Christian Laforest

Analytical and experimental comparison of six algorithms for the vertex cover problem

The vertex cover is a well-known NP-complete minimization problem in graphs that has received a lot of attention these last decades. Many algorithms have been proposed to construct vertex cover in different contexts (offline, online, list algorithms, etc.) leading to solutions of different level of quality. This quality is traditionally measured in terms of approximation ratio, that is, the worst possible ratio between the quality of the solution constructed and the optimal one. For the vertex cover problem the range of such known ratios are between 2 (conjectured as being the smallest constant ratio) and Δ, the maximum degree of the graph. Based on this measure of quality, the hierarchy is almost clear (the smaller the ratio is, the better the algorithm is).n In this article, we show that this measure, although of great importance, is too macroscopic and does not reflect the practical behavior of the methods. We prove this by analyzing (known and recent) algorithms running on a particular class of graphs: the paths. We obtain closed and exact formulas for the mean of the sizes of vertex cover constructed by these different algorithms. Then, we assess their quality experimentally in several well-chosen class of graphs (random, regular, trees, BHOSLIB benchmarks, trap graphs, etc.). The synthesis of all these results lead us to formulate a “practical hierarchy” of the algorithms. We remark that it is, more or less, the opposite to the one only based on approximation ratios, showing that worst-case analysis only gives partial information on the quality of an algorithm.

A new lower bound on the independence number of graphs

We propose a new lower bound on the independence number of a graph. We show that our bound compares favorably to recent ones (e.g. Harant (2011) [12]). We obtain our bound by using the Bhatia-Davis inequality applied with analytical results (minimum, maximum, expectation and variance) of an algorithm for the vertex coverproblem.

Implementation and comparison of heuristics for the vertex cover problem on huge graphs

We present in this paper an experimental study of six heuristics for a well-studied NP-complete graph problem: the vertex cover. These algorithms are adapted to process huge graphs. Indeed, executed on a current laptop computer, they offer reasonable CPU running times (between twenty seconds and eight hours) on graphs for which sizes are between 200 ·106 and 100 ·109 vertices and edges. n nWe have run algorithms on specific graph families (we propose generators) and also on random power law graphs. Some of these heuristics can produce good solutions. We give here a comparison and an analysis of results obtained on several instances, in terms of quality of solutions and complexity, including running times.

An Exact Algorithm to Check the Existence of (Elementary) Paths and a Generalisation of the Cut Problem in Graphs with Forbidden Transitions

A graph with forbidden transitions is a pair (G,F G ) where G: = (V G ,E G ) is a graph and F G is a subset of the set ( { ({y,x},{x,z}) in E_G^2 }.) A path in a graph with forbidden transitions (G,F G ) is a path in G such that each pair ({y,x},{x,z}) of consecutive edges does not belong to F G . It is shown in [S. Szeider, Finding paths in graphs avoiding forbidden transitions, DAM 126] that the problem of deciding the existence of a path between two vertices in a graph with forbidden transitions is Np-complete. We give an exact exponential time algorithm that decides in time O(2 n ·n 5·log(n)) whether there exists a path between two vertices of a given n-vertex graph with forbidden transitions. We also investigate a natural extension of the minimum cut problem: we give a polynomial time algorithm that computes a set of forbidden transitions of minimum size that disconnects two given vertices (while in a minimum cut problem we are seeking for a minimum number of edges that disconnect the two vertices). The polynomial time algorithm for that second problem is obtained via a reduction to a standard minimum cut problem in an associated allowed line graph.

Solving the Minimum Independent Domination Set Problem in Graphs by Exact Algorithm and Greedy Heuristic

In this paper we present a new approach to solve the Minimum Independent Dominating Set problem in general graphs which is one of the hardest optimization problem. We propose a method using a clique partition of the graph, partition that can be obtained greedily. We provide conditions under which our method has a better complexity than the complexity of the previously known algorithms. Based on our theoretical method, we design in the second part of this paper an efficient algorithm by including cuts in the search process. We then experiment it and show that it is able to solve almost all instances up to 50 vertices in reasonable time and some instances up to several hundreds of vertices. To go further and to treat larger graphs, we analyze a greedy heuristic. We show that it often gives good (sometimes optimal) results in large instances up to 60xa0u2009xa0000 vertices in less than 20 s. That sort of heuristic is a good approach to get an initial solution for our exact method. We also describe and analyze some of its worst cases.

