On the Number of Path Systems

TL;DR

This paper investigates the enumeration of consistent path systems in graphs, revealing their exponential growth and distinguishing between general and metric realizable systems, with implications for metric cone face-counts and VC-classes.

Contribution

It provides asymptotic counts for consistent path systems and differentiates those realizable as metric geodesics, advancing understanding in graph theory and metric geometry.

Findings

Number of consistent path systems is approximately n^{n^2/2}

Realizable metric path systems are exponentially fewer, about 2^{Θ(n^2)}

Results improve bounds on metric cone face-count and VC-class enumeration.

Abstract

A path system in a graph $G$ is a collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We show that the number of consistent path systems on $n$ vertices is $n^{\frac{n^2}{2}(1-o(1))}$, whereas the number of consistent path systems which are realizable as the unique geodesics w.r.t. some metric is only $2^{\Theta(n^2)}$. In addition, these insights allow us to improve known bounds on the face-count of the metric cone and shed new light on enumerating maximum-VC-classes.

Equations

$$\text{If }a\in P_{u,v}\text{ then }{P}_{u,v}\text{ is the concatenation of }{P}_{u,a}\text{ and }{P}_{a,v}.$$

$$n^{\frac{n^2}{2}(1-o(1))}\le |{\mathcal{M}}_n|\le |{\mathcal{P}}_n|\le n^{\frac{n^2}{2}},$$

$$\text{There are between }{2}^{(0.51-o(1))n^2}\text{ and }{2}^{(1.38-o(1))n^2}\text{ strictly metric n-vertex path systems.}$$

$$\mathcal{S}(G)\subset \mathcal{M}(G)\subset \mathcal{P}(G),$$

$$\mathcal{R}(\mathcal{P})=\{f:(\genfrac{}{}{0ex}{}{[n]}{2})\to [n]:f\text{ is a reˊsumeˊ of }\mathcal{P}\}.$$

$$|{\mathcal{D}}_2(G)|=\prod _{uv\notin E}|N(u)\cap N(v)|.$$

$$\log |{\mathcal{D}}_2(G)|=(1-p)\frac{n^2}{2}\log (p^{2}n)\left(1+o(1)\right).$$

$$n^{\frac{n^2}{2}(1-o(1))}\le |{\mathcal{M}}_n|\le |{\mathcal{P}}_n|\le n^{\frac{n^2}{2}},$$

$$n^{\frac{n^2}{2}(1+o(1))}=|{\mathcal{D}}_2(G)|.$$

$$w(ab)=\{\begin{array}{ll}\rho (a,b)& P_{a,b}=ab\\ \infty & \text{otherwise}\end{array}.$$

$$w(P_{a,b})=\sum _{uv\in P_{a,b}}w(uv)=\sum _{uv\in P_{a,b}}\rho (u,v)=\rho (a,b).$$

$$w(Q)=w(Q_1)+w(Q_2)=\sum _{i=1}^{k}w(x_{i-1}{x}_i)+\sum _{i=1}^{m}w(y_{i-1}{y}_i)=\sum _{i=1}^k\rho (x_{i-1},x_i)+\sum _{i=1}^m\rho (y_{i-1},y_i).$$

$$w(P_{a,b})=\rho (a,b)<\rho (a,c)+\rho (c,b)\le \sum _{i=1}^k\rho (x_{i-1},x_i)+\sum _{i=1}^m\rho (y_{i-1},y_i)=w(Q),$$

$$\begin{array}{rl}{x}_{a,c}+x_{c,b}-x_{a,b}& \ge sa,b,c\in V,c\notin P_{a,b}\\ x_{a,c}+x_{c,b}-x_{a,b}& =0a,b,c\in V,c\in P_{a,b}\\ x_{a,b}& \ge 1a,b\in [n]\end{array}$$

$$\begin{array}{rl}{x}_{a,c}+x_{c,b}-x_{a,b}& >0a,b,c\in [n],c\notin P_{a,b}\\ x_{a,c}+x_{c,b}-x_{a,b}& =0a,b,c\in [n],c\in P_{a,b}\end{array}$$

$$2x_{a,b}=(x_{a,b}+x_{b,c}-x_{a,c})+(x_{b,a}+x_{a,c}-x_{b,c})\ge 0.$$

$$\rho (z,u)\le \rho (z,v)+\rho (v,u)=\rho (z,v)\text{ and }\rho (z,v)\le \rho (z,u)+\rho (u,v)=\rho (z,u).$$

$${\text{MET}}_n=\{x\in {\mathbb{R}}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}:x_{a,b}+x_{b,c}-x_{a,c}\ge 0,\forall a,b,c\in [n]\}.$$

$$\text{The condition }T(\rho )=T(\tilde{\rho })\text{ defines an equivalence relation }\rho \sim \tilde{\rho }\text{ on this cone}.$$

$${\Delta }_t:={\Delta }_{a,b;c}:=e_{a,c}+e_{c,b}-e_{a,b}.$$

$$\sum _{s\in S}{\Delta }_s=\sum _{t\in {\left(\genfrac{}{}{0ex}{}{V}{3}\right)}^{\ast }}{\alpha }_t{\Delta }_t,$$

$$0=\sum _{s\in S}⟨{\Delta }_s,\rho ⟩=⟨\sum _{s\in S}{\Delta }_s,\rho ⟩=⟨\sum _{t\in {\left(\genfrac{}{}{0ex}{}{V}{3}\right)}^{\ast }}{\alpha }_t{\Delta }_t,\rho ⟩=\sum _{t\in {\left(\genfrac{}{}{0ex}{}{V}{3}\right)}^{\ast }}{\alpha }_t⟨{\Delta }_t,\rho ⟩.$$

$$\begin{array}{rl}⟨{\Delta }_s,x⟩=0& \forall s\in S\\ ⟨{\Delta }_t,x⟩>0& \forall t\in S^c\end{array}$$

$$\sum _{s\in S}{\beta }_s{\Delta }_s-\sum _{t\in S^c}{\lambda }_t{\Delta }_t=0.$$

$$\sum _{s\in S}{\Delta }_s=\sum _{s\in S}{\beta }_s{\Delta }_s+\sum _{s\in S}(1-{\beta }_s){\Delta }_s=\sum _{t\in S^c}{\lambda }_t{\Delta }_t+\sum _{s\in S}(1-{\beta }_s){\Delta }_s=\sum _{t\in {\left(\genfrac{}{}{0ex}{}{V}{3}\right)}^{\ast }}{\alpha }_t{\Delta }_t,$$

$$\partial (\{a,b;c\}):=e_{a,c}+e_{c,b}-e_{a,b}={\Delta }_{a,b;c}.$$

$$S=\{\{0,1;7\},\{0,3;7\},\{0,4;6\},\{1,3;6\},\{2,4;7\},\{2,5;6\},\{4,5;7\}\}$$

$$\begin{array}{rl}\sum _{s\in S}{\Delta }_s& =\frac{1}{8}{\Delta }_{0,1;6}+\frac{7}{8}{\Delta }_{0,1;7}+\frac{1}{8}{\Delta }_{0,3;6}+\frac{7}{8}{\Delta }_{0,3;7}\\ & +\frac{3}{4}{\Delta }_{0,4;6}+\frac{1}{4}{\Delta }_{0,4;7}+\frac{7}{8}{\Delta }_{1,3;6}+\frac{1}{8}{\Delta }_{1,3;7}\\ & +\frac{1}{8}{\Delta }_{2,4;6}+\frac{7}{8}{\Delta }_{2,4;7}+\frac{7}{8}{\Delta }_{2,5;6}+\frac{1}{8}{\Delta }_{2,5;7}\\ & +\frac{1}{8}{\Delta }_{4,5;6}+\frac{7}{8}{\Delta }_{4,5;7}.\end{array}$$

$$2^{0.51n^2(1-o(1))}\le |{\mathcal{S}}_n|\le 2^{1.38n^2(1-o(1))}.$$

$$|\mathcal{NS}(G)|\ge 2^{(1-p)p^2\frac{n^2}{8}(1+o(1))}.$$

Full text

In addition, these insights allow us to improve known bounds on the face-count of the metric cone and shed new light on enumerating maximum-VC-classes.

1 Introduction

This paper studies graphs as geometric objects. Geodesics offer an important perspective of a geometric space by specifying how best to move between its points. When the space in question is a connected undirected graph $G=(V,E)$, the answer is provided by a path system in $G$. To wit, given any two vertices $u,v\in V$ we have a designated path, $P_{u,v}=P_{v,u}$, between $u$ and $v$. We view $P_{u,v}$ as our ‘chosen route’ between $u$ and $v$. Metric path systems are the most well-known examples. Such a path system is defined by assigning positive weights $w:E\to\mathbb{R}_{>0}$ to $G$’s edges and letting $P_{u,v}$ be a $w$-shortest $uv$ path. In a strictly metric path system, $w$ has the property that the shortest paths are unique. In addition, a strictly metric path system is consistent in the sense that

[TABLE]

This condition can be viewed as an axiom in the geometry of path systems. Any path system that satisfies condition (1) is said to be consistent. We have discovered earlier on that consistent path systems need not be metric, and a main objective of the present article is to make this distinction quantitative, by estimating the number of consistent path systems on $n$ vertices vs. the number of metric ones. There is already a substantial body of work dealing with path systems [16, 5, 8, 7, 6].

