\in\mathbb{R}^{E_{x}}_{\geq 0}$ as follows: $y_{e}=\frac{3}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x}$ and $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$. Then $y$ can be written as a convex combination of handpicked tours of $G_{x}$ denoted by $\{\phi,\mathscr{F}\}$ with the following properties.

1. (i)

$\phi_{2}(e)=\frac{1}{2}-\frac{\alpha\theta}{2}$, for $e\in W_{x}$, 2. (ii)

$\phi_{2}(e)=\frac{x_{e}^{2}}{2}$* for all $e\in H_{x}$,* 3. (iii)

$\phi(\{2e_{v}\})=\zeta$, and 4. (iv)

$F\setminus{\delta_{F}(v)}$* induces a connected multigraph on $V\setminus{v}$ for each $F\in\mathscr{F}$.*

Notice that Lemma 3.4 implies Proposition 3.2 for $\theta$-cyclic points whose support graphs have no critical cuts. We prove Lemma 3.4 in the next section. In the remainder of this section, we show how Lemma 3.4 implies Proposition 3.2.

Proof of Proposition 3.2.

Suppose $G_{x}$ contains $t$ critical cuts. We prove the statement by induction on $t$. In fact we show that the running time of our algorithm is polynomial in $n$ and $t$. To this end, we show that the convex combination in our construction contains at most $btn^{d}$ trees, where $b$ and $d$ are constants. Notice that $t$ itself is a polynomial bounded by $n$.

If $t=0$, then $G_{x}$ does not contain a critical cut, then apply Lemma 3.4. Otherwise, find the minimal critical cut $U$ of $G_{x}$. By Observations 2.1 and 2.2, graph ${G_{x}^{U}}$ does not contain any critical cuts and ${G_{x}^{\overline{U}}}$ contains at most $t-1$ critical cuts.

Define $y^{\overline{U}}$ as follows: $y^{\overline{U}}_{e}=\frac{3}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x^{\overline{U}}}$ and $y^{\overline{U}}_{e}=\frac{x^{\overline{U}}}{2}$ for $e\in H_{x^{\overline{U}}}$. We apply the induction hypothesis on ${G_{x}^{\overline{U}}}$ to write $y^{\overline{U}}$ as convex combination of handpicked tours of ${G_{x}^{\overline{U}}}$ denoted by $\{\phi^{\overline{U}},\mathscr{F}^{\overline{U}}\}$ such that (i) $\phi^{\overline{U}}_{2}(e)=\frac{1}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x^{\overline{U}}}$, and (ii) $\phi_{2}(e)=\frac{(x^{\overline{U}}_{e})^{2}}{2}$ for all $e\in H_{x^{\overline{U}}}$. By induction $|\mathscr{F}^{\overline{U}}|\leq b(t-1)(|\overline{U}|+1)^{d}$.

Let $\zeta^{*}=\phi^{\overline{U}}(\{e_{v_{U}}\})$. Define $y^{U}_{e}=\frac{3}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x^{U}}$ and $y^{U}_{e}=\frac{x^{U}_{e}}{2}$ for $e\in H_{x^{U}}$. Applying Lemma 3.4 to $\theta$-cyclic point $x^{U}$ with $\zeta=\zeta^{*}$ we can write $y^{U}$ as convex combination of handpicked tours of ${G_{x}^{U}}$ denoted by $\{\phi^{U},\mathscr{F}^{U}\}$ such that (i) $\phi^{U}_{2}(e)=\frac{1}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x^{U}}$, (ii) $\phi^{U}_{2}(e)=\frac{(x^{U}_{e})^{2}}{2}$ for all $e\in H_{x^{U}}$, (iii) $\phi^{U}(\{2e_{v}\})=\zeta^{*}$, and (iv) $F\setminus{\delta_{F}(v_{\overline{U}})}$ induces a connected multigraph on $U$ for each $F\in\mathscr{F}^{U}$. Moreover $|\mathscr{F}^{U}|\leq b(|U|+1)^{d}$.

Now we apply Observation 3.3 to write $y$ as the desired convex combination. This convex combination contains at most $|\mathscr{F}^{U}|+|\mathscr{F}^{\overline{U}}|\leq b(|U|+1)^{d}+b(t-1)(|\overline{U}|+1)^{d}$. Note that for $d\geq 2$ this number is at most $btn^{d}$, as $|U|+|\overline{U}|=n$ , $|U|\geq 3$, and $|\overline{U}|\geq 3$. ∎

4 Finding Tours in the Base Case: Proof of

Lemma 3.4

In this section we present the proof of Lemma 3.4. We fix $G_{x}=(V_{n},E_{x})$ to be the support graph of a $\theta$-cyclic point $x$. In the base case, $G_{x}$ contains no critical cuts. (The lemmas in this section will be applied to this base case, but some hold even when $G_{x}$ is not a base case.) Moreover, we remind the reader that we assume the support of $G_{x}$ is cubic. (See discussion in the beginning of Section 3.)

A key tool we will use is that the 1-edges of $G_{x}$ can be partitioned into five induced matchings in $G_{x}$. A set $M\subset W_{x}$ is an induced matching of $G_{x}$ if $M$ is a vertex induced subgraph of $G_{x}$ and $M$ is a matching. For each induced matching $M$, we find a set of connectors $\mathcal{T}$ of $G_{x}$ where for each 1-edge $e$ in $M$, both endpoints of $e$ have degree two in every $T\in\mathcal{T}$. Thus, when $G_{x}$ has no critical cuts, a 1-edge $e$ in $M$ does not belong to any odd cuts in $T$ that are tight cuts in $G_{x}$. For each 1-edge $e$ in $M$, we can therefore reduce usage of $e$ in the parity correction from $\frac{1}{2}$ to $\frac{1-\theta}{2}$; each 1-edge saves $\frac{\theta}{2}$ exactly $\frac{1}{5}$ of the times. This yields the saving of $\frac{\theta}{10}$ on the 1-edges as stated in Lemma 3.4 with $\alpha=\frac{1}{5}$.

The induced matchings require some additional properties that we need for technical reasons as we will see later. Recall that for a vertex $u$ in $G_{x}$ we denote by $e_{u}$ the unique 1-edge incident on $u$. Let $N_{2}(u)$ denote the two vertices that are the other endpoints of the fractional-edges incident on $u$. In other words, suppose $\delta(u)=\{e_{u},f_{u},g_{u}\}$ and suppose that $w_{1}$ and $w_{2}$ are the other endpoints of $f_{u}$ and $g_{u}$, respectively. Then $N_{2}(u)=\{w_{1},w_{2}\}$. The proof of Lemma 4.1 is deferred to Section 4.5.

Lemma 4.1**.**

Suppose $G_{x}$ has no critical cuts. Let $v$ be a vertex in $V_{n}$ and let $N_{2}(v)=\{w_{1},w_{2}\}$. The set of $1$-edges in $G_{x}$, $W_{x}$, can be partitioned into five induced matchings $\{M_{1},\dots,M_{5}\}$ such that for $i\in[5]$, the following properties hold.

