On the Tree Augmentation Problem

In the Tree Augmentation problem we are given a tree T=(V,F) and a set E 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+epsilon approximate solution in time n^{{(M/epsilon^2)}^{O(1)}}. Using a simpler LP, we achieve ratio 12/7+epsilon in time ^{O(M/epsilon^2)}. This also gives ratio better than 2 for logarithmic costs, and not only for constant costs. In addition, we will show that (for arbitrary costs) the problem admits ratio 3/2 for trees of diameter <= 7. 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 suggest another natural LP-relaxation that is much simpler than the ones in previous work, and prove that it has integrality gap at most 7/4

A 4/3 Approximation for 2-Vertex-Connectivity

The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA\u2717] (improving on earlier results by Khuller and Vishkin [STOC\u2792] and Garg, Vempala and Singla [SODA\u2793]). Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work

Finding Almost Tight Witness Trees

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs

Matching Augmentation via Simultaneous Contractions

We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a 2-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of 13/8 = 1.625 improving upon an earlier 5/3-approximation. The improvement builds on a new ?-approximation preserving reduction for any ? ? 3/2 from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted

Fast Distributed Approximation for TAP and 2-Edge-Connectivity

The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph $G$ and a spanning tree $T$ for it, and the goal is to augment $T$ with a minimum set of edges $Aug$ from $G$, such that $T \cup Aug$ is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018]. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed $2$-approximation for weighted TAP which completes in $O(h)$ rounds, where $h$ is the height of $T$. When $h$ is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in $O(D+\sqrt{n}\log^*{n})$ rounds, where $n$ is the number of vertices and $D$ is the diameter of $G$. Immediate consequences of our results are an $O(D)$-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an $O(h_{MST}+\sqrt{n}\log^{*}{n})$-round 3-approximation algorithm for the weighted case, where $h_{MST}$ is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP

How to Secure Matchings Against Edge Failures

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems

An Improved Approximation Algorithm for the Matching Augmentation Problem

We present a 5/3-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A 7/4-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a 7/4-approximation algorithm," Math. Program., 182(1):315-354, 2020. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems

A PTAS for Triangle-Free 2-Matching

In the Triangle-Free (Simple) 2-Matching problem we are given an undirected graph $G=(V,E)$. Our goal is to compute a maximum-cardinality $M\subseteq E$ satisfying the following properties: (1) at most two edges of $M$ are incident on each node (i.e., $M$ is a 2-matching) and (2) $M$ does not induce any triangle. In his Ph.D. thesis from 1984, Harvitgsen presents a complex polynomial-time algorithm for this problem, with a very complex analysis. This result was never published in a journal nor reproved in a different way, to the best of our knowledge. In this paper we have a fresh look at this problem and present a simple PTAS for it based on local search. Our PTAS exploits the fact that, as long as the current solution is far enough from the optimum, there exists a short augmenting trail (similar to the maximum matching case).Comment: 27 pages, 18 figure

Complexity of bulk-robust combinatorial optimization problems

This thesis studies three robust combinatorial optimization problems on graphs. Most robust combinatorial optimization problems assume that the cost of the resources, e.g. edges in a graph, are uncertain. In this thesis, however, we study the so called bulk-robust approach which models the uncertainty in the underlying combinatorial structure. To be more precise, in the bulk-robust model we are given an explicit list of scenarios comprising a set of resources each, which fail if the corresponding scenario materializes. Additionally, we are given a property that we have to obtain such as 'containing a perfect matching', 's-t-connectivity', or 'being connected', which may arise from a fundamental combinatorial optimization problem. The goal of the bulk-robust optimization problem is to find a minimum weight set of resources such that the desired property is satisfied no matter which scenario materializes, i.e. no matter which set of resources from the list is removed. We study the bulk-robust bipartite matching problem, the bulk-robust k-edge disjoint s-t-paths problem, and the bulk-robust minimum spanning tree problem. We investigate the complexity of the three problems and show that most of them are hard to approximate even if the list of scenarios consists of singletons only. We complement these inapproximability results with polynomial-time approximation algorithms that essentially match the hardness results. Furthermore, we present FPT and XP algorithms and consider special graph classes that allow for a polynomial-time exact algorithm.In dieser Arbeit werden drei robuste kombinatorische Optimierungsprobleme auf Graphen behandelt. Die meisten robusten kombinatorischen Optimierungsprobleme nehmen an, dass die Kosten der Ressourcen, z.B. Kanten in einem Graph, unsicher sind. Diese Arbeit jedoch behandelt den Ansatz der sogenannten bulk-Robustheit, die Unsicherheiten in der kombinatorischen Struktur widerspiegelt. Genauer gesagt ist im bulk-robusten Modell eine explizite Liste von Szenarien gegegeben und jedes dieser Szenarien enthält eine Menge von Ressourcen, die ausfällt, wenn das entsprechende Szenario eintritt. Zusätzlich ist noch eine Eigenschaft gegeben, die wir erzeugen müssen. Diese Eigenschaft könnte beispielsweise sein, dass der resultierende Graph ein perfektes Matching enthält, zwei bestimmte Knoten miteinander verbindet oder zusammenhängend ist. Das Ziel des bulk-robusten Optimierungsproblems ist es, eine kosten-minimale Menge von Ressourcen zu finden, sodass die gewünschte Eigenschaft erfüllt ist, egal welches Szenario eintrifft, d.h. unabhängig davon, welche Menge von Resourcen der Liste von der Lösung entfernt wird. Diese Arbeit behandelt das bulk-robuste bipartite Matchingproblem, das bulk-robuste k-Kanten disjunkte s-t-Wege Problem und das bulk-robuste minimale Spannbaum Problem. Wir beschäftigen uns mit der Komplexität der drei Probleme und zeigen, dass die meisten sogar dann schwer zu approximieren sind, wenn die Liste der Szenarien nur aus einelementigen Mengen besteht. Wir komplementieren diese nicht-Approximierbarkeitsresultate mit effizienten Approximationsalgorithmen, deren Güte im Wesentlichen den hardness-Resultaten entspricht. Darüber hinaus präsentieren wir FPT und XP Algorithmen und entwickeln effiziente exakte Algorithmen für spezielle Graphklassen