Online Degree-Bounded Steiner Network Design

TL;DR

This paper introduces an online algorithm for the degree-bounded Steiner forest problem, achieving an optimal competitive ratio of O(log n), and establishes tight bounds and hardness results for related network design problems.

Contribution

The paper presents the first online algorithm with an optimal competitive ratio for degree-bounded Steiner forest, advancing understanding of online network design.

Findings

Achieves an O(log n) competitive ratio for the online degree-bounded Steiner forest.

Proves no algorithm can do better than o(log n) competitive ratio.

Provides hardness results for related variants like group Steiner tree.

Abstract

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem { which asks for a subgraph with minimum degree that connects a given set of vertices { is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given online but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance. We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems. Fourer and Raghavachari resolved the online variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

Equations

$$\alpha$$

$$\forall S\subseteq \mathcal{S}$$

$$\forall v\in V$$

$$\mathbf{x}(e),\alpha \in {\mathbb{R}}_{\ge 0}$$

$$\displaystyle\quad\phantom{\rule[-13.77771pt]{1.0pt}{24.11105pt}}\mathchoice{\sum_{\mathchoice{\hbox to0.0pt{\hss$\displaystyle{S\in\mathcal{S}}$\hss}}{\hbox to0.0pt{\hss$\textstyle{S\in\mathcal{S}}$\hss}}{\hbox to0.0pt{\hss$\scriptstyle{S\in\mathcal{S}}$\hss}}{\hbox to0.0pt{\hss$\scriptscriptstyle{S\in\mathcal{S}}$\hss}}}}{\sum_{S\in\mathcal{S}}}{\sum_{S\in\mathcal{S}}}{\sum_{S\in\mathcal{S}}}{\mathbf{y}(S)}$$

$$\forall e=(u,v)$$

$$\displaystyle\quad\phantom{\rule[-13.77771pt]{1.0pt}{24.11105pt}}\mathchoice{\sum_{\mathchoice{\hbox to0.0pt{\hss$\displaystyle{v}$\hss}}{\hbox to0.0pt{\hss$\textstyle{v}$\hss}}{\hbox to0.0pt{\hss$\scriptstyle{v}$\hss}}{\hbox to0.0pt{\hss$\scriptscriptstyle{v}$\hss}}}}{\sum_{v}}{\sum_{v}}{\sum_{v}}{\mathbf{z}(v)b_{v}}\leq 1$$

$$\mathbf{z}(v),\mathbf{y}(S)\in {\mathbb{R}}_{\ge 0}$$

$$\operatorname{excess}(r,b)=\sum_{v\in\Gamma_{b}(r)}\!\!\Big{(}\operatorname{deg}_{{\sf GA}}(v)-\lceil rb_{v}\rceil+2\Big{)}\enspace,$$

$$\mathrm{excess}(r,b)$$

$$=2|D(r)|+\sum _{i\in D(r)}(q_i-2)+|{\Gamma }_b(r)|$$

$$\le 2|D(r)|+3|{\Gamma }_b(r)|,$$

$$\sum _{S\in {\mathcal{S}}_{\Delta }}{\mathbf{y}}_r(S)=\sum _{S\in \mathrm{CC}(G_{\Delta }\setminus {\Gamma }_{\Delta }(r))\cap {\mathcal{S}}_{\Delta }}\frac{1}{b({\Gamma }_{\Delta }(r))}\ge \frac{|D(r)|+1}{b({\Gamma }_{\Delta }(r))}.$$

$$\sum _v{\mathbf{z}}_r(v)b_v=\sum _{v\in {\Gamma }_{\Delta }(r)}\frac{1}{b({\Gamma }_{\Delta }(r))}\cdot b_v=1.$$

$$\mathrm{excess}(r,\Delta )\ge \frac{h_{\Delta }-1/\Delta }{4{\log }_{2}n+6}\cdot b({\Gamma }_{\Delta }(r))-|{\Gamma }_{\Delta }(r)|.$$

$$U(q)$$

$$\begin{array}{rl}\mathrm{deg}(\overline{r})-\mathrm{deg}(\underline{r})& =(\lceil \overline{r}{b}_v\rceil -2)-(\lceil \underline{r}{b}_v\rceil -2)\\ & \ge (\overline{r}-\underline{r})b_v-1.\end{array}$$

$$\begin{array}{rl}\mathrm{excess}(\underline{r},\Delta )& \ge (\overline{r}-\underline{r})\cdot b({\Gamma }_{\Delta }(\overline{r}))-|{\Gamma }_{\Delta }(\overline{r})|\\ & \ge \frac{1}{2}(\overline{r}-\underline{r})\cdot b({\Gamma }_{\Delta }(\underline{r}))-|{\Gamma }_{\Delta }(\underline{r})|\\ & \ge \frac{h_{\Delta }-1/\Delta }{4{\log }_{2}n+6}\cdot b({\Gamma }_{\Delta }(\underline{r}))-|{\Gamma }_{\Delta }(\underline{r})|,\end{array}$$

$$\begin{split}h_{\Delta}&\leq(4\log_{2}n+6)\cdot\frac{\operatorname{excess}(r,\Delta)+\Gamma_{\Delta}(r)}{b(\Gamma_{\Delta}(r))}+\frac{1}{\Delta}\\ &\leq(4\log_{2}n+6)\cdot\frac{2D(r)+4\Gamma_{\Delta}(r)}{b(\Gamma_{\Delta}(r))}+\frac{1}{\Delta}\\ &\leq(4\log_{2}n+6)\cdot\Big{(}2\operatorname{\mathrm{OPT}}+4\frac{1}{\Delta}\Big{)}+\frac{1}{\Delta}\\ &\leq(24\log_{2}n+37)\cdot\operatorname{\mathrm{OPT}}\enspace.\end{split}$$

$$h\le {\ell }_H^{+}(v^{\ast })\le {\tau }_i.$$

$${\tau }_i$$

$$\le h_{\Delta }+\frac{2}{b_v}$$

$$\le h_{\Delta }+\frac{2}{\Delta }$$

$$\le h_{\Delta }+2\mathrm{OPT}$$

$$\le O(\log n)\cdot \mathrm{OPT}$$

$$T$$

$$w(\mathrm{OPT})\le w(T)=\sum _{i=1}^{r-1}{n}^i\in O(n^{r-1}).$$

$$E[w(X_r)]$$

$$E[\mathrm{deg}(root)]$$

Full text

We design a simple greedy-like algorithm that achieves a competitive ratio of $O(\log n)$ where $n$ is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio $o(\log n)$; thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.