1. (i)

$|M_{i}\cap\{e_{v},e_{w_{1}},e_{w_{2}}\}|\leq 1$. 2. (ii)

For $U\subseteq V_{n}$ such that $|\delta(U)|=3$, $|\delta(U)\cap M_{i}|\leq 1$. 3. (iii)

For $U\subseteq V_{n}$ such that $|\delta(U)|=2$, $|\delta(U)\cap M_{i}|$ is even.

For the rest of this section, let $v$ be a fixed vertex in $V_{n}$, $N_{2}(v)=\{w_{1},w_{2}\}$ and let $\{M_{1},\dots,M_{h}\}$ denote the partition of $W_{x}$ into induced matchings with the additional properties enumerated in Lemma 4.1. These properties will be used to ensure that we can save on the edges in $W_{x}$ when augmenting connectors of $G_{x}$ (with parity correctors) into tours. While Lemma 4.1 implies that $h=5$, we will use $h$ throughout this section, since if Lemma 4.1 could be proved with, say, four matchings, it would allow a larger value of $\alpha=\frac{1}{h}$ and hence directly yield a better bound in the statement of Lemma 3.4.

The proof of Lemma 3.4 consists of two main parts. First we show there is a convex combination of connectors of $G_{x}$ that satisfy certain properties. Second, we show that for each connector, we can find parity correctors such that the union of a connector and a parity corrector is a tour.

4.1 Constructing Connectors

Now we will show how to construct connectors for $G_{x}$. We will apply this when $G_{x}$ is a base case (i.e., $G_{x}$ has no critical cuts), but Definition 6 and Lemmas 4.2 and 4.3 apply even when this is not the case. Recall that $v$ is fixed vertex in $V_{n}$ and that $N_{2}(v)=\{w_{1},w_{2}\}$.

Definition 6**.**

Suppose $M\subset W_{x}$ is a subset of 1-edges of $G_{x}$. Let $\Lambda$ be a constant such that $0\leq\Lambda\leq\theta$. Suppose $x$ can be written as a convex combination of connectors of $G_{x}$ denoted by $\{\lambda,\mathcal{T}\}$. Then we say $P(v,M,\Lambda)$ holds for $\{\lambda,\mathcal{T}\}$ if it has the following properties.

$\sum_{T\in\mathcal{T}:|\delta_{T}(v)|=1}\lambda_{T}=\sum_{T\in\mathcal{T}:|\delta_{T}(v)|=3}\lambda_{T}=\Lambda$* and $\sum_{T\in\mathcal{T}:|\delta_{T}(v)|=2}\lambda_{T}=1-2\Lambda$.* 2. 2.

For each edge $st\in M$, $|\delta_{T}(s)|=|\delta_{T}(t)|=2$ for all $T\in\mathcal{T}$. 3. 3.

$T\setminus\delta_{T}(v)$* induces a connector on $V\setminus\{v\}$.*

Let us explain why the properties described above are useful in our construction. The first property allows us to control the fraction of time vertex $v$ has degree one in a connector in the convex combination $\{\lambda,\mathcal{T}\}$, which in turn will allow us to control the fraction of time a tour has the pattern $\{2e_{v}\}$ around $v$. This flexibility is required to perform the gluing procedure; it allows us to manipulate the convex combination of connectors to have the desired pattern profile for the pseudovertex (which will be $v$). The second condition ensures that no 1-edge in $M$ is part of a tight cut that is crossed an odd number of times in a connector $T\in\mathcal{T}$. Lastly, the third property guarantees that we maintain connectivity of the tours after gluing them together over critical cuts.

We defer the proofs of the next two lemmas to Section 4.3.

Lemma 4.2**.**

Suppose $M\subset W_{x}$ forms an induced matching in $G_{x}$ and edge $e_{v}\in M$. Then $x$ can be written as a convex combination of connectors of $G_{x}$ denoted by $\{\lambda,\mathcal{T}\}$ for which $P(v,M,0)$ holds.

Lemma 4.3**.**

Let $\Lambda$ be any constant such that $0\leq\Lambda\leq\theta$. Suppose $M\subset W_{x}$ forms an induced matching in $G_{x}$, $e_{v}\notin M$ and $|M\cap\{e_{w_{1}},e_{w_{2}}\}|\leq 1$. Then $x$ can be written as a convex combination of connectors of $G_{x}$ denoted by $\{\lambda,\mathcal{T}\}$ for which $P(v,M,\Lambda)$ holds.

Recall that $\{M_{1},\ldots,M_{h}\}$ is the partition of $W_{x}$ into induced matchings obtained via Lemma 4.1. Assume without loss of generality that $e_{v}\in M_{1}$. For $i=1$, let $\mathcal{T}_{1}$ be a set of connectors of $G_{x}$ and let $\{\vartheta,\mathcal{T}_{1}\}$ be a convex combination for $x$ for which $P(v,M_{1},0)$ holds (by Lemma 4.2). For $i\in\{2,\dots,h\}$, let $\mathcal{T}_{i}$ be a set of connectors of $G_{x}$ and let $\{\vartheta,\mathcal{T}_{i}\}$ be a convex combination for $x$ for which $P(v,M_{i},\frac{\Lambda}{1-\alpha})$ holds (by Lemma 4.3). Notice that $\frac{\Lambda}{1-\alpha}\leq\theta$ since $\Lambda\leq(1-\alpha)\theta$.

We can write $x$ as a convex combination of connectors from $\mathscr{T}$, by weighting each set $\mathcal{T}_{i}$ by $\alpha$. In particular, we have $x=\alpha\sum_{i=1}^{h}\sum_{T\in\mathcal{T}_{i}}\vartheta_{T}\chi^{T}$. For each $T\in\mathscr{T}$, let $\sigma_{T}=\alpha\cdot\vartheta_{T}$. Then $\{\sigma,\mathscr{T}\}$ is a convex combination for $x$. Observe that since $x_{e}=1$ for $e\in W_{x}$, we have $W_{x}\subseteq T$ for $T\in\mathscr{T}$. From Definition 6 and Lemmas 4.2 and 4.3, we observe the following.

Claim 2**.**

For each $T\in\mathscr{T}$, $T\setminus{\delta(v)}$ induces a connected, spanning subgraph on $V\setminus{\{v\}}$.

4.2 Constructing Parity Correctors

For each $T\in\mathscr{T}$, let $O_{T}$ be the set of odd degree vertices of $T$. In the second part of the proof we show that each connected subgraph $T\in\mathscr{T}$ has a “cheap” convex combination of $O_{T}$-joins.