1.1 Main Results and some Background

Among our main discoveries, we show (Theorem 3.4)

[TABLE]

where $\mathscr{P}_{n}$ and $\mathscr{M}_{n}$ is the set of all consistent resp. metric path systems on $n$ vertices.

In contrast (Theorem 5.1)

[TABLE]

Let us mention some pertinent earlier work. Perhaps most relevant is work by Graham, Yao and Yao [15] who considered similar counting problems. Their hope was that improved estimates would lead to stricter lower bounds on the computational complexity of certain graph metric problems. In particular, we manage to improve their bound on the count of faces of the metric cone. An interesting question that we study here is how many strictly metric systems a fixed graph can have. Our best construction is inspired by an idea from [15]. Namely, we introduce the notion of a monotone path system and show that certain classes of monotone path systems are strictly metric.

Perhaps more surprisingly, we discover a relation between consistent path systems and the theory of VC dimension. A paper of Alon, Moran, and Yehudayoff [1] together with subsequent work of Balogh, Mészáros, and Wagner [4] provide fairly tight bounds on the number of maximum-VC-classes. Here we place the latter results in the context of consistent path systems.

Finally, we draw connections between path systems and the notion of betweenness from ordered geometry, e.g., Coxeter’s book [9]. Inspired by the notion of colinear triples in metric spaces, we introduce the notion of a résumé. This is a concise depiction of a path system, which associates every pair of nonadjacent vertices $u,v$ in $G$ to some vertex on the chosen $uv$-path. Betweenness in metric spaces suggests a significant perspective that we develop, leading to a characterization (Theorem 4.1) of metric and strictly metric path systems.

This is a rich field of study, as manifested by the many open questions that we pose (Section 7).

2 Consistent Path Systems

A consistent path system $\mathcal{P}$ in a graph $G=(V,E)$ is a collection of simple paths in $G$ such that:

For every pair $u,v\in V$ there is exactly one $uv$-path in $\mathcal{P}$, called $P_{u,v}$. The paths $P_{u,v}$ and $P_{v,u}$ are identical. 2. 2)

The non-empty intersection of any two paths in $\mathcal{P}$ is either a single vertex or a path in $\mathcal{P}$.

A path system $\mathcal{P}$ in $G=(V,E)$ is neighborly if for every edge $uv\in E$ the $(uv)$-path in $\mathcal{P}$ is this edge. A path system $\mathcal{P}$ in $G=(V,E)$ is metric if there is a positive weight function $w:E\to\mathbb{R}_{>0}$ such that each path in $\mathcal{P}$ is a $w$-shortest path. Similarly, a path system $\mathcal{P}$ of $G=(V,E)$ is strictly metric if there is a positive weight function $w:E\to\mathbb{R}_{>0}$ such that each path in $\mathcal{P}$ is the unique $w$-shortest path.

A consistent path system $\mathcal{P}$ on a vertex set $V$ is a consistent path system defined on the complete graph on $V$. For $G=(V,E)$:

$\mathscr{P}(G)$ denotes the set of all consistent path systems in $G$.

$\mathscr{M}(G)$ denotes the set of all consistent metric path systems in $G$.

$\mathscr{S}(G)$ denotes the set of all strictly metric path systems in $G$.

Clearly,

[TABLE]

and we know by now that both inclusions are proper.

We use the shorthand $\mathscr{P}_{n}=\mathscr{P}(K_{n})$, $\mathscr{M}_{n}=\mathscr{M}(K_{n})$, $\mathscr{S}_{n}=\mathscr{S}(K_{n})$ for the set of all consistent, metric and strictly metric path systems on $n$ vertices, respectively.

Let us seek a concise way to represent a given consistent path system $\mathcal{P}$ on the vertex set $[n]$. An obvious but inefficient method is to explicitly write out all $\binom{n}{2}$ paths in $\mathcal{P}$. But there are better approaches: for each pair of vertices $1\leq u<v\leq n$, rather than specifying the entire path $P_{u,v}$, record only the vertex, $z$, which is the immediate successor of $u$ along $P_{u,v}$. By consistency, $P_{u,v}$ is the concatenation of the edge $uz$ and $P_{z,v}$. More generally, if we associate with every $u,v$ a vertex $z\in P_{u,v}\setminus\{u,v\}$, then $P_{u,v}=P_{u,z}P_{z,v}$.

Definition 2.1**.**

A résumé of a consistent path system $\mathcal{P}$ over $[n]$ is a partial function $f:\binom{[n]}{2}\to[n]$, s.t.

$\{u,v\}\in\text{dom}(f)$* if and only if $P_{u,v}\neq uv$* 2. 2.

$f(u,v)\in V(P_{u,v})\setminus\{u,v\}$* for all $\{u,v\}\in\text{dom}(f)$*

*We denote *

[TABLE]

Notice that the résumé is uniquely defined for any path system in which every path has length at most $2$. In all other cases multiple résumés exist. As we observe next, a résumé retains all the information of its corresponding path system.

Observation 2.2**.**

Let $\mathcal{P}$ be a consistent path system on $[n]$, and let $f\in\mathcal{R}(\mathcal{P})$ be any corresponding résumé. The following polynomial time algorithm recovers $\mathcal{P}$ from $f$.

Initially, no paths $P_{u,v}$ are defined. 2. 2.

For all $uv\notin\text{dom}(f)$ set $P_{u,v}\coloneqq uv$. 3. 3.

Repeat $n$ times: If $z=f(u,v)$ and $P_{u,z}$ and $P_{z,v}$ are defined set $P_{u,v}\coloneqq P_{u,z}P_{z,v}$.

The algorithm relies on the fact that if $P_{u,v}\neq uv$, then $P_{u,v}$ can be expressed as the concatenations of the two strictly shorter paths $P_{u,f(u,v)}$ and $P_{f(u,v),v}$. Initially, only single edge paths are placed in $\mathcal{P}$. After the $k$-th iteration of the loop in step 3 all the paths in $\mathcal{P}$ of length at most $k+1$ are recovered.

3 The Number of Consistent Path Systems

We now estimate the number of consistent path systems on $n$ vertices. The diameter of a path system is the largest length of a path in the system. We make the following easy observation:

Observation 3.1**.**

Every neighborly path system of diameter $2$ is necessarily consistent.

Let $\mathcal{D}_{k}(G)$ denote the collection of neighborly consistent path systems of diameter $k$ in $G$.

Lemma 3.2**.**

For any graph $G=(V,E)$

[TABLE]

Proof.

If $\text{diam}(G)\geq 3$, the statement is clearly correct, since both sides of the equation vanish. If $\mathcal{P}\in\mathcal{D}_{2}(G)$, then for every pair $u,v\in V$, $P_{u,v}$ is either an edge or path of length $2$. Since $\mathcal{P}$ is neighborly, $|P_{u,v}|=2$ if and only if $u$ and $v$ are not neighbors. But $|N(u)\cap N(v)|$ is the number of paths of length-$2$ between $u$ and $v$. Moreover, as just observed, any neighborly path system of diameter $2$ is automatically consistent. Therefore, we may choose any path of length $2$ for every non-adjacent pair to obtain a consistent path system. The conclusion follows. ∎

The next lemma speaks about the number of neighborly diameter-$2$ path systems on Erdös-Renyi $G(n,p)$ graphs using standard concentrations inequalities in this domain. Any information that we use throughout about $G(n,p)$ graphs can be found in [14].

Lemma 3.3**.**

If $G\sim G(n,p)$, where $p\geq\Omega(\frac{1}{\log n})$, then a.a.s.

[TABLE]

This yields:

Theorem 3.4**.**

We have

[TABLE]

where $\mathscr{P}_{n}$ resp. $\mathscr{M}_{n}$ is the set of all consistent resp. metric path systems on $n$ vertices.

Proof.

Clearly, $|\mathscr{M}_{n}|\leq|\mathscr{P}_{n}|$. By 2.2, $|\mathscr{P}_{n}|$ is bounded by the number of maps from $\binom{[n]}{2}$ to $[n]$, namely, $n^{\binom{n}{2}}$.

On the other hand, by lemma 3.3 for $G\sim G(n,\frac{1}{\log n})$ there holds

[TABLE]

Finally, $\mathcal{D}_{2}(G)\subset\mathscr{M}_{n}$, because any path system of diameter $2$ is induced by assigning a unit weight to every edge. ∎

4 Metrizability and Betweenness

Arguably, (strictly) metric path systems constitute the most natural instances of consistency in path systems. This means that the paths in the system can be realized as the (unique) shortest paths w.r.t. some edge-weighted $K_{n}$. We offer a straightforward but useful characterization of (strict) metrizability in terms of metrics on $V$.

We start with a definition: a pointed triple $\{a,b;c\}$ with distinct elements $a,b,c\in V$, is the set $\{a,b,c\}$ where the point $c$ is distinguished from $a,b$. Let $\binom{V}{3}^{\ast}$ denote the set of pointed triples over $[n]$ and note that $\big|\binom{V}{3}^{\ast}\big|=3\binom{|V|}{3}$. We say the (pointed) triple $\{a,b;c\}$ is colinear w.r.t. a metric $\rho:V\times V\to\mathbb{R}_{>0}$, if $\rho(a,b)=\rho(a,c)+\rho(c,b)$, i.e., the triangle inequality holds with equality for $\{a,b;c\}$. Let $T(\rho)$ denote the set of all $\rho$-colinear pointed triples. Similarly, for a consistent path system $\mathcal{P}$ let $T(\mathcal{P})=\{\{a,b;c\}\in\binom{V}{3}^{\ast}:c\in P_{a,b}\}$, i.e., $T(\mathcal{P})$ is the collection of pointed triples which are ‘colinear’ w.r.t. the paths in $\mathcal{P}$.