Fürer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA’92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

1 Introduction

The problem of satisfying connectivity demands on a graph while respecting given constraints has been a pillar of the area of network design since the early seventies [14, 13, 12, 41]. The problem of degree-bounded spanning tree, introduced in Garey and Johnson’s Black Book of NP-Completeness [37], was first investigated in the pioneering work of Fürer and Raghavachari [17] (Allerton’90). In the degree-bounded spanning tree problem, the goal is to construct a spanning tree for a graph $G=(V,E)$ with $n$ vertices whose maximal degree is the smallest among all spanning trees. Let $b^{*}$ denote the maximal degree of an optimal spanning tree. Fürer and Raghavachari [17] give a parallel approximation algorithm which produces a spanning tree of degree at most $O(\log(n)b^{*})$.

Agrawal, Klein, and Ravi ([2]) consider the following generalizations of the problem. In the degree-bounded Steiner tree problem we are only required to connect a given subset $T\subseteq V$. In the even more general degree-bounded Steiner forest problem the demands consist of vertex pairs, and the goal is to output a subgraph in which for every demand there is a path connecting the pair. They design an algorithm that obtains a multiplicative approximation factor of $O(\log(n))$. Their main technique is to reduce the problem to minimizing congestion under integral concurrent flow restrictions and to then use the randomized rounding approach due to Raghavan and Thompson ([43], STOC’85).

Shortly after the work of Agrawal et al., in an independent work in SODA’92 and later J. of Algorithms’94, Fürer and Raghavachari [18] significantly improved the result for degree-bounded Steiner forest by presenting an algorithm which produces a Steiner forest with maximum degree at most $b^{*}+1$. They show that the same guarantee carries over to the directed variant of the problem as well. Their result is based on reducing the problem to that of computing a sequence of maximal matchings on certain auxiliary graphs. This result settles the approximability of the problem, as computing an optimal solution is NP-hard even in the spanning tree case.

In this paper, we initiate the study of degree-bounded network design problems in an online setting, where connectivity demands appear over time and must be immediately satisfied. We first design a deterministic algorithm for online degree-bounded Steiner forest with a logarithmic competitive ratio. Then we show that this competitive ratio is asymptotically best possible by proving a matching lower bound for randomized algorithms that already holds for the Steiner tree variant of the problem.

In the offline scenario, the results of Fürer, Raghavachari [17, 18] and Agrawal, Klein, Ravi [2] were the starting point of a very popular line of work on various degree-bounded network design problems [34, 20, 40, 31, 27, 15]. We refer the reader to the next sections for a brief summary. One particular variant that has been extensively studied is the edge-weighted degree-bounded spanning tree. Initiated by Ravi et al. ([34], J. of Algorithms’98), in this version, we are given a weight function over the edges and a bound $b$ on the maximum degree of a vertex. The goal is to find a minimum-weight spanning tree with maximum degree at most $b$. The groundbreaking results obtained by Goemans ([20], FOCS’06) and Singh and Lau ([44], STOC’07) settle the problem by giving an algorithm that computes a minimum-weight spanning tree with degree at most $b+1$. Slightly worse result are obtained by Singh and Lau ([31], STOC’08) for the Steiner tree variant. Unfortunately, in the online setting it is not possible to obtain a comparable result. We show that for any (randomized) algorithm $\mathcal{A}$ there exists a request sequence such that $\mathcal{A}$ outputs a sub-graph that either has weight $\Omega(n)\cdot\mathrm{OPT}_{b}$ or maximum degree $\Omega(n)\cdot b$.

1.1 Our Contributions

In the online variant of degree-bounded Steiner forest, we are given the graph $G$ in advance, however, demands arrive in an online fashion. At online step $i$, a new demand $(s_{i},t_{i})$ arrives. Starting from an empty subgraph, at each step the online algorithm should augment its solution so that the endpoints of the new demand $s_{i}$ and $t_{i}$ are connected. The goal is to minimize the maximum degree of the solution subgraph. In the non-uniform variant of the problem, a degree bound $b_{v}\in\mathbb{R}^{+}$ is given for every vertex $v$. For a subgraph $H$ and a vertex $v$, let $\deg_{H}(v)$ denote the degree of $v$ in $H$. The load of a vertex is defined as the ratio ${\deg_{H}(v)}/{b_{v}}$. In the non-uniform variant of online degree-bounded Steiner forest, the goal is to find a subgraph satisfying the demands while minimizing the maximum load of a vertex.

Our algorithm is a simple and intuitive greedy algorithm. Upon the arrival of a new demand $(s_{i},t_{i})$, the greedy algorithm (GA) satisfies the demand by choosing an $(s_{i},t_{i})$-path $P_{i}$ such that after augmenting the solution with $P_{i}$, the maximum load of a vertex in $P_{i}$ is minimum. A main result of our paper is to prove that the maximum load of a vertex in the output of GA is within a logarithmic factor of $\operatorname{\mathrm{OPT}}$, the maximum load of a vertex in an optimal offline solution which knows all the demands in advance.

Theorem 1.1

*The algorithm GA produces an output with maximum load at most $O(\log n)\cdot\operatorname{\mathrm{OPT}}$. *

The crux of our analysis is establishing several structural properties of the solution subgraph. First we group the demands according to the maximum load of the bottleneck vertex at the time of arrival of the demand. We then show that for every threshold $r>0$, vertices with load at least $r$ at the end of the run of GA, form a cut set that well separates the group of demands with load at least $r$ at their bottleneck vertex. Since the threshold value can be chosen arbitrarily, this leads to a series of cuts that form a chain. The greedy nature of the algorithm indicates that each cut highly disconnects the demands. Intuitively, a cut that highly disconnects the graph (or the demands) implies a lower bound on the number of branches of every feasible solution.

We use a natural dual-fitting argument to show that for every cut set, the ratio between the number of demands in the corresponding group, over the total degree bound of the cut, is a lower bound for $\operatorname{\mathrm{OPT}}$. Hence, the problem comes down to showing that one of the cuts in the series has ratio at least $1/O(\log n)$ fraction of the maximum load $h$ of the output of GA. To this end, we first partition the range of $r\in(0,h]$ into $O(\log n)$ layers based on the total degree bound of the corresponding cut. We then show that the required cut can be found in an interval with maximum range of $r$. We analyze GA formally in Section 2.