Lemma 4.4**.**

Suppose $G_{x}$ has no critical cuts. Let $M\subset W_{x}$ be a subset of 1-edges of $G_{x}$ such that each 3-edge cut in $G_{x}$ contains at most one edge from $M$. Let $O\subseteq V$ be a subset of vertices such that $|O|$ is even and for all $e=st\in M$, neither $s$ nor $t$ is in $O$. Also suppose for any set $U\subseteq V$ such that $|\delta(U)|=2$, both $|U\cap O|$ and $|\delta(U)\cap M|$ are even. Define vector $z$ as follows: $z_{e}=\frac{1}{2}$ if $e\in W_{x}$ and $e\notin M$, $z_{e}=\frac{1-\theta}{2}$ if $e\in M$, and $z_{e}=\frac{x_{e}}{2}$ if $e\in H_{x}$. Then vector $z\in O\mathrm{\textsc{-{Join}}}(G_{x})$.

For each $i\in[h]$, define $z_{e}^{i}=\frac{1-\theta}{2}$ if $e\in{M_{i}}$ and $z_{e}^{i}=\frac{x_{e}}{2}$ otherwise. For each $T\in\mathcal{T}_{i}$, let $O_{T}\subseteq V$ be the set of odd-degree vertices of $T$. By construction, we have $V(M_{i})\cap O_{T}=\emptyset$. By Lemma 4.4, we have $z^{i}\in O_{T}\mathrm{\textsc{-{Join}}}(G_{x})$. So by Observation 2.7 we can write $z^{i}$ as a convex combination of $O_{T}$-joins of $G_{x}$ denoted by $\{\psi^{T},\cal{J}_{T}\}$ where $|J\cap\delta(u)|=1$ for $u\in O_{T}$ and $J\in\cal{J}_{T}$. This implies that $x+z^{i}$ can be written as a convex combination of tours of $G_{x}$. We denote this set of tours by $\mathcal{F}_{i}$ and we let $\mathscr{F}=\{F\in\mathcal{F}_{i}\;:\;i\in\{1,\dots,h\}\}$. Now for $F\in\mathscr{F}$ we have $F=T+J$ for some $T\in\mathscr{T}$ and $J\in\mathcal{J}_{T}$. Define $\phi_{F}=\sigma_{T}\cdot\psi_{J}^{T}$. The vector $\sum_{i=1}^{h}\alpha(x+z^{i})$ can be written as $\{\phi,\mathscr{F}\}$.

Claim 3**.**

Every tour in $\mathscr{F}$ is handpicked.

Proof..

Let $F\in\mathscr{F}$. By construction, $F=T+{J}$, where $T\in\mathscr{T}$ and $J\in\mathcal{J}_{T}$. Let $u\in V_{n}$ and $\delta(u)=\{e_{u},f_{u},g_{u}\}$. Notice that $\chi^{F}_{e_{u}}\geq 1$, since $e_{u}\in T$ for all $T\in\mathscr{T}$. Hence, we only need to show that $|\delta_{F}(u)|<6$. Suppose for contradiction that $|\delta_{F}(u)|\geq 6$. This implies that $|\delta_{T}(u)|=3$ and $|\delta_{J}(u)|=3$. However, if $|\delta_{T}(u)|=3$, then $u\in O_{T}$. From Observation 2.7, if $u\in O_{T}$ and $z(\delta(u))\leq 1$, then $|J\cap\delta(u)|=1$ which is a contradiction. $\Diamond$

Claim 4**.**

Suppose $G_{x}$ contains no critical cuts. Define vector $y\in\mathbb{R}^{E_{x}}$ as $y_{e}=\frac{3}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x}$ and $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$. Then $\{\phi,\mathscr{F}\}$ is a convex combination for $y$.

Proof..

We need to show that $y=\sum_{i=1}^{h}\alpha(x+z^{i})$. First, let $e$ be a 1-edge of $G_{x}$ and $M_{j}$ be the induced matching that contains $e$. Then, $x_{e}=1$, $z^{i}_{e}=\frac{1}{2}$ for $i\in[h]\setminus\{j\}$ and $z^{j}_{e}=\frac{1-\theta}{2}$. Hence,

[TABLE]

For a fractional $e$ of $G_{x}$, we have $z^{i}_{e}=\frac{x_{e}}{2}$ for $i\in[h]$, so $\sum_{i=1}^{h}\alpha(x_{e}+z^{i}_{e})=\frac{3}{2}x_{e}$. $\Diamond$

Now we prove some additional useful properties of the convex combination $\{\phi,\mathscr{F}\}$ for $\theta$-cyclic point $x$.

Claim 5**.**

For convex combination $\{\phi,\mathscr{F}\}$, we have $\phi_{2}(e)=\frac{1}{2}-\frac{\alpha\theta}{2}$ for $e\in W_{x}$ and $\phi_{2}(e)=\frac{(x_{e})^{2}}{2}$.

Proof..

Notice that $\phi$ defines a probability distribution on $\mathscr{F}$. We sample $F$ from $\mathscr{F}$ with the probabilities defined by $\phi$. Recall that \phi_{2}(e)=\Pr[e\text{ is doubled in F}]. Moreover, recall that $F=T+J$ where $T\in\mathscr{T}$ and $J\in\mathcal{J}_{T}$ and $T$ is associated with some matching $M\in\{M_{1},\ldots,M_{h}\}$ (i.e., $T\in\mathcal{T}_{i}$ for $i\in[h]\}$). For $e\in E_{x}$, we have

[TABLE]

For $e\in H_{x}$, this implies that $\phi_{2}(e)=x_{e}\cdot\frac{x_{e}}{2}=\frac{(x_{e})^{2}}{2}$. Also

[TABLE]

$\Diamond$

Claims 2, 4 and 5 yield Lemma 3.4. It remains to prove Lemmas 4.2, 4.3 and 4.4.

4.3 Proofs of Lemmas 4.2 and 4.3 for

Constructing Connectors

In this section, we prove Lemmas 4.2 and 4.3, which are necessary in order to write a $\theta$-cyclic point $x$ as a convex combination of connectors with property $P$ described in Definition 6.

See 4.2

Proof.

For each $st\in M$, pair the half-edges incident on $s$ and pair those incident on $t$ to obtain disjoint subsets of edges $\mathcal{P}$. Decompose $x$ into a convex combination of $\mathcal{P}$-rainbow $v$-trees $\mathcal{T}$ (i.e., $x=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{T}$) via Theorem 2.5. This is the desired convex combination since for all $T\in\mathcal{T}$, we have $|\delta_{T}(v)|=2$ and $|\delta_{T}(u)|=2$ for all endpoints $u$ of edges in $M$. Thus, the first and second conditions are satisfied. The third condition holds by definition of $v$-trees. ∎

See 4.3

Proof.

As in the proof of Lemma 4.2, for each $st\in M$, pair the half-edges incident on $s$ and pair those incident on $t$ to obtain a collection of disjoint subsets of edges $\mathcal{P}$. Apply Theorem 2.5 to obtain $\{\lambda,\mathcal{T}\}$ which is a convex combination for $x$, where $\mathcal{T}$ is a set of $\mathcal{P}$-rainbow $v$-trees (i.e., $x=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{T}$). Notice that this convex combination clearly satisfies the second requirement in Definition 6.