Theorem 4.1**.**

Let $\mathcal{P}$ be a path system on vertex set $V$

$\mathcal{P}$* is strictly metric if and only if $T(\mathcal{P})=T(\rho)$ for some metric $\rho$ on $V$.* 2. 2.

$\mathcal{P}$* is metric if and only if $T(\mathcal{P})\subseteq T(\rho)$ for some metric $\rho$ on $V$.*

Proof.

(Theorem 4.1) We only prove (1), since the proof for (2) is similar. Let $\mathcal{P}$ be a strictly metric path system that is strictly induced by a weight function $w:\binom{V}{2}\to\mathbb{R}_{>0}$. We show that $T(\mathcal{P})=T(\rho)$, where $\rho$ is the shortest path metric that $w$ induces. Namely, $\rho(a,b)=w(P_{a,b})$ for all $a,b\in V$.

Indeed, if $\{a,c;b\}\in T(\mathcal{P})$ then $b\in P_{a,c}$. In particular $\rho(a,c)=w(P_{a,c})=w(P_{a,b})+w(P_{b,c})=\rho(a,b)+\rho(b,c)$, and $\{a,c;b\}\in T(\rho)$.

If $\{a,c;b\}\notin T(\mathcal{P})$ then the $(ac)$-path $P_{a,b}P_{b,c}$ differs from $P_{a,c}$. It follows that $\rho(a,c)=w(P_{a,c})<w(P_{a,b})+w(P_{b,c})=\rho(a,b)+\rho(b,c)$ and $\{a,c;b\}\notin T(\rho)$.

For the reverse implication, let $\rho$ be a metric whose set of colinear triples coincides with $T(\mathcal{P})$. We show that $\mathcal{P}$ is the strictly metric path system that is induced by the following weight function: for $a,b\in V$

[TABLE]

As we show, every path in $\mathcal{P}$ is the unique $w$-geodesic. To this end let us make the following simple observation: If $x_{0},x_{1},\dots,x_{m}$ are elements in a metric space $(V,\Delta)$, then the following are equivalent:

1. (i)

For all $0\leq i_{0}<i_{1}\cdots<i_{k}\leq m$, $\Delta(x_{i_{0}},x_{i_{k}})=\sum_{j=1}^{k}\Delta(x_{i_{j-1}},x_{i_{j}})$ 2. (ii)

For all $0\leq i<j<k\leq m$, $\Delta(x_{i},x_{k})=\Delta(x_{i},x_{j})+\Delta(x_{j},x_{k})$

For $a,b\in V$, by (2) and the above observation

[TABLE]

Let $Q$ be any other $(ab)$-path. Clearly, $w(Q)=\infty$ unless all of $Q$’s edges are in $\mathcal{P}$. Let us turn to $Q$’s vertices. If $V(Q)\subseteq V(P_{a,b})$ but $Q\neq P$, then $E(Q)\not\subseteq E(P_{a,b})$. But then, any edge $e\in E(Q)\setminus E(P_{a,b})$ is, by consistency, not in $\mathcal{P}$, so that $w(Q)=\infty$.

Assume next that $V(Q)\not\subseteq V(P_{a,b})$. As mentioned, by assumption all of $Q$’s edges are in $\mathcal{P}$. Any vertex $c\in V(Q)\setminus V(P_{a,b})$ splits $Q$ into a concatenation of two paths $Q_{1}=x_{0}x_{1}\dots x_{k}$ and $Q_{2}=y_{0}\dots y_{m}$, where $x_{0}=a$, $x_{k}=y_{0}=c$ and $y_{m}=b$. But then

[TABLE]

But since $c\notin P_{a,b}$ the triple $(a,c,b)$ is not $\rho$-colinear, so by the triangle inequality

[TABLE]

∎

As shown in [5] and [8], the metrizability of a given path system can be decided in polynomial time using linear programming. In [5], this problem was rephrased as a multi-commodity flow problem with $O(n^{3})$ variables and $O(n^{4})$ constraints. In [8], the relevant LP has exponentially many constraints and is solved efficiently using the ellipsoid method. Here, we offer a simpler linear program, with $O(n^{2})$ variables and $O(n^{3})$ constraints.

Corollary 4.2**.**

Let $\mathcal{P}$ be a consistent path system on $V$. For every $a\neq b\in[n]$ define a variable $x_{a,b}=x_{b,a}$, and consider the following system of inequalities:

[TABLE]

When $s=0$ the above system is feasible if and only if $\mathcal{P}$ is metric. When $s=1$ the above system is feasible if and only if $\mathcal{P}$ is strictly metric.

Proof.

When $s=1$ this system feasible if and only if there exists a metric $\rho$ such that $T(\mathcal{P})=T(\rho)$. Similarly, when $s=0$ this system feasible if and only if there exists a metric $\rho$ such that $T(\mathcal{P})\subseteq T(\rho)$. The claim follows from theorem 4.1. ∎

In fact, more can be said about testing whether some path system $\mathcal{P}$ is strictly metric. Recall that a psuedometric on $V$ is a function $\rho:V\times V\to\mathbb{R}_{+}$ which satisfies all of the properties of a metric except that it allows for $\rho(x,y)=0$ even if $x\neq y$. The following characterization says that we may drop the positivity assumption when trying to determine if a path system is strictly metric. We emphasize that this is not true for the non-strict case.

Corollary 4.3**.**

The following are equivalent for a path system $\mathcal{P}$ on $V$.

1. (1)

$\mathcal{P}$* is strictly metric.* 2. (2)

There is a metric $\rho$ on $V$ s.t. $T(\rho)=T(\mathcal{P})$. 3. (3)

The following system of inequalities with variables $x_{a,b}=x_{b,a}$, $a\neq b\in[n]$, is feasible:

[TABLE]

Proof.

That (1) and (2) are equivalent is simply a restatement of theorem 4.1. Moreover, it is clear that (2) implies (3). To see that (3) implies (2), we first observe that every feasible solution to the above system of inequalities is non-negative and hence defines a pseudometric $\rho$ with $T(\mathcal{P})=T(\rho)$. Indeed,

[TABLE]

We argue that $\rho$ is actually a metric, i.e. $\rho(u,v)>0$ for all $u\neq v$. Suppose that this is not the case and $\rho(u,v)=0$ for some pair $u$ and $v$. Then for any $z\neq u,v$

[TABLE]

It follows that the above inequalities are in fact equalities, and $\{z,u;v\},\{z,v;u\}\in T(\rho)=T(\mathcal{P})$. Consequently, both $v\in P_{z,u}$ and $u\in P_{z,v}$, contrary to the assumed consistency. ∎

The (semi)metric cone on $n$ points $\text{MET}_{n}$ comprises all pseudometrics, i.e.

[TABLE]

[TABLE]

It is not hard to see that $\sim$-equivalence classes are in a one-to-one correspondence with the faces of $\text{MET}_{n}$. In particular, corollary 4.3 shows how to injectively map any strictly metric path system to a face of the metric cone.

We may further ask for what subsets $S\subset\binom{V}{3}^{\ast}$ there exists some $\rho\in\text{MET}_{n}$ with $T(\rho)=S$. For $t=\{a,b;c\}\in\binom{V}{3}^{\ast}$ let us define $\Delta_{t}\in\mathbb{R}^{\binom{V}{2}}$ as:

[TABLE]

Here, $e_{a,b}$ denotes the standard unit vector of the coordinate $\{a,b\}$. The following characterization, based on LP duality, is used in the sequel:

Lemma 4.4**.**

Given $S\subset\binom{V}{3}^{\ast}$, no pseudometric $\rho$ satisfies $S=T(\rho)$ if and only if there exist $\alpha_{t}\geq 0$, $t\in\binom{V}{3}^{\ast}$, such that

[TABLE]

and $\text{supp}(\alpha)\not\subseteq S$.

Proof.

Note that $\langle\Delta_{t},\rho\rangle\geq 0$ for any pseudometric $\rho$ and any $t\in\binom{V}{3}^{\ast}$, with equality iff $t\in T(\rho)$.

We show first that the existence of such $\alpha$ implies that $S$ is not of the form $T(\rho)$. Our assumption is that $\sum_{s\in S}\Delta_{s}=\sum_{t}\alpha_{t}\Delta_{t}$ where $\alpha_{t}\geq 0$ and $\text{supp}(\alpha)\not\subseteq S$. Now let $\rho\in\mathbb{R}^{\binom{V}{2}}$ be any pseudometric with $S\subseteq T(\rho)$, and let us show that $S\neq T(\rho)$, i.e., the inclusion is proper. Indeed,

[TABLE]

In particular, if $\alpha_{t}>0$ then $\langle\Delta_{t},\rho\rangle=0$ and $t\in T(\rho)$. But by assumption, $\text{supp}(\alpha)\not\subseteq S$, hence $S\subsetneq T(\rho)$.

