5/4-Approximation of Minimum 2-Edge-Connected Spanning Subgraph

We provide a $5/4$-approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. This improves upon the previous best ratio of $4/3$. The algorithm is based on applying local improvement steps on a starting solution provided by a standard ear decomposition together with the idea of running several iterations on residual graphs by excluding certain edges that do not belong to an optimum solution. The latter idea is a novel one, which allows us to bypass $3$-ears with no loss in approximation ratio, the bottleneck for obtaining a performance guarantee below $3/2$. Our algorithm also implies a simpler $7/4$-approximation algorithm for the matching augmentation problem, which was recently treated.Comment: The modification of 5-ears, which was both erroneous and unnecessary, is omitte

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

Approximability of Dense and Sparse Instances of Minimum 2-Connectivity, TSP and Path Problems

We study the approximability of dense and sparse instances of the following problems: the minimum 2-edge-connected (2-EC) and 2-vertex-connected (2-VC) spanning subgraph, metric TSP with distances 1 and 2 (TSP(1,2)), maximum path packing, and the longest path (cycle) problems. The approximability of dense instances of these problems was left open in Arora et al. [3]. We characterize the approximability of all these problems by proving tight upper (approximation algorithms) and lower bounds (inapproximability). We prove that 2-EC, 2-VC and TSP(1,2) are Max SNP-hard even on 3-regular graphs, and provide explicit hardness constants, under P 6= NP. We also improve the approximation ratio for 2-EC and 2-VC on graphs with maximum degree 3. These are the rst explicit hardness results on sparse and dense graphs for these problems. We apply our results to prove bounds on the integrality gaps of LP relaxations for dense and sparse 2-EC and TSP(1,2) problems, related to the famous metric TSP conjecture, due to Goemans [18]

Approximating Minimum-Size 2-Edge-Connected and 2-Vertex-Connected Spanning Subgraphs

We study the unweighted 2-edge-connected and 2-vertex-connected spanning subgraph problems. A graph is 2-edge-connected if it is connected on removal of an edge, and it is 2-vertex-connected if it is connected on removal of a vertex. The problem of finding a minimum-size 2-edge-connected (or 2-vertex-connected) spanning subgraph of a given graph is NP-hard. We present a 4/3-approximation algorithm for unweighted 2ECSS on 3-vertex-connected input graphs, which matches the best known approximation ratio due to Sebő and Vygen for the general unweighted 2ECSS problem, but our analysis is with respect to the D2 lower bound. We also give a 17/12-approximation algorithm for unweighted 2VCSS on graphs of minimum degree at least 3, which is lower than the best known ratios of 3/2 by Garg, Santosh and Singla and 10/7 by Heeger and Vygen for the general unweighted 2VCSS problem. These algorithms are accompanied by new theorems about the known lower bounds