Now let $\delta(v)=\{e_{v},f,g\}$, where $w_{1}$ and $w_{2}$ are the other endpoints of $f$ and $g$, respectively. Assume $x_{f}=\theta$ and $x_{g}=1-\theta$. Since $x=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{T}$ and $x_{e_{v}}=1$, we have $e_{v}\in T$ for $T\in\mathcal{T}$. In addition, we have $|\delta_{T}(v)|=2$ for all $T\in\mathcal{T}$ by the definition of $v$-trees. Hence, $\sum_{T\in\mathcal{T}:f\in T,g\notin T}\lambda_{T}=\theta$ and $\sum_{T\in\mathcal{T}:f\notin T,g\in T}\lambda_{T}=x_{f}=1-\theta$. Define

[TABLE]

where $\mathcal{T}_{f}\cup\mathcal{T}_{g}=\mathcal{T}$ and $\mathcal{T}_{f}\cap\mathcal{T}_{g}=\emptyset$. We can also assume that there are subsets $\mathcal{T}^{1}_{f}\subseteq\mathcal{T}_{f}$ and $\mathcal{T}^{1}_{g}\subseteq\mathcal{T}_{g}$ such that $\sum_{T\in\mathcal{T}^{1}_{f}}\lambda_{T}=\Lambda$ and $\sum_{T\in\mathcal{T}^{1}_{g}}\lambda_{T}=\Lambda$, since $\Lambda\leq\theta$. Now we consider two cases

If $e_{w_{1}}\notin M$: For $T\in\mathcal{T}_{f}^{1}$, replace $T$ with $T-f$. For $T\in\mathcal{T}_{g}^{1}$, replace $T$ with $T+f$. 2. 2.

If $e_{w_{1}}\in M$: For $T\in\mathcal{T}_{f}^{1}$, replace $T$ with $T+g$. For $T\in\mathcal{T}_{g}^{1}$, replace $T$ with $T-g$.

For all $T\in\mathcal{T}\setminus{(\mathcal{T}_{f}^{1}\cup\mathcal{T}_{g}^{1})}$, keep $T$ as is. Observe that $T\in\mathcal{T}$ is still a connector of $G_{x}$: for every $T\in\mathcal{T}_{f}$, $T-f$ is a spanning tree of $G_{x}$ and for every $T\in\mathcal{T}_{g}$, $T-g$ is spanning tree of $G_{x}$. Observe that in the first case, the degree of $e_{w_{2}}$ is preserved for every $T$, and in the second case the degree of $e_{w_{1}}$ is preserved for every $T$. We want to show that the new convex combination $\{\lambda,\mathcal{T}\}$ is the desired convex combination for $x$. Notice that in the first case,

[TABLE]

In the second case,

[TABLE]

So $x=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{T}$. Moreover, notice that for $T\in\mathcal{T}$, $T\setminus\delta_{T}(v)$ still induces a connector on $V\setminus\{v\}$ since we did not remove any edge in $T\setminus{\delta(v)}$ from the $v$-tree $T$. Finally, for each vertex $s$ with $e_{s}\in M$, we have $|\delta_{T}(s)|=2$ for all $T\in\mathcal{T}$. To observe this, notice that the initial convex combination satisfies this property for vertex $s$ (since the convex combination is obtained via Theorem 2.5). In the transformation of the convex combination we only change edges incident on $w_{1}$ and $w_{2}$, so if $s\neq w_{1},w_{2}$ the property clearly still holds after the transformation. If $s\in\{w_{1},w_{2}\}$, then as noted previously, we do not remove or add an edge incident on $s$ if $e_{s}\in M$. ∎

4.4 Proof of Lemma 4.4 for Constructing Parity Correctors

We use $O$-joins as parity correctors for each $T\in\mathscr{T}$. We now give the complete proof of Lemma 4.4.

Proof of Lemma 4.4.

Our goal is to show that $z$ belongs to $O\mathrm{\textsc{-{Join}}}(G_{x})$. By definition, $z\in[0,1]^{E_{x}}$. Now we will show that $z$ satisfies the constraint (2.2). First, we state three useful claims.

Claim 6**.**

If $z(\delta(U))\geq 1$ for $U\subset V_{n}$, then $z(\delta(U)\setminus A)-z(A)\geq 1-|A|$.

Proof..

We have $z(\delta(U)\setminus A)-z(A)=z(\delta(U))-2z(A)$. Since $z_{e}\leq\frac{1}{2}$ for all $e\in E_{x}$, we have $z(\delta(U))-2z(A)\geq 1-|A|$. $\Diamond$

Claim 7**.**

If $\delta(U)\cap M=\emptyset$, we have $z(\delta(U))\geq 1$.

Proof..

This follows from the fact that for every edge $e\notin M$, we have $z_{e}=\frac{x_{e}}{2}$. $\Diamond$

Claim 8**.**

For all $U\subset V_{n}$, $|\delta(U)\cap W_{x}|$ and $|\delta(U)|$ have same parity.

Proof..

This follows from the fact that $|\delta(U)\cap H_{x}|$ is always even since $H_{x}$ is a 2-factor of $G_{x}$. $\Diamond$

We consider the following cases. **Case 1: ** $|\delta(U)\cap W_{x}|\geq 3$, **Case 2: ** $|\delta(U)\cap W_{x}|=2$, and **Case 3: ** $|\delta(U)\cap W_{x}|=1$.

Case 1:

If $|\delta(U)\cap W_{x}|\geq 4$, then $z(\delta(U))\geq 2-2\theta\geq 1$. Thus we may assume $|\delta(U)\cap W_{x}|=3$. If $|\delta(U)\geq 4$, then $z(\delta(U))\geq\frac{3}{2}-\frac{3}{2}\theta+\frac{\theta}{2}\geq 1$, since $\theta\leq\frac{1}{2}$. If $|\delta(U)|=3$, then by assumption we have $|\delta(U)\cap M|\leq 1$. Thus, $z(\delta(U))\geq 1$. Thus, Claim 6 applies in each subcase.

Case 2:

In this case, if $|\delta(U)|\geq 4$, then $z(\delta(U))\geq(1-\theta)+\theta\geq 1$. By Claim 8, the only remaining subcase to consider is when $|\delta(U)|=2$.

By assumption, $|U\cap O|$ is even. Hence, $|A|$ must be odd, which implies that $|A|=1$. Let $\delta(U)=\{e^{\prime},e^{\prime\prime}\}$. Since $|\delta(U)|=2$, we have either $|\delta(U)\cap M|=2$ or $|\delta(U)\cap M|=0$. In both cases $z_{e^{\prime}}=z_{e^{\prime\prime}}$. Hence, $z(A)=z(\delta(U)\setminus A)$. Therefore, $z(\delta(U)\setminus A)-z(A)=0=1-|A|$.