In the reverse direction, let us consider a collection $S\subset\binom{V}{3}^{\ast}$ that is not realizable as the set of colinear triples of a pseudometric. This means that no $x\in\mathbb{R}^{\binom{V}{2}}$ satisfies the following system of inequalities:

[TABLE]

By LP-duality, (e.g., [22]), some linear combination of the above inequalities yields $0<0$. In other words, there exist real $\{\beta_{s}|s\in S\}$, and $\{\lambda_{t}\geq 0|t\in S^{c}\}$, not all zero, so that

[TABLE]

By rescaling, if necessary, we can assume that all $\beta_{s}\leq 1$, and rewrite,

[TABLE]

where $\alpha_{t}=\lambda_{t}$, $t\notin S$ and $\alpha_{t}=1-\beta_{t}$ for $t\in S$ are all non-negative and at least one $\alpha_{t}$, $t\notin S$, is non-zero. ∎

4.1 Solution Need Not Be Integral

It is also possible to view a pointed triple $\{a,b;c\}$ as an oriented triangle, where edges $\{a,c\},\{c,b\}$ are positively oriented and $\{a,b\}$ is negatively oriented. The corresponding notion of $\{a,b;c\}$’s boundary is:

[TABLE]

We can interpret lemma 4.4 as saying that a subset $S\subseteq\binom{[n]}{3}^{\ast}$ cannot be the collection of colinear triples of some pseudometric if and only if there exists another (fractional) set of pointed triples whose boundary coincides with the boundary of $S$. This formulation is very similar to Bodwin’s [5] perspective, where strictly metric path systems are presented in terms of multi-commodity network flows.

Famously, network flows satisfy an integrality condition, but this is no longer the case when we study multi-commodity network flows. This prompts us to ask whether there always exist an integral witness for non-metrizability. Namely, if the set $S\subseteq\binom{[n]}{3}^{\ast}$ is not the triple set of any pseudometric. Does there exist a multi-set $T\subseteq\binom{[n]}{3}^{\ast}$, with $T\setminus S\neq\emptyset$ and $\sum_{s\in S}\Delta_{s}=\sum_{t\in T}\Delta_{t}$? The answer to this question is no. The following set

[TABLE]

is not the triple system of any metric since

[TABLE]

On the other hand, there exists no integer witness of this fact. This can be verified by using any ILP solver, e.g., [17].

5 Strictly Metric Path Systems

We turn to estimate the number of strictly metric path systems on $n$ vertices. While the number of consistent (as well as metric) path systems on $n$ vertices is $n^{\frac{n^{2}}{2}(1-o(1))}$, the number of strictly metric systems turns out to be considerably smaller. We prove

Theorem 5.1**.**

The set $\mathscr{S}_{n}$ of all strictly metric path systems on $n$ vertices satisfies

[TABLE]

A lower bound of $2^{\frac{n^{2}}{2}(1-o(1))}$ is easy to prove, since every connected graph $G=(V,E)$ induces at least one strictly metric path system, by assigning a weight $w(e):=1+\varepsilon_{e}$ to every edge $e\in E$, where $\varepsilon_{e}$ is taken to be a small generic value.

To improve this lower bound, we consider path systems in $G(n,p)$ random graphs. While theorem 5.1 provides only a very sligt improvement over the trivial lower bound, its proof shows that almost all graphs admit $2^{\Omega(n^{2})}$ strictly metric neighborly path systems, see below. We denote by $\mathscr{NS}(G)$ the family of neighborly strictly metric path systems in the graph $G$. The reader may prefer to read first theorem 5.7 and its proof that are similar but easier.

Lemma 5.2**.**

For fixed $0<p<1$ and $G\sim G(n,p)$ there holds a.a.s.

[TABLE]

Proof.

For fixed $p$, a $G(n,p)$ graph has a.a.s. a perfect matching $M$. Let $M$’s edges be called $x_{1}y_{1},\dots,x_{k}y_{k}$, where $k=\lfloor\frac{n}{2}\rfloor$. We make the edges in $M$ ‘cheap’ and construct a corresponding family of neighborly path systems in $G$, where shortest paths naturally tend to utilize $M$-edges.

For $1\leq i<j\leq k$, we call $x_{i},x_{j}$ an admissible pair of vertices if:

$x_{i}$ and $x_{j}$ are not neighbors in $G$ 2. 2.

Both $x_{i}y_{j}$ and $x_{j}y_{i}$ are edges in $G$

The probability that the vertices $x_{i},x_{j}$ form an admissible pair is $(1-p)p^{2}$. Also, the events ”$x_{i},x_{j}$ form an admissible pair” are independent over all pairs $1\leq i<j\leq k$. By a standard concentration argument, the number of admissible pairs is a.a.s. $(1-p)p^{2}\frac{n^{2}}{8}(1+o(1))$. For each admissible pair $x_{i},x_{j}$ there are precisely two $2$-step $(x_{i}x_{j})$-paths which use an $M$-edge, namely $x_{i}y_{i}x_{j}$ and $x_{i}y_{j}x_{j}$. Clearly, there are $2^{(1-p)p^{2}\frac{n^{2}}{8}(1+o(1))}$ partial path systems where every admissible pair is connected by one of these two paths. We now argue that these systems are strictly metric. Fix one such partial system $\tilde{\mathcal{P}}$, and let $X_{u,v}$ be i.i.d. random variables over $G$’s edges $uv$. Every such a random variable is uniformly distributed on $[0,\epsilon]$, with $\epsilon>0$ some small constant. The value of $X_{u,v}$ gets added to the weight of this edge $uv$. This random noise ensures that every shortest path is unique. Concretely, here is the weight function $w:E\to\mathbb{R}_{>0}$ that we use:

[TABLE]

In words, we assign edge weights as follows: $M$-edges are the cheapest. Next come edges not in $M$ that are in one of the chosen paths in $\tilde{\mathcal{P}}$. All other edges are more expensive. By construction, an edge not in $M$ appears in at most one path in $\tilde{\mathcal{P}}$. It is not difficult to see that every path in $\tilde{\mathcal{P}}$ is the unique $w$-shortest path between its ends. Moreover, the random noise makes all the shortest paths unique. Therefore, $w$ strictly induces a path system $\mathcal{P}$ containing all the paths in $\tilde{\mathcal{P}}$. ∎

To prove the upper bound in theorem 5.1 we use the insight gained in corollary 4.3. In particular, we use lemma 4.4 to bound the number of $\sim$-equivalence classes, see eq. 3, of metrics which correspond to strictly metric systems. Note that, in particular, an upper bound on the number of faces of $\text{MET}_{n}$ yields such a bound. Graham, Yao and Yao [15] showed that this cone has at most $2^{2.72n^{2}}$ faces. Inspired by their ideas, we improve their bounds. Recall that $\Delta_{i,j;k}=e_{i,k}+e_{k,j}-e_{i,j}$ is a vector in $\mathbb{R}^{\binom{[n]}{2}}$. For all partial functions $f:\binom{[n]}{2}\to[n]$ we always assume that $f(a,b)\notin\{a,b\}$ whenever $\{a,b\}\in\text{dom}f$. We introduce the following definition:

Definition 5.3**.**

The signature $\sigma(f)\in\mathbb{Z}^{\binom{[n]}{2}}$ of a partial function $f:\binom{[n]}{2}\to[n]$ is the vector defined by

[TABLE]

The following lemma shows that signatures separate résumés of strictly metric path system from non-résumés:

Lemma 5.4**.**

Let $\mathcal{P}$ be a strictly metric path system and let $f,g:\binom{[n]}{2}\to[n]$, where $f\in\mathcal{R}(\mathcal{P})$ and $g\notin\mathcal{R}(\mathcal{P})$. Then $\sigma(f)\neq\sigma(g)$.

Proof.

The domain size of a function can be easily recovered from its signature:

[TABLE]

where $\mathbf{1}$ is the all-ones vector. Arguing by contradiction, we suppose that $f$ and $g$ are as in the theorem and yet $\sigma(f)=\sigma(g)$. As we saw, this implies that $|\text{dom}(g)|=|\text{dom}(f)|$.

We find next $\{a,b\}\in\text{dom}(g)$ such that $\{a,b;g(a,b)\}\notin T(\mathcal{P})$.

• •

If $\text{dom}(g)\neq\text{dom}(f)$ then we can take any $\{a,b\}\in\text{dom}(g)\setminus\text{dom}(f)$.

• •

If $\text{dom}(g)=\text{dom}(f)$: By assumption $g\not\in\mathcal{R}(\mathcal{P})$. Therefore, there must be a pair $\{a,b\}\in\text{dom}(g)$ for which $g(a,b)\notin P_{a,b}$.

Let us set $S=\{\{a,b;c\}\in T(P):f(a,b)\neq c\}$. We can write

[TABLE]

Since $\{u,v;g(u,v)\}\notin T(\mathcal{P})$ for some $u,v$, by lemma 4.4, $\mathcal{P}$ is not strictly metric.

∎

We now prove theorem 5.1:

Proof.

(Theorem 5.1) We start with the lower bound. As lemma 5.2 shows, a.a.s. $|\mathscr{NS}(G)|\geq 2^{\frac{n^{2}}{64}(1+o(1))}$, when $G\sim G(n,1/2)$. There is no double counting to worry about, since a neighborly path system uniquely defines its graph. Also, all $n$-vertex graphs are equally likely in $G(n,1/2)$. Consequently,

[TABLE]

It is possible to increase the coefficient of $\frac{33}{64}$ by an additive term of about $3\cdot 10^{-4}$. To this end we work in $G(n,p)$ for $p$ a little above $\frac{1}{2}$. The function to be maximized is $p^{2}(1-p)/4+H(p)-1$, where $H$ is the binary entropy function, and the maximum is attained at $p\approx 0.51037$.

