Bruno Escoffier

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

Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms

Using ideas and results from polynomial time approximation and exact computation we design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular two paradigmatic problems, max independent set and min vertex cover.

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

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.

Efficient approximation of min set cover by moderately exponential algorithms

We study the approximation of min set cover combining ideas and results from polynomial approximation and from exact computation (with non-trivial worst case complexity upper bounds) for NP-hard problems. We design approximation algorithms for min set cover achieving ratios that cannot be achieved in polynomial time (unless problems in NP could be solved by slightly super-polynomial algorithms) with worst-case complexity much lower (though super-polynomial) than those of an exact computation.

Strategy-proof mechanisms for facility location games with many facilities

This paper is devoted to the location of public facilities in a metric space. Selfish agents are located in this metric space, and their aim is to minimize their own cost, which is the distance from their location to the nearest facility. A central authority has to locate the facilities in the space, but she is ignorant of the true locations of the agents. The agents will therefore report their locations, but they may lie if they have an incentive to do it. We consider two social costs in this paper: the sum of the distances of the agents to their nearest facility, or the maximal distance of an agent to her nearest facility. We are interested in designing strategy-proof mechanisms that have a small approximation ratio for the considered social cost. A mechanism is strategy-proof if no agent has an incentive to report false information. In this paper, we design strategyproof mechanisms to locate n - 1 facilities for n agents. We study this problem in the general metric and in the tree metric spaces. We provide lower and upper bounds on the approximation ratio of deterministic and randomized strategy-proof mechanisms.

A bottom-up method and fast algorithms for MAX INDEPENDENT SET

We first propose a new method, called “bottom-up method”, that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Following the bottom-up method we propagate improvements of worst-case complexity from graphs of average degree d to graphs of average degree greater than d. Indeed, using algorithms for max independent set in graphs of average degree 3, we tackle max independent set in graphs of average degree 4, 5 and 6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree 4, 5 and 6 but also for general graphs. The best computation bounds obtained for max independent set are O*(1.1571n), O*(1.1918n) and O*(1.2071n), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O*(1.2127n) for general graphs. These results improve upon the best known polynomial space results for these cases.

Efficient Approximation of Combinatorial Problems by Moderately Exponential Algorithms

We design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular max independent set , min vertex cover and min set cover and then extend our results to max clique , max bipartite subgraph and max set packing .

An O*(1.0977 n ) exact algorithm for MAX INDEPENDENT SET in sparse graphs

We present an O*(1.0977n) search-tree based exact algorithmfor max independent set in graphs with maximum degree 3.It can be easily seen that this algorithm also works in graphs with average degree 3.

Differential approximation of min sat, max sat and related problems

We present differential approximation results (both positive and negative) for optimal satisfiability, optimal constraint satisfaction, and some of the most popular restrictive versions of them. As an important corollary, we exhibit an interesting structural difference between the landscapes of approximability classes in standard and differential paradigms.

Weighted coloring on planar, bipartite and split graphs: complexity and improved approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs We show that this problem is NP-complete in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4 Then, we prove that min weighted node coloring is NP-complete in P8-free bipartite graphs, but polynomial for P5-free bipartite graphs We next focus ourselves on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds We next deal with min weighted edge coloring in bipartite graphs We show that this problem remains strongly NP-complete, even in the case where the input-graph is both cubic and planar Furthermore, we provide an inapproximability bound of 7/6 – e, for any e > 0 and we give an approximation algorithm with the same ratio Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.