Case 3:

If $\delta(U)\cap M=\emptyset$, then by Claim 7 we have $z(\delta(U))\geq 1$. Hence, we assume $|\delta(U)\cap M|=1$. By Claim 8, we only need to consider the following cases: *Case 3i: * $|\delta(U)|=3$, and *Case 3ii: * $|\delta(U)|\geq 5$.

• Case 3i: Notice that $x(\delta(U))\geq 2$. In this case, $\delta(U)$ is either a critical cut, a vertex cut, or a degenerate tight cut. We assumed that $G_{x}$ has no critical cuts. So $U$ is either a vertex cut or a degenerate tight cut. We prove in both cases that $|U\cap O|$ is even. Then we only need to consider $|A|$ odd. If $|A|=1$, then $z(\delta(U)\setminus A)-z(A)\geq\frac{\theta}{2}\geq 0=1-|A|$. If $|A|=3$, then $z(\delta(U)\setminus A)-z(A)\geq-1+\frac{\theta}{2}\geq-2=1-|A|$.

If $U=\{u\}$ for some $u\in V_{n}$, then $u\notin O$ by assumption. Otherwise, $U$ is a degenerate tight cut. Let $\delta(U)=\{e_{u},f_{v},g_{v}\}$ where $\{f_{v},g_{v}\}=\delta(v)\cap H_{x}$ for some $v\in V_{n}$. Notice that $\delta(U\setminus\{v\})=\{e_{u},e_{v}\}$ and $e_{u}\in M$. This implies by assumption that $e_{v}\in M$, which implies that $v\notin O$. Since $|(U\setminus\{v\})\cap O|$ is even, hence $|U\cap O|$ is even.

• Case 3ii: Since $|\delta(U)|\geq 5$, if there is an edge $e\in\delta(U)\cap H_{x}$ with $z_{e}=\frac{1-\theta}{2}$, then $z(\delta(U))\geq 1-\theta+3\cdot\frac{\theta}{2}=1+\frac{\theta}{2}$. Therefore, $\theta<\frac{1}{2}$ and for all edges $e$ in $\delta(U)\cap H_{x}$, we have $z_{e}=\frac{\theta}{2}$.

Let $\mathcal{C}$ be the collection of cycles in $H_{x}$. Since $\theta<\frac{1}{2}$, every cycle in $\mathcal{C}$ is even length. Clearly, any cut crosses every cycle $C\in\mathcal{C}$ an even number of times. If all the edges in $\delta(U)\cap C$ have the same $x$ value, then $|U\cap V(C)|$ is even. We have

[TABLE]

By the argument above and the fact that $|V(C)|$ is even for all $C\in\mathcal{C}$, we conclude that $|U|$ is even. Since $G_{x}$ is a cubic graph, this implies that $|\delta(U)|$ is even. However, by Claim 8, we know that $|\delta(U)|$ is odd.

This concludes the case analysis and the proof. ∎

4.5 Proof of Lemma 4.1: Partitioning 1-edges into Induced Matchings

The goal of this section is to prove the following lemma. See 4.1

We say $\delta(U)$ is a triangular 3-cut if $|U|=3$ or $|V\setminus U|=3$, and $|\delta(U)|=3$. A bad 3-edge cut is a proper 3-edge cut that is not triangular. We construct the desired partition of $W_{x}$ into induced matchings by gluing over the bad cuts of $G_{x}$ and perform induction on the number of bad 3-edge cuts. We prove Lemma 4.1 using a two-phase induction. Claim 9 is the base case and Claims 10 and 11 are the first and second inductive steps.

Claim 9**.**

Suppose $G_{x}$ is 3-edge-connected and contains no bad 3-edge cuts. Then Lemma 4.1 holds.

Proof..

In $G_{x}$, contract every edge in $W_{x}$. We get a connected 4-regular graph $H=(W_{x},H_{x})$. An independent set in $H$ corresponds to a set of edges in $W_{x}$ that forms an induced matching in $G_{x}$. We consider two cases. If $H$ is the complete graph on five vertices, then partition the vertex set into five independent sets, which corresponds to five induced matchings in $G_{x}$. Notice that the condition (i) from Lemma 4.1 is satisfied since each induced matching contains one edge.

If $H$ is not the complete graph on five vertices, by Brook’s Theorem (see Theorem 8.4 in [BM08]) we can partition the vertices of $H$ into four independent sets where each independent set corresponds to an induced matching $\{M_{1},\ldots,M_{4}\}$ in $G_{x}$ and these four induced matchings partition $W_{x}$. If $|M_{i}\cap\{e_{r},e_{w_{1}},e_{w_{2}}\}|\leq 1$ for $i\in[4]$, then we are done. Otherwise, assume without loss of generality that $\{e_{w_{1}},e_{w_{2}}\}\in M_{4}$. Then let $M^{\prime}_{4}=M_{4}\setminus\{e_{w_{1}}\}$. The desired partition is $\{M_{1},M_{2},M_{3},M^{\prime}_{4},\{e_{w_{1}}\}\}$. Thus, condition (i) is satisfied.

Now we prove condition (ii). First, consider a vertex $u\in V$ and the cut $\delta(u)$ in $G_{x}$. Clearly $|\delta(u)\cap M_{i}|\leq|\delta(u)\cap W_{x}|\leq 1$. For a triangular 3-cut, $\delta(U)=\{e_{1},e_{2},e_{3}\}$, we cannot have $|\delta(U)\cap\{e_{1},e_{2},e_{3}\}|\geq 2$, since $\delta(U)\subseteq W_{x}$ and no pair of edges from $\delta(U)$ can belong to an induced matching. Since condition (iii) does not apply, this completes the proof of the claim. $\Diamond$

Claim 10**.**

Suppose $G_{x}$ is 3-edge-connected. Then Lemma 4.1 holds.

Proof..

Now let us consider a bad cut. In particular, consider graph $G_{x}$ with 3-edge-cut $\delta(U)=\{e_{1},e_{2},e_{3}\}$, and assume without loss of generality that $r\in U$. Let $s_{1},s_{2}$ and $s_{3}$ be the endpoints of $e_{1},e_{2}$ and $e_{3}$ that are in $U$, and $t_{1},t_{2}$ and $t_{3}$ be the other endpoints. Notice that $s_{1},s_{2},s_{3}$ (and analogously $t_{1},t_{2},t_{3}$) are distinct vertices since $G_{x}$ is 3-edge-connected. Construct graph $G_{1}=G_{x}[(V\setminus U)\cup\{s_{1},s_{2},s_{3}\}]+\{s_{1}s_{2},s_{1}s_{3},s_{2}s_{3}\}$ and, symmetrically, graph $G_{2}=G_{x}[U\cup\{t_{1},t_{2},t_{3}\}]+\{t_{1}t_{2},t_{1}t_{3},t_{2}t_{3}\}$. If both $G_{1}$ and $G_{2}$ have no bad 3-edge cuts, then we can apply Claim 9 to both $G_{1}$ and $G_{2}$. For $G_{1}$, we find induced matchings $\{M^{1}_{1},\ldots,M^{1}_{5}\}$ such that conditions (i) and (ii) hold. Similarly, for $G_{2}$, we find induced matchings $\{M^{2}_{1},\ldots,M^{2}_{5}\}$ such that (i) and (ii) hold.