As lemma 5.4 shows, we can obtain an upper bound on the number of strictly metric path systems, by bounding the total number of possible signatures for all partial functions $f:\binom{[n]}{2}\to[n]$. It is more convenient to work with vectors of the form $\sigma(f)+\mathbf{1}$. This is a vector of non-negative integers, and following up on a previous calculation, we get

[TABLE]

We bound the number of possible signatures that sum to $m+\binom{n}{2}$ for some $0\leq m\leq\binom{n}{2}$. This, in turn, does not exceed the number of integer solutions to the system

[TABLE]

which equals

[TABLE]

Summing over all $0\leq m\leq\binom{n}{2}$ we get the desired upper bound.

∎

5.1 Faces of the Metric Cone

We can use similar methods to bound the number of faces of the metric cone,

[TABLE]

An upper bound of $2^{2.72n^{2}}$ is mentioned in [15] without proof. Here, we prove a slightly better upper bound:

Theorem 5.5**.**

The order-$n$ metric cone, $\text{MET}_{n}$, has at most $2^{2.38n^{2}(1+o(1))}$ faces.

Before we prove this upper bound, we briefly discuss lower bounds for $\text{MET}_{n}$’s number of faces. Since every strictly metric path system uniquely corresponds to some face of the metric cone, the lower bound in theorem 5.1 provides as well a lower bound for the number of faces. This is the best lower bound that we are aware of. We also mention a result of Avis [3] that $\text{MET}_{n}$ has at least $2^{\binom{n}{2}(1+o(1))}$ extreme rays. His argument shows that the ”standard metric” of asymptotically almost every graph corresponds to an extreme ray of the metric cone. By ”standard metric” of a graph we mean the one where every edge is of unit length.

Let us return to the equivalence relation (3) on $\text{MET}_{n}$, where two metrics are $\sim$-equivalent if they have the same collection of collinear triples. The $\sim$-equivalence classes are in a one-to-one correspondence with $\text{MET}_{n}$’s faces. In fact, each such equivalence class coincides with the relative interior of its associated face. In the interior of $\text{MET}_{n}$ all triangle inequalities are strict, so that $T(\rho)=\emptyset$ for every $\rho\in\text{int}(\text{MET}_{n})$. This equivalence class corresponds to the full-dimensional face that is the cone’s interior. At the other extreme is the cone’s vertex, namely the origin $\{0\}$, a zero-dimensional face where $T(\rho)=\binom{[n]}{3}^{\ast}$. Every other equivalence class is contained in the boundary $\partial(\text{MET}_{n})$, and corresponds to a non-trivial face. We seek to bound the number of these equivalence classes, i.e., the cardinality of the set

[TABLE]

To get started with our proof, we extend the notion of signature to arbitrary subsets of $\binom{[n]}{3}^{\ast}$. Namely, let the signature of a set of pointed triples $S\subseteq\binom{[n]}{3}^{\ast}$ be:

[TABLE]

We also need to define $\text{cl}(S)$, the closure of a subset $S\subseteq\binom{[n]}{3}^{\ast}$. Namely,

[TABLE]

Note that $\text{cl}(S)\in\mathcal{F}$, since $\mathcal{F}$ is intersection closed. Indeed, if $F_{1}=T(\rho_{1})$ and $F_{2}=T(\rho_{2})$, then it is not difficult to see that $F_{1}\cap F_{2}=T(\rho_{1}+\rho_{2})$. We remark, that much of what we say here is not specific to the metric cone and holds just as well for any polyhedral cone. The crux of the matter seems to be that the face sets of a cone and its dual form anti-isomorphic lattices, see e.g. [22, 23]. For example, consider dual cone of $\text{MET}_{n}$. Its extreme rays are exactly the rays spanned by the vectors $\Delta_{s}$, $s\in\binom{[n]}{3}^{\ast}$.

Signature and closure are related as follows:

Lemma 5.6**.**

If $\sigma(S)=\sigma(\tilde{S})$ for some $S,\tilde{S}\subseteq\binom{[n]}{3}^{\ast}$, then $\text{cl}(S)=\text{cl}(\tilde{S})$.

Proof.

Let $\rho,\tilde{\rho}\in\text{MET}_{n}$ be such that $\text{cl}(S)=T(\rho)$, $\text{cl}(\tilde{S})=T(\tilde{\rho})$. By definition, $\langle\rho,\Delta_{s}\rangle=0$ holds for all $s\in S$. In particular,

[TABLE]

But $\langle\rho,s\rangle\geq 0$ for all $s\in\binom{[n]}{3}^{\ast}$, and therefore $\langle\rho,s\rangle=0$ for all $s\in\tilde{S}$. This implies $\tilde{S}\subseteq T(\rho)$ and hence $T(\tilde{\rho})=\text{cl}(\tilde{S})\subseteq T(\rho)$. By the same argument, $T(\rho)\subseteq T(\tilde{\rho})$. ∎

Now we can prove theorem 5.5.

Proof.

(Theorem 5.5) For every $F\in\mathcal{F}$ we fix a subset $B\subseteq F$ such that $\{\Delta_{t}:t\in B\}$ is a basis of the subspace spanned by $\{\Delta_{t}:t\in F\}$. Note that $\text{cl}(B)=F$. So, by lemma 5.6 all these sets have distinct signatures. It therefore suffices to bound the number of such signatures.

Clearly, $0\leq|B|\leq\binom{n}{2}$, since $B$ is a basis for some subspace of $\mathbb{R}^{\binom{[n]}{2}}$. How many distinct signatures $\sigma(S)$ are there of sets $S\subset\binom{[n]}{3}^{\ast}$ of given cardinality $0\leq m\leq\binom{n}{2}$? For every $t\in S$, the vector $\Delta_{t}$ has two $+1$ entries and one $-1$ entry. Therefore, it is possible to express $\sigma(S)$ as $v_{p}-v_{n}$, where $v_{p},v_{n}$ are ${\binom{[n]}{2}}$-dimensional vectors of non-negative integer entries such that $\langle v_{n},\mathbf{1}\rangle=m$, and $\langle v_{p},\mathbf{1}\rangle=2m$. The number of such vectors $v_{p}$ does not exceed the number of integer solutions to the system

[TABLE]

which equals

[TABLE]

Similarly, the number of vectors $v_{n}$ is bounded by

[TABLE]

Therefore, the number of signatures $\sigma(S)$ with $|S|=m$ is bounded by

[TABLE]

Summing over $0\leq m\leq\binom{n}{2}$ yields the result. ∎

5.2 Strictly Metric Systems in a Fixed Graph

As before, $\mathscr{NS}(G)$ denotes the number of neighborly strictly metric path systems of a given graph $G$. We seek the maximum of $|\mathscr{NS}(G)|$ over all $n$-vertex graphs $G$. As theorem 5.1 and lemma 5.2 show, $|\mathscr{NS}(G)|=2^{O(n^{2})}$ for every $n$-vertex graph, yet $|\mathscr{NS}(G)|=2^{\Theta(n^{2})}$ for asymptotically almost all of them. We ask for the largest $\alpha>0$ for which there exist $n$-vertex graphs with $|\mathscr{NS}(G)|\geq 2^{\alpha n^{2}}$. As it turns out, balanced complete bipartite graphs yield $\alpha\geq\frac{1}{8}$.

Theorem 5.7**.**

There holds $|\mathscr{NS}(K_{\frac{n}{2},\frac{n}{2}})|\geq 2^{\frac{n^{2}}{8}}$.

Proof.

This proof can be viewed as a simpler version of the proof of lemma 5.2. Let $X=\{x_{1},\dots,x_{\frac{n}{2}}\}$ and $Y=\{y_{1},\dots,y_{\frac{n}{2}}\}$ be the two parts of $K_{\frac{n}{2},\frac{n}{2}}$’s vertex set. We introduce the perfect matching $M$ of all edges $x_{i}y_{i}$ for $i=1,\ldots,\frac{n}{2}$, and think of the edges in $M$ as ‘cheap’, so that shortest paths tend to use them. For every $1\leq i<j\leq\frac{n}{2}$, there are precisely two $2$-hop paths between $x_{i}$ and $x_{j}$ that use edges from $M$, namely, $x_{i}y_{i}x_{j}$ and $x_{i}y_{j}x_{j}$. For every pair $x_{i},x_{j}$ we may (independently) choose between one of these two paths. We associate with every edge $x_{i}y_{j}$ an i.i.d. random variable $X_{i,j}$ that is uniformly distributed on $[0,\epsilon]$, for some small constant $\epsilon>0$. These random variables act as random noise to make every shortest path unique. Define the following weight function $w:E\to\mathbb{R}_{>0}$ on $K_{\frac{n}{2},\frac{n}{2}}$:

[TABLE]

It’s not difficult to see that every chosen path is uniquely shortest w.r.t. these edge weights. There are $\binom{\frac{n}{2}}{2}$ pairs $x_{i},x_{j}$ and $2$ choices for each such pair. The conclusion follows. ∎

5.2.1 Monotone path systems

We now consider a family of neighborly diameter-$2$ path systems $\mathcal{P}$ in $\mathcal{J}_{n}$, the join of a clique and an anti-clique. To wit, $V(\mathcal{J}_{n})$ is the disjoint union of the sets $\{x_{1},\ldots,x_{n}\}$ and $\{y_{1},\ldots,y_{n}\}$, and the only pairs of non-neighbors are $x_{i},x_{j}$ with $1\leq i<j\leq n$. The path system $\mathcal{P}$ that we are considering is neighborly and diameter $2$, so we only need to specify, for every pair $1\leq i<j\leq n$, the index $1\leq k\leq n$, such that

[TABLE]

It is convenient to represent the (unique) résumé of $\mathcal{P}$ as a symmetric matrix $M$ where $m_{ij}=m_{ji}=k$ when eq. 4 holds (and $m_{ii}=\ast$ for all $i$). We call $\mathcal{P}$ monotone if the entries in every row, ignoring the diagonal entry, are non-decreasing. By symmetry, the same holds when we go down columns. In terms of résumés, if $\mathcal{P}$ is monotone and say, $f(x_{i},x_{j})=y_{a}$ and $f(x_{i},x_{k})=y_{b}$ then $j\leq k$ implies $a\leq b$.

