Improved Approximation Algorithms for Steiner Connectivity Augmentation Problems

The Weighted Connectivity Augmentation Problem is the problem of augmenting the edge-connectivity of a given graph by adding links of minimum total cost. This work focuses on connectivity augmentation problems in the Steiner setting, where we are not interested in the connectivity between all nodes of the graph, but only the connectivity between a specified subset of terminals. We consider two related settings. In the Steiner Augmentation of a Graph problem ($k$-SAG), we are given a $k$-edge-connected subgraph $H$ of a graph $G$. The goal is to augment $H$ by including links and nodes from $G$ of minimum cost so that the edge-connectivity between nodes of $H$ increases by 1. In the Steiner Connectivity Augmentation Problem ($k$-SCAP), we are given a Steiner $k$-edge-connected graph connecting terminals $R$, and we seek to add links of minimum cost to create a Steiner $(k+1)$-edge-connected graph for $R$. Note that $k$-SAG is a special case of $k$-SCAP. All of the above problems can be approximated to within a factor of 2 using e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this work, we leverage the framework of Traub and Zenklusen to give a $(1 + \ln{2} +\varepsilon)$-approximation for the Steiner Ring Augmentation Problem (SRAP): given a cycle $H = (V(H),E)$ embedded in a larger graph $G = (V, E \cup L)$ and a subset of terminals $R \subseteq V(H)$, choose a subset of links $S \subseteq L$ of minimum cost so that $(V, E \cup S)$ has 3 pairwise edge-disjoint paths between every pair of terminals. We show this yields a polynomial time algorithm with approximation ratio $(1 + \ln{2} + \varepsilon)$ for $2$-SCAP. We obtain an improved approximation guarantee of $(1.5+\varepsilon)$ for SRAP in the case that $R = V(H)$, which yields a $(1.5+\varepsilon)$-approximation for $k$-SAG for any $k$

Finding a Highly Connected Steiner Subgraph and its Applications

Given a (connected) undirected graph G, a set X ? V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ? X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness). Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph

Pruning 2-Connected Graphs

Given an edge-weighted undirected graph $G$ with a specified set of terminals, let the emph{density} of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If $G$ is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of $G$? We answer this question in the affirmative by giving an algorithm to emph{prune} $G$ and find such subgraphs of any desired size, at the cost of only a logarithmic increase in density (plus a small additive factor). We apply the pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the kv problem, we are given an undirected graph $G$ with edge costs and an integer $k$; the goal is to find a minimum-cost 2-vertex-connected subgraph of $G$ containing at least $k$ vertices. In the bv problem, we are given the graph $G$ with edge costs, and a budget $B$; the goal is to find a 2-vertex-connected subgraph $H$ of $G$ with total edge cost at most $B$ that maximizes the number of vertices in $H$. We describe an $O(log n log k)$ approximation for the kv problem, and a bicriteria approximation for the bv problem that gives an $O(frac{1}{eps}log^2 n)$ approximation, while violating the budget by a factor of at most $3+eps$

Approximation Algorithms for Flexible Graph Connectivity

We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), and IPCO 2020: pp. 13-26). Let $k\geq 1$, $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges $S$ and a set of unsafe edges $U$, and nonnegative costs $c: E\to\Re$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any subset $F'$ of unsafe edges with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible solution $F$ that minimizes $c(F) = \sum_{e \in F} c_e$. We present a simple $2$-approximation algorithm for the $(1,1)$-FGC problem via a reduction to the minimum-cost rooted $2$-arborescence problem. This improves on the $2.527$-approximation algorithm of Adjiashvili et al. Our $2$-approximation algorithm for the $(1,1)$-FGC problem extends to a $(k+1)$-approximation algorithm for the $(1,k)$-FGC problem. We present a $4$-approximation algorithm for the $(p,1)$-FGC problem, and an $O(q\log|V|)$-approximation algorithm for the $(p,q)$-FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted $(1,1)$-FGC problem by presenting a $16/11$-approximation algorithm. The $(p,q)$-FGC problem is related to the well-known Capacitated $k$-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of Capacitated Network Design. We give a $\min(k,2 u_{max})$-approximation algorithm for the Cap-k-ECSS problem, where $u_{max}$ denotes the maximum capacity of an edge.Comment: 23 pages, 1 figure, preliminary version in the Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), December 15-17, (LIPIcs, Volume 213, Article No. 9, pp. 9:1-9:14), see https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9. Related manuscript: arXiv:2102.0330

A Variant of the Maximum Weight Independent Set Problem

We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph $G=(V,E)$, a weight function $w: V \rightarrow \mathbb{R^+}$, a budget function $b: V \rightarrow \mathbb{Z^+}$, and a positive integer $B$. The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A $k$-budgeted independent set in $G$ is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most $k$. The goal is to find a $B$-budgeted independent set in $G$ such that its weight is maximum among all the $B$-budgeted independent sets in $G$. We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for $d+1$-claw free graphs whose approximation ratio ($d$) is competitive with the approximation ratio ($\frac{d}{2}$) of MWIS (unweighted). Furthermore, we extend Baker's technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201