We complement our first theorem by giving an example for a special case of online degree-bounded Steiner tree in which no online (randomized) algorithm can achieve a competitive ratio better than $\Omega(\log n)$.

Theorem 1.2

Any (randomized) online algorithm for the degree bounded online Steiner tree problem has (multiplicative) competitive ratio $\Omega(\log n)$. This already holds when $b_{v}=1$ for every node.

We further investigate the following extensions of the online degree bounded Steiner tree problem. First, we consider the edge-weighted variant of the degree-bounded Steiner tree problem. Second, we consider the group Steiner tree version in which each demand consists of a subset of vertices, and the goal is to find a tree that covers at least one vertex of each demand group. The following theorems show that one cannot obtain a non-trivial competitive ratio for these versions in their general form.111Our lower bound results imply that one needs to restrict the input in order to achieve competitiveness. In particular for the edge-weighted variant, our proof does not rule out the existence of a competitive algorithm when the edge weights are polynomially bounded.

Theorem 1.3

*Consider the edge weighted variant of online degree-bounded Steiner tree. For any (randomized) online algorithm $\mathcal{A}$, there exists an instance and a request sequence such that either $\operatorname{E}\left[\operatorname{maxdegree}(\mathcal{A})\right]\geq\Omega(n)\cdot b$ or $\operatorname{E}\left[\operatorname{weight}(\mathcal{A})\right]\geq\Omega(n)\cdot\operatorname{\mathrm{OPT}}_{b}$, where $\operatorname{\mathrm{OPT}}_{b}$ denotes the minimum weight of a Steiner tree with maximum degree $b$. *

Theorem 1.4

*There is no deterministic algorithm with competitive ratio $o(n)$ for the degree-bounded group Steiner tree problem. *

1.2 Related Degree-Bounded Connectivity Problems

The classical family of degree-bounded network design problems have various applications in broadcasting information, package distribution, decentralized communication networks, etc. (see e.g. [19, 23]). Ravi et al. ([34], J. Algorithms’98), first considered the general edge-weighted variant of the problem. They give a bi-criteria $(O(\log n),O(\log n)\cdot b)$-approximation algorithm, i.e., the degree of every node in the output tree is $O(\log n)\cdot b$ while its total weight is $O(\log n)$ times the optimal weight. A long line of work (see e.g. [28], STOC’00 and [29], SIAM J. C.) was focused on this problem until a groundbreaking breakthrough was obtained by Goemans ([20], FOCS’06); his algorithm computes a minimum-weight spanning tree with degree at most $b+2$. Later on, Singh and Lau ([44], STOC’07) improved the degree approximation factor by designing an algorithm that outputs a tree with optimal cost while the maximum degree is at most $b+1$.

In the degree-bounded survivable network design problem, a number $d_{i}$ is associated with each demand $(s_{i},t_{i})$. The solution subgraph should contain at least $d_{i}$ edge-disjoint paths between $s_{i}$ and $t_{i}$. Indeed this problem has been shown to admit bi-criteria approximation algorithms with constant approximation factors (e.g. [31], STOC’08). We refer the reader to a recent survey in [30]. This problem has been recently considered in the node-weighted variant too (see e.g. [40, 15]). The degree-bounded variant of several other problems such as $k$-MST and $k$-arborescence has also been considered in the offline setting for which we refer the reader to [27, 9] and references therein.

1.3 Related Online Problems

Online network design problems have attracted substantial attention in the last decades. The online edge-weighted Steiner tree problem, in which the goal is to find a minimum-weight subgraph connecting the demand nodes, was first considered by Imase and Waxman ([24], SIAM J. D. M.’91). They showed that a natural greedy algorithm has a competitive ratio of $O(\log n)$, which is optimal up to constants. This result was generalized to the online edge-weighted Steiner forest problem by Awerbuch et al. ([7], SODA’96) and Berman and Coulston ([10], STOC’97). Later on, Naor, Panigrahi, and Singh ([39]), FOCS’11) and Hajiaghayi, Liaghat, and Panigrahi ([22], FOCS’13), designed poly-logarithmic competitive algorithms for the more general node-weighted variant of Steiner connectivity problems. This line of work has been further investigated in the prize-collecting version of the problem, in which one can ignore a demand by paying its given penalty. Qian and Williamson ([42], ICALP’11) and Hajiaghayi, Liaghat, and Panigrahi ([21], ICALP’14) develop algorithms with a poly-logarithmic competitive algorithms for these variants.

The online $b$-matching problem is another related problem in which vertices have degree bounds but the objective is to maximize the size of the solution subgraph. In the worst case model, the celebrated result of Karp et al. ([26], STOC’90) gives a $(1-1/e)$-competitive algorithm. Different variants of this problem have been extensively studied in the past decade, e.g., for the random arrival model see [16, 25, 32, 36], for the full information model see [33, 38], and for the prophet-inequality model see [5, 3, 4]. We also refer the reader to the comprehensive survey by Mehta [35].

Many of the problems mentioned above can be described with an online packing or covering linear program. Initiated by work of Alon et al. [6] on the online set cover problem, Buchbinder and Naor developed a strong framework for solving packing/covering LPs fractionally online. For the applications of their general framework in solving numerous online problems, we refer the reader to the survey in [11]. Azar et al. [8] generalize this method for the fractional mixed packing and covering LPs. In particular, they show an application of their method for integrally solving a generalization of capacitated set cover. Their result is a bi-criteria competitive algorithm that violates the capacities by at most an $O(\log^{2}n)$ factor while the cost of the ouput is within $O(\log^{2}n)$ factor of optimal. We note that although the fractional variant of our problem is a special case of online mixed packing/covering LPs, we do not know of any online rounding method for Steiner connectivity problems.

1.4 Preliminaries

Let $G=(V,E)$ denote an undirected graph of size $n$ ($|V|=n$). For every vertex $v\in V$, let $b_{v}\in\mathbb{R}^{+}$ denote the degree bound of $v$. In the degree-bounded Steiner forest problem, we are given a sequence of connectivity demands. The $i^{th}$ demand is a pair of vertices $(s_{i},t_{i})$ which we call the endpoints of the demand. An algorithm outputs a subgraph $H\subseteq G$ in which for every demand its endpoints are connected. The load of a vertex $v$ w.r.t. $H$ is defined as $\ell_{H}(v)={\deg_{H}(v)}/{b_{v}}$. We may drop the subscript $H$ when it is clear from the context. The goal is to find a subgraph $H$ that minimizes the maximum load of a vertex. Observe that one can always find an optimal solution without a cycle, hence the name Steiner forest. Furthermore, without loss of generality222One can add a dummy vertex for every vertex $v\in V$ connected to $v$. We then interpret a demand between two vertices as a demand between the corresponding dummy vertices. The degree bound of a dummy vertex can be set to infinity. This transformation can increase the degree of any node by at most a factor of $2$, which does not affect our asymptotic results., we assume that the endpoints of the demands are distinct vertices with degree one in $G$ and degree bound infinity. We denote the maximum load of a vertex in an optimal subgraph by $\operatorname{\mathrm{OPT}}=\min_{H}\max_{v}\ell_{H}(v)$.