Theorem 5.8**.**

Every monotone path system as above, is strictly metric.

We start with some technical comments on consistent diameter-$2$ path systems. Below we use the convention that $\alpha_{a,a;c}=\alpha_{a,b;a}=0$.

Lemma 5.9**.**

Let $f:\binom{[n]}{2}\to[n]$ be the résumé of a diameter-$2$, consistent $n$-vertex path system $\mathcal{P}$. Then $\mathcal{P}$ is strictly metric if and only if there exists $x\in\mathbb{R}^{\binom{[n]}{2}}$ satisfying the following system of linear inequalities,

[TABLE]

Before we go into the proof, let us emphasize that we do not limit ourselves to non-negative $x$, as the proof amply illustrates.

Proof.

(Lemma 5.9) Clearly, if $\mathcal{P}$ is strictly metric, then the system (5) is feasible, even with positive $x$.

Next we make some comments on feasible solutions to the system (5). We express such solutions in terms of $w\in\mathbb{R}^{\binom{[n]}{2}}$, and assume w.l.o.g. that $w_{u,v}=\infty$ for every edge $uv$ that does not appear in $\mathcal{P}$. We also observe that feasible solutions are translation invariant. Namely, for any real constant $K$, the function $w$ is a feasible solution if and only if $w+K\mathbf{1}$ is.

To proceed, let us choose $K>0$ much bigger than $|w_{u,v}|$ for every edge $uv\in\mathcal{P}$. We argue that all paths in $\mathcal{P}$ are uniquely shortest on the complete graph with edge weights $\tilde{w}=w+K\mathbf{1}$. For $P_{u,v}\in\mathcal{P}$ we need to show that every competitor $uv$-path $Q$ has strictly larger weight. We can clearly assume that $Q$ uses only edges in $\mathcal{P}$ or else $\tilde{w}(Q)=\infty$. If $P_{u,v}$ is an edge, then necessarily any other $uv$-path $Q$ has $2$ or more edges, whence $\tilde{w}(P_{u,v})=K+O_{K}(1)<2K+O_{K}(1)<\tilde{w}(Q)+O_{K}(1)$.

If $P_{u,v}$ is a $2$-edge path, then $uv\notin\mathcal{P}$ and so $\tilde{w}(uv)=\infty$. If $Q$ is a $uv$-path of $3$ or more edges, then $\tilde{w}(P_{u,v})=2K+O_{K}(1)<3K+O_{K}(1)<\tilde{w}(Q)+O_{K}(1)$. Finally, if $Q=uzv$ is another $uv$-path of $2$ edges then

[TABLE]

since $\tilde{w}$ is a feasible solution of the above system of inequalities. ∎

Lemma 5.9 yields the following result which we use to prove theorem 5.8. It is also possible to accomplish this using lemma 4.4.

Lemma 5.10**.**

Let $f:\binom{[n]}{2}\to[n]$ be the résumé of a diameter-$2$, consistent $n$-vertex path system $\mathcal{P}$. Then $\mathcal{P}$ is not strictly metric if and only if there exist coefficients $\alpha_{a,b;c}\geq 0$ such that for all $a,b\in\text{dom}(f)$, $\sum_{i=1}^{n}\alpha_{a,b;i}=1$ and

[TABLE]

and $\alpha_{a,b;f(a,b)}<1$ for some $a,b\in\text{dom}(f)$.

Proof.

This claim follows from lemma 5.10 and some form of LP duality, such as Farkas Lemma or Gordan’s Lemma [10]. Indeed, the system

[TABLE]

is not feasible if and only if there exists $\beta_{a,b;c}\geq 0$ not all zero such that

[TABLE]

where we set $\beta_{a,b;f(a,b)}=0$. By scaling if necessary, we can assume that $\sum_{i=1}^{n}\beta_{a,b;i}<1$ for all $a,b$. Adding $\sum_{a,b\in\text{dom}(f)}\left(1-\sum_{i=1}^{n}\beta_{a,b;i}\right)(e_{a,f(a,b)}+e_{f(a,b),b})$ to both sides we get

[TABLE]

where

[TABLE]

and $\alpha_{a,b;f(a,b)}<1$ for at least one $a,b$.

∎

Now we prove theorem 5.8

Proof.

In this proof we also refer to the (monotone) matrix $M$ mentioned after (4), which may help the reader’s intuition.

If $\mathcal{P}$ is not strictly metric, then by lemma 5.10, this is witnessed by some non-negative $\alpha_{x_{i},x_{j};y_{k}}$ for which $\sum_{k=1}^{n}\alpha_{x_{i},x_{j};y_{k}}=1$ and

[TABLE]

and $\alpha_{x_{i},x_{j};f(x_{i},x_{j})}<1$ for some $i<j$. Notice that the coefficient of $e_{x_{a},y_{c}}$ counts the paths in $\mathcal{P}$ which start at $x_{a}$ and go via $y_{c}$. In particular, for fixed $x_{a}$ and $y_{c}$

[TABLE]

To simplify our notation, we write below $\{a,b;c\}$ rather than $\{x_{a},x_{b};y_{c}\}$ and $f(a,b)=c$ instead of $f(x_{a},x_{b})=y_{c}$.

Let $(a,b)$ with $a<b$ be the lexicographically first pair of indices such that $\alpha_{a,b;f(a,b)}<1$. As discussed above, by lemma 5.10 we have

[TABLE]

since both sides count the occurrences of $f(a,b)$ in row $a$ of $M$.

By the minimality assumption, $\alpha_{a,j;f(a,j)}=1$ for $j<b$ and $\alpha_{a,j;k}=0$ whenever $j<b$, $k\neq f(a,j)$. So we can write

[TABLE]

where both sides count the occurrences of $f(a,b)$ in row $a$ of $M$ starting from column $b$.

Since $M$ is monotone, if two entries in the same row are equal, then all entries in between them are equal to them as well.

In other words, there exists some $c>b$ such that $f(a,j)=f(a,b)$ for all $b\leq j<c$ and $f(a,j)>f(a,b)$ for $j\geq c$. We can therefore write

[TABLE]

Since $\alpha_{i,j;k}\leq 1$ and $\alpha_{a,b;f(a,b)}<1$ we have $\sum_{j=b}^{c-1}\alpha_{a,j;f(a,b)}<c-b$. In particular, there is some $l\geq c$ such that $\alpha_{a,l;f(a,b)}>0$. Now, it must also be the case that

[TABLE]

By the minimality of $a$ we have that $\alpha_{i,l,f(i,l)}=1$ for $i<a$ and $\alpha_{i,l,c}=0$ for $i<a$, $c\neq f(i,l)$. Therefore,

[TABLE]

As before, both sides count the occurrences of $f(a,b)$ in column $l$ starting from row $a$. On the other hand, since $f(a,l)>f(a,b)$, by monotonicity $f(i,l)\geq f(a,l)>f(a,b)$ for all $i\geq a$. Therefore,

[TABLE]

a contradiction. ∎

Monotone path systems yield a slight improvement of our previous bound.

Proposition 5.11**.**

The number of monotone path systems in the $n$-vertex graph $\mathcal{J}_{\frac{n}{2}}$ is $2^{\alpha n^{2}(1+o(1))}$. In particular,

[TABLE]

where $\alpha=\frac{3}{16}\left(3\log_{2}3-4\right)\approx.141$

The proof of proposition 5.11 builds on the notion of plane partitions. These generalization of integer partitions were introduced in [19]. Let us recall some of the essentials. A plane partition of the positive integer $n$ is a two-dimensional array of non-negative integers $\pi_{i,j}$ that is non-increasing in both indices and which sums up to $n$, i.e. $\pi_{i+1,j}\leq\pi_{i,j}$, $\pi_{i,j+1}\leq\pi_{i,j}$ and $\sum_{i,j}\pi_{i,j}=n$. An $(r,s,t)$-boxed plane partition is a plane partition such that $\pi_{i,j}\leq t$ with $\pi_{i,j}$ non-zero only when $i\leq r$ and $j\leq s$. In other words, it is an $r\times s$ matrix with entries in $\{0,1\dots,t\}$ whose rows and columns are non-increasing. For example, here is a $(3,4,5)$-boxed plane partition of $21$:

[TABLE]

MacMahon [19] found a formula for $N(r,s,t)$, the number of $(r,s,t)$-boxed plane partitions.

