arXiv subjects

Zeev Nutov

Publications and source records attributed to Zeev Nutov.

At least 37 records · Page 2Linked to original sources

arXiv2021

Practical Budgeted Submodular Maximization

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $α=1-1/e\approx 0.632$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko impractical as it leads to a time complexity of $O(n^5)$ (which can be slightly improved using the thresholding technique of Badanidiyuru & Vondrak (2014), but only to roughly $O(n^4)$). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses, we get the same optimal approximation ratio of $α$ with an improved time complexity of roughly $O(n^3)$. Furthermore, by making only a single guess, we get an almost as good approximation ratio of $0.6174>0.9767α$ in roughly $O(n^2)$ time. Prior to our work, the only algorithms that were known to obtain an approximation ratio close to $α$ for BSM were the algorithm of Sviridenko and an algorithm of Ene & Nguyen (2019) that achieves $(α-ε)$-approximation. However, the algorithm of Ene & Nguyen requires ${(1/ε)}^{O(1/ε^4)}n\log^2 n$ time, and hence, is of theoretical interest only as ${(1/ε)}^{O(1/ε^4)}$ is huge even for moderate values of $ε$. In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (2020) studied a simple greedy algorithm that already has a long research history, and proved that its approximation ratio is at least 0.405. We improve over this result, and show that the approximation ratio of this algorithm is within the range [0.427, 0.462].

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arXiv2020

Approximation algorithms for connectivity augmentation problems

In Connectivity Augmentation problems we are given a graph $H=(V,E_H)$ and an edge set $E$ on $V$, and seek a min-size edge set $J \subseteq E$ such that $H \cup J$ has larger edge/node connectivity than $H$. In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by $1$. In the Block-Tree Augmentation problem $H$ is connected and $H \cup S$ should be $2$-connected. In Leaf-to-Leaf Connectivity Augmentation problems every edge in $E$ connects minimal deficient sets. For this version we give a simple combinatorial approximation algorithm with ratio $5/3$, improving the previous $1.91$ approximation that applies for the general case. We also show by a simple proof that if the Steiner Tree problem admits approximation ratio $α$ then the general version admits approximation ratio $1+\ln(4-x)+ε$, where $x$ is the solution to the equation $1+\ln(4-x)=α+(α-1)x$. For the currently best value of $α=\ln 4+ε$ this gives ratio $1.942$. This is slightly worse than the best ratio $1.91$, but has the advantage of using Steiner Tree approximation as a "black box", giving ratio $< 1.9$ if ratio $α\leq 1.35$ can be achieved. In the Element Connectivity Augmentation problem we are given a graph $G=(V,E)$, $S \subseteq V$, and connectivity requirements $\{r(u,v):u,v \in S\}$. The goal is to find a min-size set $J$ of new edges on $S$ such that for all $u,v \in S$ the graph $G \cup J$ contains $r(u,v)$ $uv$-paths such that no two of them have an edge or a node in $V \setminus S$ in common. The problem is NP-hard even when $\max_{u,v \in S} r(u,v)=2$. We obtain approximation ratio $3/2$, improving the previous ratio $7/4$.

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arXiv2019

An $\tilde{O}(\log^2 n)$-approximation algorithm for $2$-edge-connected dominating set

In the Connected Dominating Set problem we are given a graph $G=(V,E)$ and seek a minimum size dominating set $S \subseteq V$ such that the subgraph $G[S]$ of $G$ induced by $S$ is connected. In the $2$-Edge-Connected Dominating Set problem $G[S]$ should be $2$-edge-connected. We give the first non-trivial approximation algorithm for this problem, with expected approximation ratio $\tilde{O}(\log^2n)$.

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arXiv2019

Bounded Degree Group Steiner Tree Problems

We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that $T$ satisfies a certain property and has minimal maximum degree. - In the Min-Degree Group Steiner Tree problem we are given a collection ${\cal S}$ of groups (subsets of $V$) and $T$ should contain a node from every group. - In the Min-Degree Steiner $k$-Tree problem we are given a set $R$ of terminals and an integer $k$, and $T$ should contain at least $k$ terminals. We show that if the former problem admits approximation ratio $ρ$ then the later problem admits approximation ratio $ρ\cdot O(\log k)$. For bounded treewidth graphs, we obtain approximation ratio $O(\log^3 n)$ for Min-Degree Group Steiner Tree. In the more general Bounded Degree Group Steiner Tree problem we are also given edge costs and degree bounds $\{b(v):v \in V\}$, and $T$ should obey the degree constraints $deg_T(v) \leq b(v)$ for all $v \in V$. We give a bicriteria $(O(\log N \log |{\cal S}|),O(\log^2 n))$-approximation algorithm for this problem on tree inputs, where $N$ is the size of the largest group, generalizing the approximation of Garg, Konjevod, and Ravi for the case without degree bounds.

