Approximating subset kk-connectivity problems

A subset $T \subseteq V$ of terminals is $k$-connected to a root $s$ in a directed/undirected graph $J$ if $J$ has $k$ internally-disjoint $vs$-paths for every $v \in T$; $T$ is $k$-connected in $J$ if $T$ is $k$-connected to every $s \in T$. We consider the {\sf Subset $k$-Connectivity Augmentation} problem: given a graph $G=(V,E)$ with edge/node-costs, node subset $T \subseteq V$, and a subgraph $J=(V,E_J)$ of $G$ such that $T$ is $k$-connected in $J$, find a minimum-cost augmenting edge-set $F \subseteq E \setminus E_J$ such that $T$ is $(k+1)$-connected in $J \cup F$. The problem admits trivial ratio $O(|T|^2)$. We consider the case $|T|>k$ and prove that for directed/undirected graphs and edge/node-costs, a $\rho$-approximation for {\sf Rooted Subset $k$-Connectivity Augmentation} implies the following ratios for {\sf Subset $k$-Connectivity Augmentation}: (i) $b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k})$; (ii) $\rho \cdot O(\frac{|T|}{|T|-k} \log k)$, where b=1 for undirected graphs and b=2 for directed graphs, and $H(k)$ is the $k$th harmonic number. The best known values of $\rho$ on undirected graphs are $\min\{|T|,O(k)\}$ for edge-costs and $\min\{|T|,O(k \log |T|)\}$ for node-costs; for directed graphs $\rho=|T|$ for both versions. Our results imply that unless $k=|T|-o(|T|)$, {\sf Subset $k$-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

A 1.751.75 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.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279

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) $\psi(S,v)$ from $S$ to any node $v$ is at least the demand $d_v$ of $v$. In many SL problems $\psi(S,v)=d_v$ 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

Approximating minimum power covers of intersecting families and directed edge-connectivity problems

AbstractGiven a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Let G=(V,E) be a graph with edge costs {c(e):e∈E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r∈V, and require that G contains k pairwise edge-disjoint rv-paths for all v∈V−r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k=1, these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln|V|) approximation threshold. For k=Ω(nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O(kln|V|). Our algorithms are based on a more general O(ln|V|)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X∩Y,X∪Y∈F for any intersecting X,Y∈F, and an edge set I covers F if for every X∈F there is an edge in I entering X

Listing minimal edge-covers of intersecting families with applications to connectivity problems

AbstractLet G=(V,E) be a directed/undirected graph, let s,t∈V, and let F be an intersecting family on V (that is, X∩Y,X∪Y∈F for any intersecting X,Y∈F) so that s∈X and t∉X for every X∈F. An edge set I⊆E is an edge-cover of F if for every X∈F there is an edge in I from X to V−X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I⊆E the minimal member of the residual family FI of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k-connected and k-outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time

Data Structures for Node Connectivity Queries

Let $\kappa(s,t)$ denote the maximum number of internally disjoint paths in an undirected graph $G$. We consider designing a data structure that includes a list of cuts, and answers the following query: given $s,t \in V$, determine whether $\kappa(s,t) \leq k$, and if so, return a pointer to an $st$-cut of size $\leq k$ (or to a minimum $st$-cut) in the list. A trivial data structure that includes a list of $n(n-1)/2$ cuts and requires $\Theta(kn^2)$ space can answer each query in $O(1)$ time. We obtain the following results. In the case when $G$ is $k$-connected, we show that $n$ cuts suffice, and that these cuts can be partitioned into $(2k+1)$ laminar families. Thus using space $O(kn)$ we can answers each min-cut query in $O(1)$ time, slightly improving and substantially simplifying a recent result of Pettie and Yin. We then extend this data structure to subset $k$-connectivity. In the general case we show that $(2k+1)n$ cuts suffice to return an $st$-cut of size $\leq k$,and a list of size $k(k+2)n$ contains a minimum $st$-cut for every $s,t \in V$. Combining our subset $k$-connectivity data structure with the data structure of Hsu and Lu for checking $k$-connectivity, we give an $O(k^2 n)$ space data structure that returns an $st$-cut of size $\leq k$ in $O(\log k)$ time, while $O(k^3 n)$ space enables to return a minimum $st$-cut

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 $O(\ln^2 k)/\epsilon$ when $m \geq (1+\epsilon)k$; 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 $m \geq k$