Proposition 5.12**.**

For all $r,s,t\geq 1$ we have

[TABLE]

A symmetric $(r,t)$ plane partitions is an $(r,r,t)$-boxed plane partition s.t. $\pi_{i,j}=\pi_{j,i}$ for all $i$ and $j$. In other words, this is a symmetric $r\times r$ matrix with entries in $\{0,1,\dots,t\}$ with non-increasing rows (and columns). Proving a conjecture of MacMahon, Andrews [2] found a formula for $S(r,t)$, the number of symmetric $(r,t)$ plane partitions.

Proposition 5.13**.**

For all $r,t\geq 1$ we have

[TABLE]

Let us return to path systems. A matrix $M$ as in section 5.2.1 is an $(n,n-1)$ symmetric plane partition, with the only restriction that $m_{ii}=\ast$ for all $i$. This allows us to reduce the counting problem of such monotone systems to the asymptotics of $S(n,n)$, because the additional factor that comes from all the possible diagonals of $M$ is only a lower order term.

Lemma 5.14**.**

We have

[TABLE]

Proof.

Put differently, we are claiming that

[TABLE]

where $\beta=\frac{3}{4}\left(3\ln 3-4\ln 2\right)$. By propositions 5.12 and 5.13 we have

[TABLE]

and it follows that $2\lim_{n\to\infty}\frac{\ln S(n,n)}{n^{2}}=\lim_{n\to\infty}\frac{\ln N(n,n,n)}{n^{2}}$. Therefore, it suffices to prove that the latter limit exists and is equal to $2\beta$. First we can rewrite proposition 5.12 as

[TABLE]

Therefore,

[TABLE]

where in the second line we have applied Stirling’s approximation. For every $a\geq 0$, the function $f_{a}(x)=(a+x)\ln(a+x)$ is increasing with $x\geq 1$, and therefore

[TABLE]

In particular,

[TABLE]

We turn to determine this integral, starting with the indefinite one

[TABLE]

and hence the definite integral yields

[TABLE]

It follows that

[TABLE]

∎

Finally, we prove proposition 5.11.

Proof.

(Proposition 5.11) For ease of notation we prove the claim for the $2n$-vertex graph $\mathcal{J}_{n}$, rather than the $n$-vertex $\mathcal{J}_{\frac{n}{2}}$. As already mentioned, this amounts to showing that after taking the log, the number of monotone systems in $\mathcal{J}_{n}$ and the number of $(n,n)$ symmetric plane partitions agree up to $1+o_{n}(1)$ factor. Let $\mathcal{M}$ denote the set of monotone path systems in $\mathcal{J}_{n}$. There is an injection from $\mathcal{M}$ to set of symmetric $(n,n-1)$ plane partition. Indeed, if $f:\binom{[n]}{2}\to[n]$ is the résumé of a monotone system we associate with it a plane partition $\pi_{i,j}$ such that $n-f(i,j)=\pi_{i,j}=\pi_{j,i}$ and $\pi_{i,i}=\pi_{i+1,i}$. It follows that $|\mathcal{M}|\leq S(n,n-1)$. On the other hand, given a symmetric $(n,n-1)$ plane partition $\pi_{i,j}$, the résumé $f(i,j)=n-\pi_{i,j}$ defines a monotone path system. The only reason that this mapping is not injective is that modifications of the diagonal entries $\pi_{i,i}$ leave the induced path system unchanged. Therefore, we arrive at the inequality $S(n,n-1)\leq|\mathcal{M}|n^{n}$. Observe that

[TABLE]

and

[TABLE]

The claim then follows from lemma 5.14. ∎

Remark 5.15**.**

The constant in proposition 5.11 can be modestly improved by considering $\mathcal{B}_{n,\gamma}$, a slightly different $n$-vertex graph. It is the join of an anti-clique of size $\beta n$ with a clique of size $(1-\gamma)n$, where $0<\gamma<1$. An identical argument shows that monotone systems in such a graph are also strictly metric. The analysis is similar and shows that $\gamma\approx 0.63$ maximizes the number of monotone systems and yields $\mathscr{NS}(\mathcal{B}_{n,\gamma})\geq 2^{\xi n^{2}(1+o(1))}$ where $\xi\approx 0.154$.

6 Maximum VC Classes

We now take a different perspective of path systems provided by the theory of VC dimension. We first recall the fundamentals of this domain. Let $\mathcal{F}\subseteq 2^{[n]}$ be a set system for some $n\geq 1$. We say that $\mathcal{F}$ shatters a set $S\subseteq[n]$ if $\mathcal{F}\cap S\coloneqq\{S\cap F:F\in\mathcal{F}\}=2^{S}$. The VC dimension of $\mathcal{F}$ is

[TABLE]

The well known Perles-Sauer-Shelah bound ([21]) states that if $\mathcal{F}\subseteq 2^{[n]}$ and $\dim_{VC}\mathcal{F}=d$, then $|\mathcal{F}|\leq\sum_{i=0}^{d}\binom{n}{i}$. In this view, a class $\mathcal{F}\subseteq 2^{[n]}$ of VC dimension $d$ is said to be a maximum class if $|\mathcal{F}|=\sum_{i=0}^{d}\binom{n}{d}$.

Let $\mathcal{P}$ be a consistent path system on $[n]$ and let $\mathcal{F}\subset 2^{[n]}$,

[TABLE]

In other words, $\mathcal{F}_{\mathcal{P}}$ is a set system containing all the singletons, the empty-set (i.e., a full 1-dimensional skeleton) as well as the vertex set of each path in $\mathcal{P}$. We have the following observation:

Observation 6.1**.**

For any consistent path system $\mathcal{P}$, the set system $\mathcal{F}_{\mathcal{P}}$ is an intersection-closed maximum class of VC dimension $2$.

Proof.

That $\mathcal{F}_{\mathcal{P}}$ is intersection closed follows from the consistency of $\mathcal{P}$. Also, it is clear that $|\mathcal{F}_{\mathcal{P}}|=\binom{n}{2}+\binom{n}{1}+\binom{n}{0}$. It remains to show that this class has VC dimension $2$. Indeed, $\mathcal{F}_{\mathcal{P}}$ shatters every set $\{u,v\}$ of cardinality $2$, because $V(P_{u,v})$, $\{v\}$, $\{u\}$ and $\emptyset$ are all contained in $\mathcal{F}_{\mathcal{P}}$.

Finally, $\mathcal{F}_{\mathcal{P}}$ cannot shatter any $3$-vertex set $S=\{a,b,c\}$. Otherwise, some path $P\in\mathcal{P}$ contains all three vertices. Say they appear in $P$ in the order $a,b,c$. By consistency, any path in $\mathcal{P}$ that contains the vertices $a$ and $c$ must also contain $b$, contrary to the condition of shattering. ∎

Let $m(n,d)$ denote the number of maximum classes $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension $d$. The following result is due to Alon, Moran and Yehudayoff [1]:

Proposition 6.2**.**

For fixed $d\geq 1$,

[TABLE]

As shown in [4], the upper bound in proposition 6.2 tight. Their lower bound on $m(n,d)$ is attained by appealing to certain induced matchings of the boolean cube. Here, we reprove this lower bound from a different perspective, by generalizing the path system construction in theorem 3.4. Explicitly, the statement is:

Theorem 6.3**.**

For fixed $d\geq 1$,

[TABLE]

Moreover, the same holds even for the number of intersection closed maximum classes of VC dimension $d$.

Before proving theorem 6.3 we consider the special case when $d=2$. Note that the upper bounds in proposition 6.2 along with 6.1 reprove the upper bound in theorem 3.4. Moreover, for $d=2$ theorem 6.3 follows from the lower bound in theorem 3.4, and the fact that $|\mathscr{P}_{n}|\leq m(n,2)$. Recall that in the lower bound proof of theorem 3.4 we considered the number of diameter-$2$ path systems in random $G(n,p)$ graphs. Likewise, in the proof of theorem 6.3 we apply a similar idea to Linial-Meshulam random complexes, [18, 20]. Let us recall the fundamentals of this randomized model. $Y_{k}(n,p)$ is a distribution over $k$-dimensional simplicial complexes with vertex set $[n]$. A complex $Y\sim Y_{k}(n,p)$ sampled from this distribution has a full $(k-1)$-dimensional skeleton, i.e., $Y$ contains all subsets of $[n]$ of size $\leq k$. Every subset of size $k+1$ (a ”$k$-dimensional face”) is placed in $Y$ independently with probability $p$. The set of $Y$’s $k$-dimensional faces is denoted by $E(Y)$. Note that $Y_{1}(n,p)$ is identical with $G(n,p)$. Here we’ll work with $k=d-1$.

Let $Y$ be a $k$-dimensional simplicial complex on vertex set $[n]$ and let $S\in\binom{[n]}{k+1}\setminus Y$. We say that a vertex $a\not\in S$ is $Y$-compatible to $S$ if $S\cup\{a\}\setminus\{x\}\in Y$ for every $x\in S$. We call $S\cup\{a\}$ a $Y$-compatible extension of the base $S$. Notice that the extended set $S\cup\{a\}$ indeed uniquely defines its base $S$, because for any $x\neq a$ in $S\cup\{a\}$, the set $S\cup\{a\}\setminus\{x\}$ belongs to $Y$.