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arXiv2019

Approximating $k$-connected $m$-dominating sets

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln k)$ to $\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k)$ -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln^2 k)/ε$ when $m \geq (1+ε)k$; furthermore, we obtain ratio $\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k)$ for uniform weights. These results are obtained by showing the same ratios for the Subset $k$-Connectivity problem when the set $T$ of terminals is an $m$-dominating set with $m \geq k$.

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arXiv2019

A $(4+ε)$-approximation for $k$-connected subgraphs

We obtain approximation ratio $2(2+\frac{1}{\ell})$ for the (undirected) $k$-Connected Subgraph problem, where $\ell \approx \frac{1}{2} (\log_k n-1)$ is the largest integer such that $2^{\ell-1} k^{2\ell+1} \leq n$. For large values of $n$ this improves the $6$-approximation of Cheriyan and Végh when $n =Ω(k^3)$, which is the case $\ell=1$. For $k$ bounded by a constant we obtain ratio $4+ε$. For large values of $n$ our ratio matches the best known ratio $4$ for the augmentation version of the problem, as well as the best known ratios for $k=6,7$. Similar results are shown for the problem of covering an arbitrary crossing supermodular biset function.

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arXiv2018

Approximating activation edge-cover and facility location problems

What approximation ratio can we achieve for the Facility Location problem if whenever a client $u$ connects to a facility $v$,the opening cost of $v$ is at most $θ$ times the service cost of $u$? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph $G=(V,E)$, a set $R \subseteq V$ of terminals, and thresholds $\{t^e_u,t^e_v\}$ for each $uv$-edge $e \in E$. The goal is to find an assignment ${\bf a}=\{a_v:v \in V\}$ to the nodes minimizing $\sum_{v \in V} a_v$, such that the edge set $E_{\bf a}=\{e=uv: a_u \geq t^e_u, a_v \geq t^e_v\}$ activated by ${\bf a}$ covers $R$. We obtain ratio $1+ω(θ) \approx \ln θ-\ln \ln θ$ for the problem, where $ω(θ)$ is the root of the equation $x+1=\ln(θ/x)$ and $θ$ is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get that the above variant of Facility Location admits ratio $1+ω(θ)$; if for each facility all service costs are identical then we show a better ratio $\displaystyle 1+\max_{k \geq 1} \frac{H_k-1}{1+k/θ}$, where $H_k=\sum_{i=1}^k 1/i$. For the Min-Power Edge-Cover problem we improve the ratio $1.406$ of Calinescu et. al. (achieved by iterative randomized rounding) to $1+ω(1)<1.2785$. For unit thresholds we improve the ratio $73/60 \approx 1.217$ to $\frac{1555}{1347} \approx 1.155$.

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arXiv2018

On the Tree Augmentation Problem

In the Tree Augmentation problem we are given a tree $T=(V,F)$ and a set $E \subseteq V \times V$ of edges with positive integer costs $\{c_e:e \in E\}$. The goal is to augment $T$ by a minimum cost edge set $J \subseteq E$ such that $T \cup J$ is $2$-edge-connected. We obtain the following results. Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the $2$-approximation barrier for instances when the maximum cost $M$ of an edge in $E$ is bounded by a constant; his algorithm computes a $1.96418+ε$ approximate solution in time $n^{{(M/ε^2)}^{O(1)}}$. Using a simpler LP, we achieve ratio $\frac{12}{7}+ε$ in time $2^{O(M/ε^2)} poly(n)$.This gives ratio better than $2$ for logarithmic costs, and not only for constant costs. One of the oldest open questions for the problem is whether for unit costs (when $M=1$) the standard LP-relaxation, so called Cut-LP, has integrality gap less than $2$. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most $28/15=2-2/15$. In addition, we will prove that another natural LP-relaxation, that is much simpler than the ones in previous work, has integrality gap at most $7/4$.