The following mixed packing/covering linear program ($\mathbb{P}$) is a relaxation for the natural integer program for degree-bounded Steiner forest. Let $\mathcal{S}$ denote the collection of subsets of vertices that separate the endpoints of at least one demand. For a set of vertices $S$, let $\delta(S)$ denote the set of edges with exactly one endpoint in $S$. In $\mathbb{P}$, for an edge $e$, $\mathbf{x}(e)=1$ indicates that we include $e$ in the solution while $\mathbf{x}(e)=0$ indicates otherwise. The variable $\alpha$ indicates an upper bound on the load of every vertex. The first set of constraints ensures that the endpoints of every demand is connected. The second set of constraints ensures that the load of a vertex is upper bounded by $\alpha$. The program $\mathbb{D}$ is the dual of the LP relaxation $\mathbb{P}$.

[TABLE]

[TABLE]

In the online variant of the problem, $G$ and the degree bounds are known in advance, however, the demands are given one by one. Upon the arrival of the $i^{th}$ demand, the online algorithm needs to output a subgraph $H_{i}$ that satisfies all the demands seen so far. The output subgraph can only be augmented, i.e., for every $j<i$, $H_{j}\subseteq H_{i}$. The competitive ratio of an online algorithm is then defined as the worst case ratio of ${\max_{v}\ell_{H}(v)}/{\operatorname{\mathrm{OPT}}}$ over all possible demand sequences where $H$ is the final output of the algorithm.

2 Online Degree-Bounded Steiner Forest

Consider an arbitrary online step where a new demand $(s,t)$ arrived. Let $H$ denote the online output of the previous step. In order to augment $H$ for connecting $s$ and $t$, one can shortcut through the connected components of $H$. We say an edge $e=(u,v)$ is an extension edge w.r.t. $H$, if $u$ and $v$ are not connected in $H$. Let $P$ denote an $(s,t)$-path in $G$. The extension part $P^{*}$ of $P$ is defined as the set of extension edges of $P$. Observe that augmenting $H$ with $P^{*}$ satisfies the demand $(s,t)$. For a vertex $v$, we define $\ell^{+}_{H}(v)=\ell_{H}(v)+{2}/{b_{v}}$ to be the uptick load of $v$. We slightly misuse the notation by defining $\ell^{+}_{H}(P^{*})=\max_{v\in V(P^{*})}\ell^{+}_{H}(v)$ as the uptick load of $P^{*}$, where $V(P^{*})$ is the set of endpoints of edges in $P^{*}$.

The greedy algorithm (GA) starts with an empty subset $H$. Upon arrival of the $i$-th demand $(s_{i},t_{i})$, we find a path $P_{i}$ with smallest uptick load $\ell^{+}_{H}(P_{i}^{*})$ where $P_{i}^{*}$ is the extension part of $P_{i}$. We break ties in favor of the path with a smaller number of edges. Note that the path $P_{i}$ can be found efficiently using Dijkstra-like algorithms. We define $\tau_{i}=\ell^{+}_{H}(P_{i}^{*})$ to be the arrival threshold of the $i$-th demand. We satisfy the new demand by adding $P_{i}^{*}$ to the edge set of $H$ (see Algorithm 1).

2.1 Analysis

We now use a dual-fitting approach to show that GA has an asymptotically tight competitive ratio of $O(\log(n))$. In the following we use GA to also refer to the final output of our greedy algorithm. We first show several structural properties of GA. We then use these combinatorial properties to construct a family of feasible dual vectors for $\mathbb{D}$. We finally show that there always exists a member of this family with an objective value of at least a ${1}/{O(\log(n))}$ fraction of the maximum load of a vertex in GA. This in turn proves the desired competitive ratio by using weak duality.

For a real value $r\geq 0$, let $\Gamma(r)$ denote the set of vertices with $\ell^{+}_{{\sf GA}}(v)\geq r$. Let $D(r)$ denote the indices of demands $i$ for which the arrival threshold $\tau_{i}$ is at least $r$. For a subgraph $X$, let $\operatorname{CC}(X)$ denote the collection of connected components of $X$. For ease of notation, we may use the name of a subgraph to denote the set of vertices of that subgraph as well. Furthermore, for a graph $X$ and a subgraph $Y\subseteq X$, let $X\setminus Y$ denote the graph obtained from $X$ by removing the vertices of $Y$.

Recall that $\mathcal{S}$ is the collection of subsets that separate the endpoints of at least one demand. The following lemma, intuitively speaking, implies that $\Gamma(r)$ well-separates the endpoints of $D(r)$.

Lemma 2.1

For any threshold $r>0$, we have $\left|\operatorname{CC}(G\setminus\Gamma(r))\cap\mathcal{S}\right|\geq|D(r)|+1$.

**Proof. **For a vertex $v\in G\setminus\Gamma(r)$ we use $\operatorname{CC}(v)$ to denote its connected component. Observe that, since the endpoints of demands are nodes with infinite degree bound, the endpoints are not contained in $\Gamma(r)$, and, hence, are in $G\setminus\Gamma(r)$.

We construct a graph $F$ that has one node for every connected component in $G\setminus\Gamma(r)$ that contains an endpoint of a demand in $D(r)$. Edges in $F$ are defined as follows. For every demand $i\in D(r)$ between $s_{i}$ and $t_{i}$, we add an edge that connects the components $\operatorname{CC}(s_{i})$ and $\operatorname{CC}(t_{i})$. In the following we argue that $F$ does not contain cycles. This gives the lemma since in a forest $|D(r)|+1=|E_{F}|+1\leq|V_{F}|\leq|\operatorname{CC}(G\setminus\Gamma(r))\cap\mathcal{S}|$.