Self-stabilizing Algorithms for Connected Vertex Cover and Clique Decomposition Problems

In many wireless networks, there is no fixed physical backbone nor centralized network management. The nodes of such a network have to self-organize in order to maintain a virtual backbone used to route messages. Moreover, any node of the network can be a priori at the origin of a malicious attack. Thus, in one hand the backbone must be fault-tolerant and in other hand it can be useful to monitor all network communications to identify an attack as soon as possible. We are interested in the minimum Connected Vertex Cover problem, a generalization of the classical minimum Vertex Cover problem, which allows to obtain a connected backbone. Recently, Delbot et al. [11] proposed a new centralized algorithm with a constant approximation ratio of 2 for this problem. In this paper, we propose a distributed and self-stabilizing version of their algorithm with the same approximation guarantee. To the best knowledge of the authors, it is the first distributed and fault-tolerant algorithm for this problem. The approach followed to solve the considered problem is based on the construction of a connected minimal clique partition. Therefore, we also design the first distributed self-stabilizing algorithm for this problem, which is of independent interest.

Trees in Graphs with Conflict Edges or Forbidden Transitions

In a recent paper [Paths, trees and matchings under disjunctive constraints, Darmann et. al., Discr. Appl. Math., 2011] the authors add to a graph G a set of conflicts, i.e. pairs of edges of G that cannot be both in a subgraph of G. They proved hardness results on the problem of constructing minimum spanning trees and maximum matchings containing no conflicts. A forbidden transition is a particular conflict in which the two edges of the conflict must be incident. We consider in this paper graphs with forbidden transitions. We prove that the construction of a minimum spanning tree without forbidden transitions is still ({ensuremath{mathcal{NP}}})-Hard, even if the graph is a complete graph. We also consider the problem of constructing a maximum tree without forbidden transitions and prove that it cannot be approximated better than n 1/2 − e for all e > 0 even if the graph is a star. We strengthen in this way the results of Darmann et al. concerning the minimum spanning tree problem. We also describe sufficient conditions on forbidden transitions (conflicts) to ensure the existence of a spanning tree in complete graphs. One of these conditions uses graphic sequences.

New Approximation Algorithms for the Vertex Cover Problem

The vertex cover is a classical NP-complete problem that has received great attention these last decades. A conjecture states that there is no c-approximation polynomial algorithm for it with c a constant strictly less than 2. In this paper we propose a new algorithm with approximation ratio strictly less than 2 (but non constant). Moreover we show that our algorithm has the potential to return any optimal solution.

Domination problems with no conflicts

Abstract Domination problems have been studied in graph theory for decades. In most of them, it is NP-complete to find an optimal solution, while it is easy (and even trivial in some cases) to find a solution in polynomial time, regardless of its size. In recent works, authors added conflicts to classical discrete optimization problems. In this paper, a conflict is a pair of vertices that cannot be both in a solution. Set of conflicts can be viewed as edges of a so called conflict graph. An instance is then a support graph and a conflict graph. With these new constraints, the existence of a solution (dominating set or independent dominating set) with no conflicts is no more guaranteed. We explore this subject and we prove that it is NP-complete to decide the existence of a solution even in very restricted classes of graphs and conflicts (sparse or dense). We also propose polynomial algorithms for some sub-cases, using deterministic finite automata.

Nash-Williams-type and Chvátal-type Conditions in One-Conflict Graphs

Nash-Williams and Chvatal conditions (1969 and 1972) are well known and classical sufficient conditions for a graph to contain a Hamiltonian cycle. In this paper, we add constraints, called conflicts. A conflict is a pair of edges of the graph that cannot be both in a same Hamiltonian path or cycle. Given a graph G and a set of conflicts, we try to determine whether G contains such a Hamiltonian path or cycle without conflict. We focus in this paper on graphs in which each vertex is part of at most one conflict, called one-conflict graphs. We propose Nash-Williams-type and Chvatal-type results in this context.