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arXiv2017

Improved approximation algorithms for $k$-connected $m$-dominating set problems

A graph is $k$-connected if it has $k$ internally-disjoint paths between every pair of nodes. A subset $S$ of nodes in a graph $G$ is a $k$-connected set if the subgraph $G[S]$ induced by $S$ is $k$-connected; $S$ is an $m$-dominating set if every $v \in V \setminus S$ has at least $m$ neighbors in $S$. If $S$ is both $k$-connected and $m$-dominating then $S$ is a $k$-connected $m$-dominating set, or $(k,m)$-cds for short. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. We consider the case $m \geq k$ and obtain the following approximation ratios. For unit disc-graphs we obtain ratio $O(k\ln k)$, improving the previous ratio $O(k^2 \ln k)$. For general graphs we obtain the first non-trivial approximation ratio $O(k^2 \ln n)$.

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arXiv2015

Iterative rounding approximation algorithms for degree-bounded node-connectivity network design

We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$ where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity requirement $k$, our algorithm computes a solution that is a $4$-approximation on the cost, and the degree of each node $v$ in the solution is at most $4b(v)+O(k)$. These ratios improve the previous $O(\log k)$-approximation on the cost and $O(2^k b(v))$ approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k$-out-connectivity, directed and undirected $k$-connectivity, and undirected rooted $k$-connectivity and subset $k$-connectivity.

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arXiv2015

A simplified 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2

The Tree Augmentation Problem (TAP) is: given a connected graph $G=(V,{\cal E})$ and an edge set $E$ on $V$ find a minimum size subset of edges $F \subseteq E$ such that $(V,{\cal E} \cup F)$ is $2$-edge-connected. In the conference version \cite{EFKN-APPROX} was sketched a $1.5$-approximation algorithm for the problem. Since a full proof was very complex and long, the journal version was cut into two parts. In the first part \cite{EFKN-TALG} was only proved ratio $1.8$. An attempt to simplify the second part produced an error in \cite{EKN-IPL}. Here we give a correct, different, and self contained proof of the ratio $1.5$, that is also substantially simpler and shorter than the previous proofs.

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arXiv2015

A $1.75$ LP approximation for the Tree Augmentation Problem

In the Tree Augmentation Problem (TAP) the goal is to augment a tree $T$ by a minimum size edge set $F$ from a given edge set $E$ such that $T \cup F$ is $2$-edge-connected. The best approximation ratio known for TAP is $1.5$. In the more general Weighted TAP problem, $F$ should be of minimum weight. Weighted TAP admits several $2$-approximation algorithms w.r.t. to the standard cut LP-relaxation, but for all of them the performance ratio of $2$ is tight even for TAP. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than $2$ is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than $2$, which is a long time open problem even for TAP. In this paper we introduce such an LP-relaxation and give an algorithm that computes a feasible solution for TAP of size at most $1.75$ times the optimal LP value. This gives some hope to break the ratio $2$ for the weighted case. Our algorithm computes some initial edge set by solving a partial system of constraints that form the integral edge-cover polytope, and then applies local search on $3$-leaf subtrees to exchange some of the edges and to add additional edges. Thus we do not need to solve the LP, and the algorithm runs roughly in time required to find a minimum weight edge-cover in a general graph.

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arXiv2014

Approximating Source Location and Star Survivable Network Problems

In Source Location (SL) problems the goal is to select a mini-mum cost source set $S \subseteq V$ such that the connectivity (or flow) $ψ(S,v)$ from $S$ to any node $v$ is at least the demand $d_v$ of $v$. In many SL problems $ψ(S,v)=d_v$ if $v \in S$, namely, the demand of nodes selected to $S$ is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node $v$ gets a "bonus" $p_v \leq d_v$ if it is selected to $S$. Fukunaga showed that for undirected graphs one can achieve ratio $O(k \ln k)$ for his variant, where $k=\max_{v \in V}d_v$ is the maximum demand. We improve this by achieving ratio $\min\{p^*\lnk,k\}\cdot O(\ln (k/q^*))$ for a more general version with node capacities, where $p^*=\max_{v \in V} p_v$ is the maximum bonus and $q^*=\min_{v \in V} q_v$ is the minimum capacity. In particular, for the most natural case $p^*=1$ considered by Fukunaga, we improve the ratio from $O(k \ln k)$ to $O(\ln^2k)$. We also get ratio $O(k)$ for the edge-connectivity version, for which no ratio that depends on $k$ only was known before. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio $O(\min\{\ln n,\ln^2 k\})$ for this variant, improving over the best ratio known for the general case $O(k^3 \ln n)$ of Chuzhoy and Khanna.