Assume for contradiction that the sequence $(e_{i_{0}},\dots,e_{i_{k-1}})$, $k\geq 2$ forms a (minimal) cycle in $F$, where $e_{i_{j}}$ corresponds to the demand between vertices $s_{i_{j}}$ and $t_{i_{j}}$ (see Figure 1 in the Appendix). Assume wlog. that $e_{i_{k-1}}$ is the edge of the cycle for which the corresponding demand appears last, i.e., $i_{k-1}\geq i_{j}$ for every $j<k$. Let $H$ denote the online solution at the time of arrival of the demand $i_{k-1}$. We can augment $H$ to connect each $t_{i_{j}}$ to $s_{i_{j+1\bmod k}}$, $0\leq j\leq k-1$ without using any node from $\Gamma(r)$, since these nodes are in the same component in $G\setminus\Gamma(r)$. But then we have a path $P$ between $s_{i_{k-1}}$ and $t_{i_{k-1}}$ and the extension part $P^{*}$ does not contain any vertex from $\Gamma(r)$. This is a contradiction since the arrival threshold for the demand $i_{k-1}$ is at least $r$. $\Box$

Lemma 2.1 shows that cutting $\Gamma(r)$ highly disconnects the demands in $D(r)$. Indeed this implies a bound on the toughness of the graph. Toughness, first defined by Chvátal [14] and later generalized by Agrawal et al. [1], is a measure of how easy it is to disconnect a graph into many components by removing a few vertices. More formally, the toughness of a graph is defined as $\min_{X\subseteq V}\frac{|X|}{|\operatorname{CC}(G\setminus X)|}$. For the spanning tree variant of the problem, it is not hard to see that $\operatorname{\mathrm{OPT}}$ is at least the reciprocal of toughness. Although it is more involved, we can still establish a similar relation for the non-uniform Steiner forest problem (see Lemma 2.3). However, first we need to show a lower bound on the number of demands separated by $\Gamma(r)$.

We establish a lower bound for $|D(r)|$ with respect to the load of vertices in $\Gamma(r)$. For any $r,b>0$ we define $\Gamma_{b}(r):=\{\ell_{{\sf GA}}^{+}(v)\geq r\wedge b_{v}\geq b\}$, as the set of nodes with degree bound at least $b$ that have uptick load at least $r$ in the final online solution. We further define

[TABLE]

which sums the (absolute) load that nodes in $\Gamma_{b}(r)$ receive after their uptick load became $r$ or larger. The following lemma relates $|D(r)|$ to $\operatorname{excess}(r,b)$.

Lemma 2.2

*For any $r,b>0$, $\operatorname{excess}(r,b)\leq 2|D(r)|+3|\Gamma_{b}(v)|$. *

**Proof. **Consider some online step $i$. Let $H$ denote the output of the previous step. Let $P^{*}_{i}$ be the extension part of the path selected by GA and let $V(P^{*}_{i})$ denote the endpoints of edges of $P^{*}_{i}$. Since in GA we break ties in favor of the path with the smaller number of edges, we can assume that the path does not go through a connected component of $H$ more than once, i.e., for every $C\in\operatorname{CC}(H)$, $|V(P^{*}_{i})\cap C|\leq 2$.

Consider the variable quantity $\delta(r,b):=\sum_{v:\ell^{+}_{H}(v)\geq r\wedge b_{v}\geq b}(\deg_{H}(v)-\lceil rb_{v}\rceil+2)$ throughout the steps of GA where $H$ denotes the output of the algorithm at every step. Intuitively, at each step $\delta(r,b)$ denotes the total increment in degrees of those vertices in $\Gamma_{b}(r)$ that already reached uptick load $r$. In particular, at the end of the run of GA, $\delta(r,b)=\operatorname{excess}(r,b)$.

Now suppose at step $i$ adding the edges $P^{*}_{i}$ to $H$ increases $\delta(r,b)$ by $q_{i}$. There are two reasons for such an increase. On the one hand, if the demand $i$ is from $D(r)$ it may simply increase $\operatorname{deg}_{H}(v)$ for some node with uptick load at least $r$. On the other hand also if the demand is not from $D(r)$ it may cause a node to increase its uptick load to $r$ in which case it could contribute to the above sum with at most 1 (in a step the degree may increase by 2; the first increase by 1 could get the uptick load to $\geq r$ and the second increase contributes to the sum).

Clearly increases of the second type can accumulate to at most $|\Gamma_{b}(r)|$. In order to derive a bound on the first type of increases recall that we assume that endpoints of demands are distinct vertices with degree one and thus $s_{i}$ and $t_{i}$ are outside any connected component of $H$. Since $V(P^{*}_{i})$ contains at most two vertices of a connected component of $H$, we can assert that the path selected by GA is passing through at least $q_{i}/2$ components of $H$ that contain some vertices of $\Gamma_{b}(r)$. Hence, after adding $P^{*}_{i}$ to the solution, the number of connected components of the solution that contain some vertices of $\Gamma_{b}(r)$ decreases by at least $(q_{i}-2)/2$. However, throughout all the steps, the total amount of such decrements cannot be more than the number of vertices in $\Gamma_{b}(r)$. Therefore

[TABLE]

and the lemma follows. $\Box$

Let $\Delta>0$ denote the minimum degree bound of a vertex with non-zero degree in an optimal solution. Note that this definition implies that $\operatorname{\mathrm{OPT}}\geq 1/\Delta$. For a set of vertices $\Gamma$, let $b(\Gamma)=\sum_{v\in\Gamma}b_{v}$. We now describe the main dual-fitting argument for bounding the maximum load in GA.

Lemma 2.3

For any $r>0$, ${|D(r)|}/{b(\Gamma_{\Delta}(r))}\leq\operatorname{\mathrm{OPT}}$.

**Proof. **Let $G_{\Delta}$ denote the graph obtained from $G$ by removing vertices with degree bound below $\Delta$. Similarly, let $\mathcal{S}_{\Delta}$ denote the collection of sets that separate a demand in $G_{\Delta}$. Consider a slightly modified dual program $\text{\ref{LP:dualsf}}_{\Delta}$ defined on $G_{\Delta}$ and $\mathcal{S}_{\Delta}$, i.e., we obtain $\text{\ref{LP:dualsf}}_{\Delta}$ from the original dual program by removing all vertices with degree bound below $\Delta$. We note that the objective value of a dual feasible solution for $\text{\ref{LP:dualsf}}_{\Delta}$ is still a lower bound for $\operatorname{\mathrm{OPT}}$. In the remainder of the proof, we may refer to $\text{\ref{LP:dualsf}}_{\Delta}$ as the dual program. Recall that in a feasible dual solution, we need to define a dual value $\mathbf{y}(S)$ for every cut $S\in\mathcal{S}_{\Delta}$ such that for every edge $e=(u,v)$ the total value of cuts that contain $e$ does not exceed $\mathbf{z}(u)+\mathbf{z}(v)$. On the other hand, $\sum_{v}\mathbf{z}(v)b_{v}$ cannot be more than one.

