Jérôme Monnot

Approximation algorithms and hardness results for labeled connectivity problems

Abstract Let G=(V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function ℒ:E→ℕ. In addition, each label ℓ∈ℕ has a non-negative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I⊆ℕ such that the edge set {e∈E:ℒ(e)∈I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s–t path problem (MinLP) the goal is to identify an s–t path minimizing the combined cost of its labels. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP.

Local search for the minimum label spanning tree problem with bounded color classes

In the Minimum Label Spanning Tree problem, the input consists of an edge-colored undirected graph, and the goal is to find a spanning tree with the minimum number of different colors. We investigate the special case where every color appears at most r times in the input graph. This special case is polynomially solvable for r=2, and NP- and APX-complete for any fixed r>=3. We analyze local search algorithms that are allowed to switch up to k of the colors used in a feasible solution. We show that for k=2 any local optimum yields an (r+1)/2-approximation of the global optimum, and that this bound is tight. For every k>=3, there exist instances for which some local optima are a factor of r/2 away from the global optimum.

The path partition problem and related problems in bipartite graphs

We prove that it is NP-complete to decide whether a bipartite graph of maximum degree three on nk vertices can be partitioned into n paths of length k. Finally, we propose some approximation and inapproximation results for several related problems.

Reoptimization of minimum and maximum traveling salesman's tours

In this paper, reoptimization versions of the traveling salesman problem (TSP) are addressed. Assume that an optimum solution of an instance is given and the goal is to determine if one can maintain a good solution when the instance is subject to minor modifications. We study the case where nodes are inserted in, or deleted from, the graph. When inserting a node, we show that the reoptimization problem for MinTSP is approximable within ratio 4/3 if the distance matrix is metric. We show that, dealing with metric MaxTSP, a simple heuristic is asymptotically optimum when a constant number of nodes are inserted. In the general case, we propose a 4/5-approximation algorithm for the reoptimization version of MaxTSP

Weighted Node Coloring: When Stable Sets Are Expensive

A version of weighted coloring of a graph is introduced: each node v of a graph G = (V, E) is provided with a positive integer weight w(v) and the weight of a stable set S of G is w(S) = max{w(v) : v ? V ? S}. A k-coloring S = (S1, . . . , Sk) of G is a partition of V into k stable sets S1, . . . , Sk and the weight of S is w(S1) + . . . + w(Sk). The objective then is to find a coloring S = (S1, . . . , Sk) of G such that w(S1) + . . . + w(Sk) is minimized. Weighted node coloring is NP-hard for general graphs (as generalization of the node coloring problem). We prove here that the associated decision problems are NP-complete for bipartite graphs, for line-graphs of bipartite graphs and for split graphs. We present approximation results for general graphs. For the other families of graphs dealt, properties of optimal solutions are discussed and complexity and approximability results are presented.

Time slot scheduling of compatible jobs

A version of weighted coloring of a graph is introduced which is motivated by some types of scheduling problems: each node v of a graph G corresponds to some operation to be processed (with a processing time w(v)), edges represent nonsimultaneity requirements (incompatibilities). We have to assign each operation to one time slot in such a way that in each time slot, all operations assigned to this slot are compatible; the length of a time slot will be the maximum of the processing times of its operations. The number k of time slots to be used has to be determined as well. So, we have to find a k-coloring

On Strong Equilibria in the Max Cut Game

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Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

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