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arXiv2013

Approximating {0,1,2}-Survivable Networks with Minimum Number of Steiner Points

We consider low connectivity variants of the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set $R$ of terminals in a metric space (M,d), a subset $B \subseteq R$ of "unstable" terminals, and connectivity requirements {r_{uv}: u,v \in R}, find a minimum size set $S \subseteq M$ of additional points such that the unit-disc graph of $R \cup S$ contains $r_{uv}$ pairwise internally edge-disjoint and $(B \cup S)$-disjoint $uv$-paths for all $u,v \in R$. The case when $r_{uv}=1$ for all $u,v \in R$ is the {\sf Steiner Tree with Minimum Number of Steiner Points} (ST-MSP) problem, and the case $r_{uv} \in \{0,1\}$ is the {\sf Steiner Forest with Minimum Number of Steiner Points} (SF-MSP) problem. Let $Δ$ be the maximum number of points in a unit ball such that the distance between any two of them is larger than 1. It is known that $Δ=5$ in $\mathbb{R}^2$ The previous known approximation ratio for {\sf ST-MSP} was $\lfloor (Δ+1)/2 \rfloor+1+ε$ in an arbitrary normed space \cite{NY}, and $2.5+ε$ in the Euclidean space $\mathbb{R}^2$ \cite{cheng2008relay}. Our approximation ratio for ST-MSP is $1+\ln(Δ-1)+ε$ in an arbitrary normed space, which in $\mathbb{R}^2$ reduces to $1+\ln 4+ε< 2.3863 +ε$. For SN-MSP with $r_{uv} \in \{0,1,2\}$, we give a simple $Δ$-approximation algorithm. In particular, for SF-MSP, this improves the previous ratio $2Δ$.

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arXiv2012

Steiner Forest Orientation Problems

We consider connectivity problems with orientation constraints. Given a directed graph $D$ and a collection of ordered node pairs $P$ let $P[D]=\{(u,v) \in P: D {contains a} uv{-path}}$. In the {\sf Steiner Forest Orientation} problem we are given an undirected graph $G=(V,E)$ with edge-costs and a set $P \subseteq V \times V$ of ordered node pairs. The goal is to find a minimum-cost subgraph $H$ of $G$ and an orientation $D$ of $H$ such that $P[D]=P$. We give a 4-approximation algorithm for this problem. In the {\sf Maximum Pairs Orientation} problem we are given a graph $G$ and a multi-collection of ordered node pairs $P$ on $V$. The goal is to find an orientation $D$ of $G$ such that $|P[D]|$ is maximum. Generalizing the result of Arkin and Hassin [DAM'02] for $|P|=2$, we will show that for a mixed graph $G$ (that may have both directed and undirected edges), one can decide in $n^{O(|P|)}$ time whether $G$ has an orientation $D$ with $P[D]=P$ (for undirected graphs this problem admits a polynomial time algorithm for any $P$, but it is NP-complete on mixed graphs). For undirected graphs, we will show that one can decide whether $G$ admits an orientation $D$ with $|P[D]| \geq k$ in $O(n+m)+2^{O(k\cdot \log \log k)}$ time; hence this decision problem is fixed-parameter tractable, which answers an open question from Dorn et al. [AMB'11]. We also show that {\sf Maximum Pairs Orientation} admits ratio $O(\log |P|/\log\log |P|)$, which is better than the ratio $O(\log n/\log\log n)$ of Gamzu et al. [WABI'10] when $|P|<n$. Finally, we show that the following node-connectivity problem can be solved in polynomial time: given a graph $G=(V,E)$ with edge-costs, $s,t \in V$, and an integer $\ell$, find a min-cost subgraph $H$ of $G$ with an orientation $D$ such that $D$ contains $\ell$ internally-disjoint $st$-paths, and $\ell$ internally-disjoint $ts$-paths.