For a real value $r>0$, we construct a dual vector $(\mathbf{y}_{r},\mathbf{z}_{r})$ as follows. For every $v\in\Gamma_{\Delta}(r)$, set $\mathbf{z}_{r}(v)={1}/{b(\Gamma_{\Delta}(r))}$; for other vertices set $\mathbf{z}_{r}(v)=0$. For every component $S\in\operatorname{CC}(G_{\Delta}\setminus\Gamma_{\Delta}(r))\cap\mathcal{S}_{\Delta}$, set $\mathbf{y}_{r}(S)={1}/{b(\Gamma_{\Delta}(r))}$; for all other sets set $\mathbf{y}_{r}(S)=0$. We prove the lemma by showing the feasibility of the dual vector $(\mathbf{y}_{r},\mathbf{z}_{r})$ for $\text{\ref{LP:dualsf}}_{\Delta}$.

Consider an arbitrary component $C\in\operatorname{CC}(G\setminus\Gamma(r))$. By definition, $C$ separates at least one demand. Let $i$ be such a demand and let $t_{i}\in C$ denote an endpoint of it. Removing vertices with degree bound below $1/\Delta$ from $C$ may break $C$ into multiple smaller components. However, an endpoint of a demand has degree bound infinity, and, hence, the component that contains $t_{i}$ belongs to $\mathcal{S}_{\Delta}$. Therefore $|\operatorname{CC}(G\setminus\Gamma(r))\cap\mathcal{S}|\leq|\operatorname{CC}(G_{\Delta}\setminus\Gamma_{\Delta}(r))\cap\mathcal{S}_{\Delta}|$; which by Lemma 2.1 leads to $|\operatorname{CC}(G_{\Delta}\setminus\Gamma_{\Delta}(r))\cap\mathcal{S}_{\Delta}|\geq|D(r)|+1$. Therefore the dual objective for the vector $(\mathbf{y}_{r},\mathbf{z}_{r})$ is at least

[TABLE]

Thus we only need to show that $(\mathbf{y}_{r},\mathbf{z}_{r})$ is feasible for Program $\text{\ref{LP:dualsf}}_{\Delta}$. First, by construction we have

[TABLE]

Now consider an arbitrary edge $e=(u,v)$. If $e$ does not cross any of the components in $\operatorname{CC}(G_{\Delta}\setminus\Gamma_{\Delta}(r))$, then $\sum_{S:e\in\delta(S)}\mathbf{y}_{r}(S)=0$ and we are done. Otherwise, $\sum_{S:e\in\delta(S)}\mathbf{y}_{r}(S)={1}/{b(\Gamma_{\Delta}(r))}$. However, exactly one endpoint of $e$ is in $\Gamma_{\Delta}(r)$. Thus $\mathbf{z}_{r}(u)+\mathbf{z}_{r}(v)={1}/{b(\Gamma_{\Delta}(r))}$, which implies that the dual vector is feasible. $\Box$

We now have all the ingredients to prove the main theorem.

**Proof of Theorem 1.1: **Let GA denote the final output of the greedy algorithm. Let $h$ denote the maximum load of a vertex in ${\sf GA}$, i.e., $h=\max_{v}\ell_{{\sf GA}}(v)$. Furthermore, let $h_{\Delta}=\max_{v:b_{v}\geq\Delta}\ell_{{\sf GA}}(v)$. In the following we first show that $h_{\Delta}\leq O(\log n)\cdot\operatorname{\mathrm{OPT}}$. For this we use the folowing claim.

Claim 2.4

There exists an $r>0$ such that

[TABLE]

**Proof. **Recall that $\Gamma_{\Delta}(r)$ is the set of vertices with uptick load at least $r$ in GA and degree bound at least $\Delta$. The function $b(\Gamma_{\Delta}(r))$ is non-increasing with $r$ since for every $r_{1}<r_{2}$, $\Gamma_{\Delta}(r_{2})\subseteq\Gamma_{\Delta}(r_{1})$. For $r=1/\Delta$, $b(\Gamma_{\Delta}(r))\leq n(n+1)\Delta$ as a node with degree bound $b_{v}>(n+1)\Delta$ will have uptick load at most $(n+1)/b_{v}<{1}/{\Delta}$, and, hence, will not be in $\Gamma_{\Delta}(r)$. Also, $r<h_{\Delta}$ implies that $b(\Gamma_{\Delta}(r))\geq\Delta$ as there exists a node with load $h_{\Delta}$ in $\Gamma_{\Delta}(r)$ and this node has degree bound at least $\Delta$.

We now partition the range of $r$ (from $1/\Delta$ to $h_{\Delta}$) into logarithmically many intervals. We define the $q$-th interval by

[TABLE]

for $q\in\{0,\dots,\lceil\log_{2}((n+1)n)\rceil\}$. We further set $\overline{r}(q)=\max U(q)$ and $\underline{r}(q)=\min U(q)$, and call $\overline{r}(q)-\underline{r}(q)$ the length of the $q$-th interval. Since there are only $2\log_{2}n+3$ possible choices for $q$ there must exist an interval of length at least $(h_{\Delta}-1/\Delta)/(2\log_{2}n+3)$.

Consider a node $v$ that is contained in $\Gamma_{\Delta}(\overline{r})$ and, hence, also in $\Gamma_{\Delta}(\underline{r})$. This node starts contributing to $\operatorname{excess}(\underline{r},\Delta)$, once its uptick load reached $\underline{r}$ and contributes at least until its uptick load reaches $\overline{r}$. Hence, the total contribution is at least $\operatorname{deg}(\overline{r})-\operatorname{deg}(\underline{r})$, where $\operatorname{deg}(\overline{r})$ and $\operatorname{deg}(\underline{r})$ denotes the degree of node $v$ when reaching uptick load $\overline{r}$ and $\underline{r}$, respectively. We have,

[TABLE]

Summing this over all nodes in $\Gamma_{\Delta}(\overline{r})$ gives

[TABLE]

where the second inequality uses the fact that $|\Gamma_{\Delta}(\overline{r})|\leq|\Gamma_{\Delta}(\underline{r})|\leq|\Gamma_{\Delta}(\overline{r})|/2$. $\Box$

Claim 2.5

$h_{\Delta}\leq(24\log_{2}n+37)\operatorname{\mathrm{OPT}}$.