Notice that for each edge $e\in\{e_{1},e_{2},e_{3}\}$, there is exactly one induced matching in $\{M^{1}_{1},\ldots,M^{1}_{5}\}$ and in $\{M^{2}_{1},\ldots,M^{2}_{5}\}$ that contains $e_{1}$. Without loss of generality, suppose $M^{1}_{i}$ and $M^{2}_{i}$ each contain edge $e_{i}$ for $i\in[3]$. Then let $M_{i}=M^{1}_{i}\cup M^{2}_{i}$ for $i\in[5]$ and notice that $M_{i}$ is an induced matching in $G_{x}$. We conclude by induction on the number of bad cuts in $G_{x}$, since both $G_{1}$ and $G_{2}$ have fewer bad 3-edge cuts than does $G_{x}$. $\Diamond$

Claim 11**.**

Suppose $G_{x}$ is 2-edge-connected. Then Lemma 4.1 holds.

Proof..

We proceed by induction on the number of 2-edge cuts of $G_{x}$. If $G_{x}$ does not contain any 2-edge cuts then $G_{x}$ is 3-edge-connected, so by Claim 10 the claim follows.

For the induction step, consider 2-edge cut $\delta(U)=\{e_{1},e_{2}\}$. Since $x$ is a half-cycle point, note that $e_{1},e_{2}\in W_{x}$. Let $s_{1}$ and $s_{2}$ be the endpoints of $e_{1}$ and $e_{2}$ that are in $U$ and let $t_{1}$ and $t_{2}$ be the other endpoints. (Observe that neither $s_{1}s_{2}$ nor $t_{1}t_{2}$ is an edge in $G_{x}$; otherwise $G_{x}$ would contain a cut of $x$-value less than 2.) Consider graphs $G_{1}=G[U]+s_{1}s_{2}$ and $G_{2}=G[V\setminus U]+t_{1}t_{2}$. The set of 1-edges of $G_{1}$ is $\{W_{x}\cap E(G_{1})\}\cup\{s_{1}s_{2}\}$, and the set of 1-edges of $G_{2}$ is $\{W_{x}\cap E(G_{2})\}\cup\{t_{1}t_{2}\}$.

Without loss of generality, assume $r\in S$. Apply induction on $G_{1}$ to find induced matchings $\{M^{1}_{1},\ldots,M^{1}_{5}\}$ where $s_{1}s_{2}\in M^{1}_{1}$, and on $G_{2}$ to obtain induced matchings $\{M^{2}_{1},\ldots,M^{2}_{5}\}$ where $t_{1}t_{2}\in M^{2}_{1}$. Set $M_{1}=\{M^{1}_{1}\cup M^{2}_{1}\cup\{e_{1},e_{2}\}\}\setminus\{s_{1}s_{2},t_{1}t_{2}\}$ and set $M_{i}=M^{1}_{i}\cup M^{2}_{i}$ for $i\in\{2,\ldots,5\}$. Then $\{M_{1},\ldots,M_{5}\}$ partition $W_{x}$ into induced matchings and satisfy conditions (i), (ii) and (iii). $\Diamond$

The proof of Lemma 4.1 follows from Claim 11.

5 Construction of Tours for Uniform Points

Recall the definition of $\frac{2}{k}$-uniform point from Section 1.1.2. In this section we prove Theorem 1.5 regarding $\frac{2}{3}$-uniform points and then we prove Theorem 1.7 concerning $\frac{2}{4}$-uniform points.

We would like to remark that a key idea in the proofs of both theorems is to first write the uniform point as a convex combination of points containing a perfect matching of 1-edges. In the case of a $\frac{2}{3}$-uniform point, this results in a convex combination of $\frac{1}{2}$-cyclic points. Using our main result (Theorem 1.3), we can save on 1-edges, which results on a uniform saving over all edges in the convex combination. In the case of a $\frac{2}{4}$-uniform point, when we have an even number of vertices, we can obtain a convex combination of points where each vertex is incident to a 1-edge and three fractional $\frac{1}{3}$-edges. Then, if we have no tight 4-edge cuts, which occurs when the graph is essentially 6-edge-connected, we can save on 1-edges. (Notice that the graph corresponding to a $\frac{2}{4}$-uniform point does not contain any 5-edge cuts.) This could serve as a base case if (i) the odd case is solved, and (ii) we could glue tours over tight 4-edge cuts.

5.1 TSP on $\frac{2}{3}$-Uniform Points

We start by the following lemma reducing TSP on $\frac{2}{3}$-uniform points to TSP on $\frac{1}{2}$-cyclic points.

Lemma 5.1**.**

If for any $\frac{1}{2}$-cyclic point $x$ the vector $y$ defined as: $y_{e}=\frac{3}{2}-\epsilon$ for $e\in W_{x}$ and $y_{e}=\frac{3}{4}-\delta$ for $e\in H_{x}$ and $y_{e}=0$ for $e\notin E_{x}$ for constants $\epsilon,\delta\geq 0$ belongs to $\operatorname{{{TSP}}}(K_{n})$, then for any $\frac{2}{3}$-uniform point $z$ we have $(\frac{3}{2}-\frac{\epsilon}{2}-\delta)z\in\operatorname{{{TSP}}}(K_{n})$.

Proof.

Let $z$ be a $\frac{2}{3}$-uniform point, and let $G_{z}=(V_{n},E_{z})$ be its support. Notice that $z\in\emptyset\mathrm{\textsc{-{Join}}}(G_{z})$. Hence $z$ can be written as a convex combination of $\emptyset$-joins of $G_{z}$ denoted by $\{\lambda,\mathscr{C}\}$. Observe that each $\emptyset$-join $\mathcal{C}\in\mathscr{C}$ is in fact a 2-factor of $G_{z}$ since $z(\delta(u))=2$ and $|\mathcal{C}\cap\delta(u)|\leq 2$ for $u\in V_{n}$. For $\mathcal{C}\in\mathscr{C}$, we define $p^{\mathcal{C}}$ to be such that $p^{\mathcal{C}}_{e}=1$ for $e\in\mathcal{C}$ and $p^{\mathcal{C}}_{e}=\frac{1}{2}$ for $e\in E_{z}\setminus\mathcal{C}$ and $p^{\mathcal{C}}_{e}=0$ for $e\in E_{n}\setminus E_{z}$. Notice that $p^{\mathcal{C}}$ is a $\frac{1}{2}$-cyclic point. Define $y^{\mathcal{C}}$ as follows: for $e\in E_{n}$ let $y^{\mathcal{C}}_{e}=\frac{3}{2}-\epsilon$ if $e\in W_{p^{\mathcal{C}}}$, and $y^{\mathcal{C}}_{e}=\frac{3}{4}-\delta$ if $e\in H_{p^{\mathcal{C}}}$ and $y_{e}=0$ otherwise. By assumption, we have $y^{\mathcal{C}}\in\operatorname{{{TSP}}}(K_{n})$. Therefore,