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arXiv2012

Approximating minimum-cost edge-covers of crossing biset-families

An ordered pair $\hat{S}=(S,S^+)$ of subsets of $V$ is called a {\em biset} if $S \subseteq S^+$; $(V-S^+,V-S)$ is the co-biset of $\hat{S}$. Two bisets $\hat{X},\hat{Y}$ intersect if $X \cap Y \neq \emptyset$ and cross if both $X \cap Y \neq \emptyset$ and $X^+ \cup Y^+ \neq V$. The intersection and the union of two bisets $\hat{X},\hat{Y}$ is defined by $\hat{X} \cap \hat{Y} = (X \cap Y, X^+ \cap Y^+)$ and $\hat{X} \cup \hat{Y} = (X \cup Y,X^+ \cup Y^+)$. A biset-family ${\cal F}$ is crossing (intersecting) if $\hat{X} \cap \hat{Y}, \hat{X} \cup \hat{Y} \in {\cal F}$ for any $\hat{X},\hat{Y} \in {\cal F}$ that cross (intersect). A directed edge covers a biset $\hat{S}$ if it goes from $S$ to $V-S^+$. We consider the problem of covering a crossing biset-family ${\cal F}$ by a minimum-cost set of directed edges. While for intersecting ${\cal F}$, a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing ${\cal F}$ is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently. Let us say that a biset-family ${\cal F}$ is $k$-regular if $\hat{X} \cap \hat{Y}, \hat{X} \cup \hat{Y} \in {\cal F}$ for any $\hat{X},\hat{Y} \in {\cal F}$ with $|V-(X \cup Y)| \geq k+1$ that intersect. In this paper we obtain an $O(\log |V|)$-approximation algorithm for arbitrary crossing ${\cal F}$; if in addition both ${\cal F}$ and the family of co-bisets of ${\cal F}$ are $k$-regular, our ratios are: $O(\log \frac{|V|}{|V|-k})$ if $|S^+ \setminus S|=k$ for all $\hat{S} \in {\cal F}$, and $O(\frac{|V|}{|V|-k} \log \frac{|V|}{|V|-k})$ if $|S^+ \setminus S| \leq k$ for all $\hat{S} \in {\cal F}$. Using these generic algorithms, we derive approximation algorithms for some network design problems.

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arXiv2012

Small $\ell$-edge-covers in $k$-connected graphs

Let $G=(V,E)$ be a $k$-edge-connected graph with edge costs $\{c(e):e \in E\}$ and let $1 \leq \ell \leq k-1$. We show by a simple and short proof, that $G$ contains an $\ell$-edge cover $I$ such that: $c(I) \leq \frac{\ell}{k}c(E)$ if $G$ is bipartite, or if $\ell |V|$ is even, or if $|E| \geq \frac{k|V|}{2} +\frac{k}{2\ell}$; otherwise, $c(I) \leq (\frac{\ell}{k}+\frac{1}{k|V|})c(E)$. The particular case $\ell=k-1$ and unit costs already includes a result of Cheriyan and Thurimella, that $G$ contains a $(k-1)$-edge-cover of size $|E|-\lfloor |V|/2 \rfloor$. Using our result, we slightly improve the approximation ratios for the {\sf $k$-Connected Subgraph} problem (the node-connectivity version) with uniform and $β$-metric costs. We then consider the dual problem of finding a spanning subgraph of maximum connectivity $k^*$ with a prescribed number of edges. We give an algorithm that computes a $(k^*-1)$-connected subgraph, which is tight, since the problem is NP-hard.

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arXiv2011

Combinatorial Algorithms for Capacitated Network Design

We focus on designing combinatorial algorithms for the Capacitated Network Design problem (Cap-SNDP). The Cap-SNDP is the problem of satisfying connectivity requirements when edges have costs and hard capacities. We begin by showing that the Group Steiner tree problem (GST) is a special case of Cap-SNDP even when there is connectivity requirement between only one source-sink pair. This implies the first poly-logarithmic lower bound for the Cap-SNDP. We next provide combinatorial algorithms for several special cases of this problem. The Cap-SNDP is equivalent to its special case when every edge has either zero cost or infinite capacity. We consider a special case, called Connected Cap-SNDP, where all infinite-capacity edges in the solution are required to form a connected component containing the sinks. This problem is motivated by its similarity to the Connected Facility Location problem [G+01,SW04]. We solve this problem by reducing it to Submodular tree cover problem, which is a common generalization of Connected Cap-SNDP and Group Steiner tree problem. We generalize the recursive greedy algorithm [CEK] achieving a poly-logarithmic approximation algorithm for Submodular tree cover problem. This result is interesting in its own right and gives the first poly-logarithmic approximation algorithms for Connected hard capacities set multi-cover and Connected source location. We then study another special case of Cap-SNDP called Unbalanced point-to-point connection problem. Besides its practical applications to shift design problems [EKS], it generalizes many problems such as k-MST, Steiner Forest and Point-to-Point Connection. We give a combinatorial logarithmic approximation algorithm for this problem by reducing it to degree-bounded SNDP.

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