**Proof. **If we use the $r$ from the previous claim and solve for $h_{\Delta}$ we obtain

[TABLE]

Here the second inequality is due to Lemma 2.2, the third uses Lemma 2.3 and the fact that $b(\Gamma_{\Delta}(r))\geq\Delta|\Gamma_{\Delta}(r)|$. The last inequality uses $\operatorname{\mathrm{OPT}}\geq 1/\Delta$. $\Box$

We now bound the maximum load $h$ in terms of the restricted maximum load $h_{\Delta}$. Consider a vertex $v^{*}$ with maximum load $\ell_{{\sf GA}}(v^{*})=h$. If $b_{v^{*}}\geq\Delta$ then $h_{\Delta}=h$ and we are done. Otherwise consider the last iteration $i$ in which the degree of $v^{*}$ is increased in the solution. Let $H$ denote the output of the algorithm at the end of the previous iteration $i-1$. The degree of $v^{*}$ in the online solution is increased by at most two at iteration $i$. Hence

[TABLE]

Recall that in our greedy algorithm, $\tau_{i}$ is the minimum uptick load of a path that satisfies the new demand. Let $P$ denote a path that connects $s_{i}$ and $t_{i}$ in an optimal solution. Recall that by the definition, $b_{v}\geq\Delta$ for every vertex $v$ of $P$. For every vertex $v$ in $P$ we have

[TABLE]

Therefore leading to $h\leq O(\log n)\cdot\operatorname{\mathrm{OPT}}$. $\Box$

3 An Asymptotically Tight Lower Bound

In the following we show a lower bound for online degree-bounded Steiner tree. Consider a graph $G=(X\uplus Z,E)$, with $|Z|=2^{\ell}$ and $|X|=\binom{2^{\ell}}{2}$. For every pair $\{z_{1},z_{2}\}$ of nodes from $Z$ there exists an edge $\{z_{1},z_{2}\}\in E$ and a node $x\in X$ that is connected to $z_{1}$ and $z_{2}$. An arbitrary node from $Z$ acts as root for the Steiner tree problem.

For a node $z\in Z$, an algorithm $A$, and a request sequence $\sigma$ (consisting of nodes from $X$) we use $\operatorname{deg}_{A,\sigma}(z)$ to denote the number of neighbors of $z$ among all nodes from X in the Steiner tree obtained when running algorithm $A$ on request sequence $\sigma$. Similarly, we use $\operatorname{deg}^{\prime}_{A,\sigma}(z)$ to denote the number of those neighbors of $z$ that also appear in $\sigma$. Note that $\operatorname{deg}(\cdot)$ and $\operatorname{deg}^{\prime}(\cdot)$ ignore edges between nodes from $Z$ as an algorithm (online or offline) can simply connect all nodes in $Z$ in a cycle which increases the degree of any node by only 2.

Lemma 3.1

Fix a possibly randomized online algorithm $A$. For any subset $S\subseteq Z$, $|S|=2^{s}$, $0\leq s\leq\ell$ there exists a request sequence $\sigma_{S}$ consisting of terminals from $X$ s.t.

• •

for a node $x\in\sigma_{S}$ both its neighbors in $G$ are from set $S$;

• •

there exists a node $z^{*}\in S$ with $\operatorname{E}[\operatorname{deg}^{\prime}_{A,\sigma}(z^{*})]\geq s/2$;

• •

there exists an offline algorithm $\mathrm{OFF}$ for servicing requests in $\sigma$ with $\max_{z\in S}\{\operatorname{deg}_{\mathrm{OFF},\sigma}(z)\}\leq 1$, and $\operatorname{deg}_{\mathrm{OFF},\sigma}(z)=0$ for $z\in(Z\setminus S)\cup\{z^{*}\}$.

**Proof. **We prove the lemma by induction over $s$. The base case $s=0$ holds trivially when choosing the empty request sequence. For the induction step consider an arbitrary subset $S\subseteq Z$ with $|S|=2^{s+1}$. Partition $S$ into two disjoint subsets $S_{1}$ and $S_{2}$ of cardinality $2^{s}$ each.

Let $\sigma_{1}$ be the request sequence that exists due to induction hypothesis for set $S_{1}$. Hence, there is a node $z_{1}^{*}\in S_{1}$ with $\operatorname{E}[\operatorname{deg}^{\prime}_{A,\sigma_{1}}(z_{1}^{*})]\geq s/2$. Now, let $\tilde{A}$ behave like algorithm $A$ after it already received a request sequence $\sigma_{1}$ (hence, it starts with all edges that are chosen when running $A$ on $\sigma_{1}$; note, however, that $\operatorname{deg}^{\prime}_{\tilde{A},\sigma_{2}}(\cdot)$ only takes into account edges incident to nodes from $\sigma_{2}$). Due to induction hypothesis for $\tilde{A}$ and set $S_{2}$ there exists a request sequence $\sigma_{2}$ such that $\operatorname{E}[\operatorname{deg}^{\prime}_{\tilde{A},\sigma_{2}}(z_{2}^{*})]\geq s/2$ for a node $z_{2}^{*}\in S_{2}$. Hence, the request sequence $\sigma=\sigma_{1}\circ\sigma_{2}$ fulfills $\operatorname{E}[\operatorname{deg}^{\prime}_{A,\sigma}(z_{1}^{*})]\geq s/2$ and $\operatorname{E}[\operatorname{deg}^{\prime}_{A,\sigma}(z_{2}^{*})]\geq s/2$.

We extend the request sequence by appending the node $x$ that is connected to $z_{1}^{*}$ and $z_{2}^{*}$ in $G$. After serving the request one of the edges $\{x,z_{1}\}$ or $\{x,z_{2}\}$ must be chosen with probability at least 1/2 by $A$. Hence, afterwards either $\operatorname{E}[\operatorname{deg}^{\prime}_{A,\sigma}(z_{1}^{*})]$ or $\operatorname{E}[\operatorname{deg}^{\prime}{\delta}_{A,\sigma}(z_{2}^{*})]$ must be at least $(s+1)/2$.