The case $k=1$ provides an interesting illustration. Here $Y$ is a graph, and $S=\{u,v\}$ comprises a pair of $Y$-non-neighbors. A vertex $a\neq u,v$ is $Y$-compatible to $S$, if it is a $Y$-neighbor of both $u$ and $v$.

Standard concentration results yield:

Lemma 6.4**.**

Let $Y\sim Y_{k}(n,p)$ for some $k\geq 0$. If $p\geq\Omega(\frac{1}{\log n})$ then a.a.s.

$|E(Y)|=p\binom{n}{k+1}(1+o(1))$** 2. 2.

The number of $Y$-compatible vertices to any $S\in\binom{[n]}{k+1}\setminus Y$ is $p^{k+1}n(1+o(1))$.

We now prove theorem 6.3.

Proof.

(Theorem 6.3)

As already mentioned, we wish to show that the upper bound in proposition 6.2 is tight. We construct a large collection of maximal families $\mathcal{F}$ of VC-dimension $d$. In particular, each $\mathcal{F}$ is a set systems with $\sum_{i=0}^{d}\binom{n}{i}$ sets. To start, we sample $Y\sim Y_{d-1}(n,p)$ with $p=\frac{1}{\log n}$. We may and will assume (with asymptotic almost certainty) that $Y$ satisfies the conclusions of lemma 6.4. The family $\mathcal{F}$ comprises all the faces of $Y$, as well as a $Y$-compatible extension of every set in $\binom{[n]}{d}\setminus E(Y)$. Clearly, $\mathcal{F}$ contains $\sum_{i=0}^{d}\binom{n}{i}$ sets, as planned.

By lemma 6.4, any set in $\binom{[n]}{d}\setminus E(Y)$ has $p^{d}n(1+o(1))$ $Y$-compatible extensions. Since $|\binom{[n]}{d}\setminus E(Y)|=(1-p)\binom{n}{d}(1+o(1))$ we get a total of

[TABLE]

distinct set systems $\mathcal{F}$ of the desired cardinality. Moreover, $p=\frac{1}{\log n}=o(1)$ and $p^{d}=n^{-d\frac{\log\log n}{\log n}}=n^{-o(1)}$ and the above expression simplifies to

[TABLE]

There is no double-counting here, because, as mentioned, the extended sets in $\mathcal{F}$ uniquely split to base-plus-extension. It remains to show that the VC dimension of each such system is $d$. It is at least $d$, since clearly all members of $E(Y)$ are shattered. On the other hands, suppose that one such system $\mathcal{F}$ shatters some set $Z$ of cardinality $d+1$. Necessarily, $Z\in\mathcal{F}$ because all sets in $\mathcal{F}$ have cardinality $\leq d+1$. By construction, $Z$ must be a $Y$-compatible extension, i.e., $Z=S\cup\{a\}$ for some $S\not\in T$. It follows that $\mathcal{F}$ does not shatter $Z$.

∎

7 Open Questions

Numerous interesting open questions suggest themselves. Here are some of them

• •

We have determined the asymptotics of $\log(|\mathscr{M}_{n}|)$ as well as $\log(|\mathscr{P}_{n}|)$, viz.,

[TABLE]

where $\mathscr{P}_{n}$ and $\mathscr{M}_{n}$ is the set of all consistent, resp. metric, path systems on $n$ vertices. It is reasonable to conjecture that $|\mathscr{M}_{n}|=o_{n}(|\mathscr{P}_{n}|)$, and even ask about the rate at which $|\mathscr{M}_{n}|/|\mathscr{P}_{n}|$ tends to zero with $n$.

We note that $|\mathscr{P}_{n}\setminus\mathscr{M}_{n}|=n^{\frac{n^{2}}{2}(1+o(1))}$. Recall that the Petersen graph $H$ admits a non-metric neighborly path system $\mathcal{P}_{H}$ [8]. Let us fix a copy of $H$ inside a $G(n,\frac{1}{\log n})$ graph $G$. Each of the following $n^{\frac{n^{2}}{2}(1+o(1))}$ path systems in $G$ is neighborly, consistent and non-metric. On $H$ all these systems coincide with $\mathcal{P}_{H}$. Every other pair of non-adjacent vertices in $G$ is connected by any of their $\frac{(1+o(1))n}{\log^{2}n}$ length-$2$ paths. The sub-system $\mathcal{P}_{H}$ makes all of them non-metric.

• •

Almost all bounds in the paper are about the asymptotics of the logarithm. It is desirable, of course, to upgrade these estimates to direct asymptotics.

• •

There are various natural measures of disagreement $\Delta$ between pairs of path systems, $\mathcal{P}$ and $\mathcal{Q}$ on the same vertex set $[n]$. Here are two, but numerous other measures suggest themselves:

The number of pairs $1\leq i<j\leq n$ for which $P_{i,j}\neq Q_{i,j}$. 2. 2.

The number of triples $i,j,k\in[n]$ where $k$ resides on one of the two paths $P_{i,j}$, $Q_{i,j}$, but not on both.

This induces a notion of distance between an $n$-vertex path system $\mathcal{P}$ and a class $\mathcal{A}$ of $n$-vertex path systems, namely

[TABLE]

We have considered various classes of path systems, e.g., consistent, metric, strictly metric, neighborly and more. Given two such classes of path systems $\mathcal{A}$ and $\mathcal{B}$, it is interesting to

Determine $\min\text{dist}(\mathcal{P},\mathcal{A})$ over $\mathcal{P}\in\mathcal{B}$, or 2. 2.

Understand how $\text{dist}(\mathcal{P},\mathcal{A})$ is distributed over $\mathcal{P}\in\mathcal{B}$.

• •

It is particularly intriguing to compare between metric and strictly metric path systems. Here is a possible perspective on this question. Define $B$ to be the cube $(0,1)^{\binom{n}{2}}$, where we view a point $w\in B$ as a map $\binom{n}{2}\to(0,1)$. Let $\mathcal{Q}$ be any strictly metric path system on vertex set $[n]$. Corresponding to $\mathcal{Q}$ is a set of linear inequalities in the values $w(i,j)$ that uniquely defines $\mathcal{Q}$. Namely, it is the intersection of all half spaces that are defined by an inequality of the form $w(Q_{xy})<w(Q^{\prime})$, over all pairs of vertices $x,y$, where $Q_{xy}$ is the $xy$-path in $\mathcal{Q}$ and $Q^{\prime}\neq Q_{xy}$ is any other $xy$ path. Here, if $P=(v_{1},v_{2},\ldots)$ is a path, then $w(P):=\sum_{i}w(v_{i},v_{i+1})$. This collection of linear inequalities defines a full-dimensional polytope $C_{\mathcal{Q}}$. The collection of polytopes $C_{\mathcal{Q}}$, ranging over all strictly metric $n$-point path systems $\mathcal{Q}$ partition the cube $B$ minus a measure zero set into a finite collection of polytopes. What do these polytopes typically look like? What are their vertex sets?

This measure-zero set comprises the hyperplane arrangement $\mathcal{A}$ defined by all homogeneous linear equations of the form $w(P)=w(Q)$ for some two distinct paths $P,Q$ with the same pair of end vertices.

Many questions suggest themselves: Given $w\in\mathcal{A}$, which (non-strict, but consistent) path systems does it correspond to? How to sort them out? Can we decide which pairs of strictly metric path systems share a face? Of which dimension? Which non-strict metric path systems border on a given strict one? This discussion naturally assigns a dimension to every non-strict metric path system $\mathcal{P}$, namely, the highest dimension of a face of $\mathcal{A}$ on which $\mathcal{P}$ resides. What can be said about this dimension? Does every face of $\mathcal{A}$ contain some non-strict metric path system?

• •

The Graham, Yao and Yao paper [15] does not speak about path systems but rather about connection matrices, for the definition of which we refer the reader to their paper. The relation between the two theories is still not fully understood.

• •

These questions are also closely related to the quest of the $f$-vector of the metric cone. For this as well as the study of the metric polytope, see [12, 11, 13]

• •

For the number of $n$-vertex strictly metric path systems we have the estimates:

[TABLE]

Is it possible to narrow the gap between the bounds? Perhaps even derive the exact constant in the exponent.

• •

In particular, we still do not know how many strictly metric path systems a fixed graph can have. Neither do we know the answer for a random $G(n,\frac{1}{2})$ graph, i.e., for almost all graphs.

• •

In theorem 5.8 we find certain classes of monotone path systems that are strictly metric. We conjecture that this is a special case of a broader phenomenon. Let $f:\binom{[n]}{2}\to[n]$ be the (unique) résumé of a diameter-$2$ path system $\mathcal{P}$. We say $\mathcal{P}$ is monotone if $f(i,j)\leq f(i,k)$ whenever $\{i,j\},\{i,k\}\in\text{dom}f$ and $j\leq k$. We conjecture:

Conjecture 7.1**.**

Every monotone consistent path system of diameter $2$ is strictly metric.

• •

We recall that we still do not have a satisfactory theory of neighborly path systems. To mention one specific question in this direction: Given a graph $G$, how hard is it to decide whether $G$ has a consistent neighborly path system that is non-metric?