[TABLE]

Observe that for $e\in E_{x}$

[TABLE]

Finally, for $e\in E_{n}\setminus E_{x}$, we have $\hat{z}_{e}=0$. ∎

A consequence of Theorem 1.3 is that for $\frac{2}{3}$-uniform point $x\in\mathbb{R}^{E_{n}}$, we have $(\frac{3}{2}-\frac{1}{40})x\in\operatorname{{{TSP}}}(K_{n})$.

Haddadan, Newman and Ravi [HNR21] used the following theorem in [BIT13] to obtain the first factor below $\frac{3}{2}$ for approximating TSP on $\frac{2}{3}$-uniform point.

Theorem 5.2** ([BIT13]).**

Let $G=(V,E)$ be a bridgeless cubic graph. Then $G$ has a 2-factor that covers all 3-edge cuts and 4-edge cuts of $G$.

We can combine the ideas in the proof of Theorem 1 of [HNR21] with Theorem 1.3 to prove the following.

Theorem 5.3**.**

Let $x$ be a $\frac{2}{3}$-uniform point and $G_{x}=(V_{n},E_{x})$ its support graph. Then $\frac{17}{12}x\in\operatorname{{{TSP}}}(K_{n})$. If $G_{x}$ is Hamiltonian, then $\frac{87}{68}x\in\operatorname{{{TSP}}}(K_{n})$.

Proof.

By Theorem 5.2, $G_{x}$ has a 2-factor $\mathcal{C}$ that covers all 3-edge cuts and 4-edge cuts of $G_{x}$. Define vector $y^{1}$ as follows: $y^{1}_{e}=1$ for $e\in\mathcal{C}$ and $y^{1}_{e}=\frac{4}{5}$ for $e\in E_{x}\setminus\mathcal{C}$ and $y^{1}_{e}=0$ for $e\in E_{n}\setminus E_{x}$.

Claim 12**.**

We have $y^{1}\in\operatorname{{{TSP}}}(K_{n})$.

Proof..

Notice graph $G^{\prime}=G_{x}/\mathcal{C}$ is a 5-edge-connected graph. We can assume without loss of generality that $G^{\prime}$ is also 5-regular666Replace every vertex $v$ of degree more than 5 with a doubled cycle of length $|\delta(v)|$ and connect each vertex in the cycle to a neighbor of $v$ in $G^{\prime}$.. For any vertex $r$ of $G^{\prime}$ we have $\frac{2}{5}\chi^{E(G^{\prime})}\in r\mathrm{\textsc{-Tree}}(G^{\prime})$. So the vector $\frac{2}{5}\chi^{E(G^{\prime})}$ can be written as a convex combination of $r$-trees of $G^{\prime}$ denoted by $\{\lambda,\mathcal{T}\}$. For $T\in\mathcal{T}$ the multigraph $F_{T}=\mathcal{C}+2T$ is a tour of $G_{x}$ and therefore $K_{n}$. Finally, $y^{1}=\sum_{T\in\mathcal{T}}\lambda_{T}\chi^{F_{T}}$. $\Diamond$

On the other hand, we can define $z$ where $z_{e}=\frac{1}{2}$ for $e\in\mathcal{C}$ and $z_{e}=1$ for $e\in E_{x}\setminus\mathcal{C}$ for $z_{e}=0$ for $e\in E_{n}\setminus E_{x}$. Vector $z$ is a $\frac{1}{2}$-cyclic point, hence we can apply Theorem 1.3 to obtain vector $y^{2}\in\operatorname{{{TSP}}}(K_{n})$ such that $y^{2}_{e}=\frac{3}{4}$ for $e\in\mathcal{C}$, $y^{2}_{e}=\frac{3}{2}-\frac{1}{20}$ for $e\in E_{x}\setminus\mathcal{C}$ and $y^{2}_{e}=0$ for $e\in E_{n}\setminus E_{x}$. Notice that $\frac{7}{9}y^{1}+\frac{2}{9}y^{2}\in\operatorname{{{TSP}}}(G)$ and is equal to $\frac{17}{12}x$.

If $G_{x}$ is Hamiltonian, we can assume $\mathcal{C}$ is the Hamiltonian cycle of $G_{x}$. Hence $\chi^{\mathcal{C}}\in\operatorname{{{TSP}}}(K_{n})$. In this case $\frac{7}{17}\cdot\chi^{\mathcal{C}}+\frac{10}{17}\cdot y^{2}\in\operatorname{{{TSP}}}(K_{n})$ and is equal to $\frac{87}{68}x$. ∎

5.2 A Base Case for $\frac{2}{4}$-Uniform Points

Due to the fact that we can glue over critical cuts, we observed that TSP on a $\theta$-cyclic point $x$ is essentially equivalent to the problem with the assumption that $G_{x}$ contains no critical cuts. Analogously, in the case of a $\frac{2}{4}$-uniform point $x$, Theorem 1.7 could serve as the base case if we were able to glue over the proper minimum cuts of $G_{x}$. However, the difference here is that (1) the gluing arguments we presented for $\theta$-cyclic points can not easily be extended to this case (due to the increased complexity of the distribution of patterns), and (2) we require an even number of vertices for our arguments.

See 1.7

Proof.

We prove the claim by showing that there is a distribution of tours that satisfies the properties. It is easy to see that the proof yields a convex combination of tours of $G$. Since $G$ does not have a proper 4-edge cut and since it is Eulerian, a proper cut of $G$ has at least 6 edges.

Define $y_{e}=\frac{1}{4}$ for all $e\in E$. Vector $y$ is in the perfect matching polytope of $G$ and can be written as a convex combination of perfect matchings of $G$. Choose a perfect matching $M$ at random from the distribution defined be the convex multipliers of this convex combination.

Let $r\in V$. Define vector $z$ as follows: $z_{e}=1$ if $z\in M$ and $z_{e}=\frac{1}{3}$ for $z\in E\setminus M$. Observe that $z\in r\mathrm{\textsc{-Tree}}(G_{x})$ for any $v\in V$: $z(\delta(U))\geq\frac{1}{3}\cdot|\delta(U)|\geq 2$ if $|U|\geq 2$ and $|V\setminus U|\geq 2$ and $z(\delta(r))=2$.