It remains to argue that there exists a good offline algorithm. Combining the offline algorithms $\mathrm{OFF}_{1}$ and $\mathrm{OFF}_{2}$ for $\sigma_{1}$ and $\sigma_{2}$ gives an offline algorithm for $\sigma_{1}\circ\sigma_{2}$ that has $\max_{z}\{\operatorname{deg}_{\mathrm{OFF},\sigma_{1}\circ\sigma_{2}}(z)\}\leq 1$ and $\operatorname{deg}_{\mathrm{OFF},\sigma_{1}\circ\sigma_{2}}(z)\}=0$ for $z\notin S_{1}\cup S_{2}$ and for $z\in\{z_{1}^{*},z_{2}^{*}\}$. Now, when the node $x$ connected to $z_{1}^{*}$ and $z_{2}^{*}$ is appended to the request sequence the offline algorithm can serve this request by either buying edge $\{x,z_{1}\}$ or $\{x,z_{2}\}$, and can therefore gurantee that $z_{*}$ (either $z_{1}$ or $z_{2}$) has degree [math]. $\Box$

**Proof of Theorem 1.2: **Choosing $s=\log n$ in the above lemma gives our lower bound. $\Box$

Appendix A Figures

Appendix B Lower Bounds for Other Degree-Bounded Steiner Connectivity Problems

In this section we consider strong lower bounds for the general form of two variants of degree-bounded network design problems.

B.1 Online Degree-Bounded Edge-weighted Steiner Tree

Below we present a graph instance $G=(V,E)$ for online bounded-degree edge-weighted Steiner tree in which for any (randomized) online algorithm $A$ there exists a demand sequence for which $A$ either violates the degree bound by a large factor or generates a much larger weight than the optimum.

Consider a graph $G$ as shown in Figure 2(a), which has $n=2k+1$ nodes and $3k$ edges. Every node $i$ ($1\leq i\leq k$) is connected to the root with a zero-weight edge and to node $k+i$ with weight $n^{i}$. In addition, there exist zero-weight edges connecting node $i$ ($k+1\leq i<2k$) to node $i+1$, and there exists a zero-weight edge that connects node $2k$ to the root. We assume that all node weights $b_{v}$ are equal to one and the degree bound $b$ for $\operatorname{\mathrm{OPT}}$ is equal to $3$.

**Proof of Theorem 1.3: **The adversary consecutively presents terminals starting from node $1$. At each step $i$ the algorithm $A$ adds some edges to its current solution such that the $i^{th}$ terminal $t_{i}=i$ gets connected to the $\mathit{root}$.

We use $X_{i}$ to denote the subset of edges chosen by $A$ after step $i$. We also use $0\leq p_{ij}\leq 1$ and $0\leq q_{ij}\leq 1$ to denote the probability that the edges $\{i,\mathit{root}\}$ and $\{i,k+i\}$, respectively, are in $X_{j}$ . After each step $j$, all terminals $t_{i}$ ($i\leq j$) must be connected to the $\mathit{root}$ by at least one of $\{i,\mathit{root}\}$ or $\{i,k+i\}$. Therefore $p_{ij}+q_{ij}\geq 1$. In addition, for every $j_{1}\leq j_{2}$ we have $q_{ij_{1}}\leq q_{ij_{2}}$ and $p_{ij_{1}}\leq p_{ij_{2}}$, because $X_{j_{1}}\subseteq X_{j_{2}}$. The adversary stops the sequence at the first step $r$ for which $q_{rr}>{1}/{2}$. If this never happens the sequence is stopped after requesting $k$ nodes.

We use $\operatorname{\mathrm{OPT}}_{b}$ to denote the weight of a minimum Steiner tree with maximum degree $b=3$. In order to find an upper bound for $\operatorname{\mathrm{OPT}}_{b}$, consider the following tree $T$:

[TABLE]

As we can observe in Figure 2(b), $T$ meets the degree bounds and connects the first $r$ terminals to the root, so forms a valid solution. We have

[TABLE]

Back to algorithm $A$, the adversary causes one of the following two cases:

1. (Case a)

The process stops at step $r$ with $q_{rr}>\frac{1}{2}$. Then

[TABLE]

Hence, $\operatorname{E}[w(X)]\geq\Omega(n)\cdot w(\operatorname{\mathrm{OPT}})$. 2. (Case b)

The process continues until step $k$, i.e., for every $i\leq k$, $q_{ii}\leq\frac{1}{2}$.

[TABLE]

Hence, $\operatorname{E}[\operatorname{deg}(\mathit{root})]\geq\Omega(n)\cdot b$.

This shows that for any online algorithm $A$ there is a demand sequence on which $A$ either generates a large weight or violates the degree bound by a large factor. $\Box$

B.2 Online Degree-Bounded Group Steiner Tree

In this section, presenting an adversary scenario, we show there is no deterministic algorithm for online degree-bounded group Steiner tree with competitive ratio $o(n)$ even if $G$ is a star graph.

**Proof of Theorem 1.4: **For any integer $n>1$, we provide a graph instance $G$ of size $n$ and an online scenario, in which no deterministic algorithm can obtain a competitive ratio better than $n-1$. Let $G$ be a star with $n-1$ leaves $v_{2}$ to $v_{n}$, and $v_{1}$ be the internal node. For an algorithm $\mathcal{A}$, we describe the adversary scenario as follows.

Let $v_{1}$ be the root. Let the first demand group be the set of all leaves. Whenever $\mathcal{A}$ connects a node $v_{i}$ to $v_{1}$ in $H$, adversary removes the selected nodes $v_{i}$ from the next demand groups, until all leaves are connected to $v_{1}$ in $H$. In particular let $C$ denote the set of all nodes connected to $v_{1}$ in $H$ so far. Let the demand group be the set of all leaves in $\{v_{2},v_{3},\ldots,v_{n}\}\setminus C$. We do this until $\{v_{2},v_{3},\ldots,v_{n}\}\setminus C=\emptyset$, which means every leaf is connected to $v_{1}$ in $H$. $G$ is a star, thus a leaf $v_{i}$ is connected to $v_{1}$ in $H$ iff $H$ includes the edge between $v_{i}$ and $v_{1}$. Thus after all demands $\mathcal{A}$ has added all edges in $G$ to $H$. Hence $deg(v_{1})=n-1$.

Now consider the optimal offline algorithm. Let $g_{i}$ denote the $i$-th demand group. Assume we have $k$ demand groups. By construction of the demand groups $g_{k}\subset g_{k-1}\subset\ldots\subset g_{1}$. Thus there is a single node that exists in all group demands. Hence the optimal offline algorithms only needs to connect that node to the root in $H$. Therefore, the degree of each node is at most $1$ and the competitive ratio is $n-1$. $\Box$