Approximation and Hardness Results for Label Cut and Related Problems

Abstract

We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first non-trivial approximation and hardness results for the Label Cut problem. Firstly, we present an O(√ m)-approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NP-hard to approximate Label Cut within 2 log 1−1/ log log c n n for any constant c < 1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions).

Random Contractions and Sampling for Hypergraph and Hedge Connectivity

“…In contrast, previous work by Zhang et al [24] for the s-t min-cut problem in hedge graphs showed that it is NP-hard to obtain an approximation better than 2 log 1−1/ log log c n n for any constant c < 1/2, and gave an O( √ m)-approximation algorithm. Moreover, Fellows et al [5] showed that even in graphs with constant pathwidth, the problem is W [2]-hard when parameterized by m, and W [1]-hard when parameterized by k.…”

“…Previously, it was shown by Zhang et al [24] that the problem of finding a minimum s-t hedge cut -a hedge-cut of minimum value that separates a given pair of vertices s and t -is NP-hard. 2 In sharp contrast, we use randomized contraction techniques to give a quasi-polynomial time algorithm for finding a global min-hedge-cut, thereby providing strong evidence that this problem is in P. We also provide a quasi-polynomial bound on the number of min-hedge-cuts in a graph.…”

Approximate tradeoffs on weighted labeled matroids

“…It has been studied in several papers for spanning trees, paths, Hamiltonian cycles, cuts, etc. [4,23,7,5,9,36,31]. The generalization where the colors have a non-negative cost often aims to minimize the sum of the costs of colors used (also known as the chromatic price in [24]).…”

The Parameterized Complexity of Some Minimum Label Problems

“…Given a set of communication network nodes, the problem of finding a connected communication network using as few types of communication media (i.e., labels/colors) as possible is exactly the Minimum Label Spanning Tree problem, in which the property Π is the property of being a spanning tree of G (see [5,14] for more details). Among the minimum label problems that have been extensively studied, we mention the Minimum Label Spanning Tree problem [1,2,3,5,9,11,14,15,18,19,20], the Minimum Label Path problem [2,4,9,17,21] (where Π is the property of being a path between two designated vertices), the Minimum Label Cut problem [10,21] (where Π is the property of being a cut between two designated vertices), and the Minimum Label Perfect Matching problem [12] (where Π is the property of being a perfect matching).…”