Applying Brook’s theorem (similar to the proof of Lemma 4.1) we can find collection $\{M_{1},\ldots,M_{7}\}$ of induced matchings of $G$ that partition $M$. Choose $i\in[7]$ uniformly at random. For each $e=st\in M_{i}$, include the three edges incident on $s$ in one set and the three edges incident to $t$ in another set. Notice all six edge are distinct since $G$ has no proper 4-edge cuts. Apply Theorem 2.5 to decompose $z$ into a convex combination of rainbow $r$-trees of $G$ with respect to this partition. Take a random $r$-tree $T$ from this convex combination using the distribution defined by the convex multipliers. Let $O$ be the set of odd degree vertices of $T$. Note that for each $e=st\in M_{i}$, $s,t\notin O$ by construction. Define vector $p$ to be such that $p_{e}=\frac{1}{2}$ for $e\in M\setminus\{M_{i}\}$ and $p_{e}=\frac{1}{6}$ otherwise. We have $p\in O\mathrm{\textsc{-{Join}}}(G)$. Therefore, we can write $p$ as convex combination of $O$-joins of $G$. Choose one of the $O$-joins at random from the convex combination and label it $J$. Note that $F=T+J$ is a tour of $G$. For an edge $e\in M$ we have

[TABLE]

If $e\notin M$, then we have $\Pr[e\in J|e\notin M]=\frac{1}{6}$. Hence,

[TABLE]

Observe that $\mathbb{E}[z_{e}]=1\cdot\Pr[e\in M]+\frac{1}{3}\cdot Pr[e\notin M]=\frac{1}{2}$. Therefore, $\Pr[e\in T]=\Pr[e\notin T]=\frac{1}{2}$.

[TABLE]

Thus, each edge $e\in E$ is used to an extent $(\frac{3}{2}-\frac{1}{42})\cdot\frac{1}{2}$. This concludes the proof. ∎

6 Concluding Remarks

In this paper, we showed how to improve the multiplicative approximation factor of Christofides algorithm on the 1-edges of $\theta$-cyclic points from $\frac{3}{2}$ to $\frac{3}{2}-\frac{\theta}{10}$. Approaching Conjecture 1 from this angle, we propose the following open problem, which is implied by the four-thirds conjecture.

Open Problem 1**.**

Let $x\in\mathbb{R}^{E_{n}}$ be a $\theta$-cyclic point. Define vector $y$ as follows: $y_{e}=\frac{4}{3}$ for $e\in W_{x}$, $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$ and $y_{e}=0$ for $e\notin E_{x}$. Can we show $y\in\operatorname{{{TSP}}}(K_{n})$?

In fact, the bound above is tight: for $\epsilon>0$, there exists a $\frac{1}{2}$-cyclic point $x^{\epsilon}$ such that vector $y^{\epsilon}$ defined as $y^{\epsilon}_{e}=\frac{4}{3}-\epsilon$ for $e\in W_{x^{\epsilon}}$, $y_{e}=\frac{3}{2}x^{\epsilon}_{e}$ for $e\in H_{x^{\epsilon}}$ and $y_{e}=0$ for $e\notin E_{x^{\epsilon}}$. Then $y^{\epsilon}\notin\operatorname{{{TSP}}}(K_{n})$ (See Figure 6.1).

This makes the problem above intriguing even when restricted to $\frac{1}{2}$-cyclic points. For a $\frac{1}{2}$-cyclic point $x$ where $H_{x}$ is disjoint union of 3-cycles, Boyd and Carr [BC11] achieved this bound. Notice that this class includes $x^{\epsilon}$ in Figure 6.1. Interestingly, a construction very similar to that of Boyd and Sebő [BS21] implies that for a $\frac{1}{2}$-cyclic point $x$ where $H_{x}$ is a union of vertex-disjoint 4-cycles, we can go beyond this factor.

Theorem 6.1**.**

Let $x\in\mathbb{R}^{E_{n}}$ be a $\frac{1}{2}$-cyclic point where $H_{x}$ is a union of vertex-disjoint 4-cycles. Define vector $y$ as follows: $y_{e}=\frac{5}{4}$ for $e\in W_{x}$, $y_{e}=\frac{3}{2}x_{e}$ for $e\in H_{x}$ and $y_{e}=0$ for $e\notin E_{x}$. We have $y\in\operatorname{{{TSP}}}(K_{n})$.

Proof Sketch.

In this case, Boyd and Sebő [BS21] showed that $G_{x}$ has a Hamilton cycle $H$ such that $H\subset W_{x}$ and $H$ intersects each 4-cycle of $H_{x}$ at opposite edges.

They also show that $x$ can be written as convex combination of connectors of $G_{x}$ denoted by $\{\lambda,\mathcal{T}\}$ such that for $T\in\mathcal{T}$ we have $|T\cap C|=2$ and $|T\cap(C\cap H)|=1$ for each 4-cycle $C\in H_{x}$ [BS21].

For a $T\in\mathcal{T}$ define $O_{T}$ be the odd degree vertices of $T$ and define vector $z^{T}$ as follows: $z^{T}_{e}=\frac{1}{3}$ for $e\in W_{x}$, $z^{T}_{e}=\frac{1}{6}$ for $e\in H_{x}\cap H$ and $z^{T}_{e}=\frac{1}{2}$ for $e\in H_{x}\setminus H$. For $T\in\mathcal{T}$, we can show $z^{T}\in O_{T}\mathrm{\textsc{-{Join}}}(G_{x})$ by following essentially the same arguments as Boyd and Sebő. This implies that $\chi^{T}+z^{T}\in\operatorname{{{TSP}}}(K_{n})$. Therefore, $p=\sum_{T\in\mathcal{T}}\lambda_{T}(\chi^{T}+z^{T})\in\operatorname{{{TSP}}}(K_{n})$. In addition, we have $\chi^{H}\in\operatorname{{{TSP}}}(K_{n})$. We conclude that $y=\frac{1}{4}\chi^{H}+\frac{3}{4}p\in\operatorname{{{TSP}}}(K_{n})$. ∎

Since the proof of Theorem 6.1 is essentially the same as that in [BS21], it does not seem to extend to $\theta$-cyclic points in which the fractional edges form $4$-cycles.

For $\frac{2}{4}$-uniform points it would be interesting to find a way to apply the gluing approach, perhaps yielding further improvements to the factors presented in [KKO20] and [GLL*+*21]. For $\frac{2}{k}$-uniform points with $k\geq 5$ nothing is known for TSP or 2EC beyond Christofides $\frac{3}{2}$-approximation and the tiny step below $\frac{3}{2}$ implied by the recent work of Karlin et al.

Acknowledgements

We thank R. Ravi for his comments on the presentation of this paper. The work of A. Haddadan is supported by the U.S. Office of Naval Research under award number N00014-18-1-2099 and the U. S. National Science Foundation under award number CCF-1527032. The work of A. Newman is supported in part by IDEX-IRS SACRE.