On rooted kk-connectivity problems in quasi-bipartite digraphs

We consider the directed Rooted Subset $k$-Edge-Connectivity problem: given a set $T \subseteq V$ of terminals in a digraph $G=(V+r,E)$ with edge costs and an integer $k$, find a min-cost subgraph of $G$ that contains $k$ edge disjoint $rt$-paths for all $t \in T$. The case when every edge of positive cost has head in $T$ admits a polynomial time algorithm due to Frank, and the case when all positive cost edges are incident to $r$ is equivalent to the $k$-Multicover problem. Recently, [Chan et al. APPROX20] obtained ratio $O(\ln k \ln |T|)$ for quasi-bipartite instances, when every edge in $G$ has an end in $T+r$. We give a simple proof for the same ratio for a more general problem of covering an arbitrary $T$-intersecting supermodular set function by a minimum cost edge set, and for the case when only every positive cost edge has an end in $T+r$

Spider covers for prize-collecting network activation problem

In the network activation problem, each edge in a graph is associated with an activation function, that decides whether the edge is activated from node-weights assigned to its end-nodes. The feasible solutions of the problem are the node-weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this paper, we consider a prize-collecting version of the network activation problem, and present first non- trivial approximation algorithms. Our algorithms are based on a new LP relaxation of the problem. They round optimal solutions for the relaxation by repeatedly computing node-weights activating subgraphs called spiders, which are known to be useful for approximating the network activation problem

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}

Approximating survivable networks with β-metric costs

AbstractThe Survivable Network Design (SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β-metric costs when c(xz)⩽β[c(xy)+c(yz)] for all x,y,z∈V, which varies from uniform costs (β=1/2) to metric costs (β=1).For the k-Connected Subgraph (k-CS) problem our ratios are: 1+2βk(1−β)−12k−1 for undirected graphs, and 1+4β3k(1−3β2)−12k−1 for directed graphs and 12⩽β<13. For undirected graphs this improves the ratios β1−β of Böckenhauer et al. (2008) [3] and 2+βkn of Kortsarz and Nutov (2003) [11] for all k⩾4 and 12+3k−22(4k2−7k+2)⩽β⩽k2(k+1)2−2. We also show that SND admits the ratios 2β1−β for undirected graphs, and 4β31−3β2 for directed graphs with 1/2⩽β<1/3. For two important particular cases of SND, so-called Subset k-CS and Rooted SND, our ratios are 2β31−3β2 for directed graphs and β1−β for subset k-CS on undirected graphs

Approximation Algorithms for (S,T)-Connectivity Problems

We study a directed network design problem called the $k$-$(S,T)$-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$-vertex connected spanning subgraph problem is a special case of the $k$-$(S,T)$-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For $k=1$, we call the problem the $(S,T)$-connectivity problem. We study three variants of the problem: the standard $(S,T)$-connectivity problem, the relaxed $(S,T)$-connectivity problem, and the unrestricted $(S,T)$-connectivity problem. We give hardness results for these three variants. We design a $2$-approximation algorithm for the standard $(S,T)$-connectivity problem. We design tight approximation algorithms for the relaxed $(S,T)$-connectivity problem and one of its special cases. For any $k$, we give an $O(\log k\log n)$-approximation algorithm, where $n$ denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost $k$-vertex connected spanning subgraph problem which is $O(\log k\log\frac{n}{n-k})$ due to Nutov in 2009