We describe recent work on a variant of a distance labeling problem in graphs, called the forbidden-set labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given

This article investigates complexity and approximability properties of combinatorial optimization problems yielded by the notion of Shared Risk Resource Group (SRRG). SRRG has been introduced in order to capture network survivability issues where a failure may break a whole set of resources, and has been formalized as colored graphs, where a set of resources is represented by a set of edges with same color. We consider here the analogous of classical problems such as determining paths or cuts with the minimum numbers of colors or color disjoint paths. These optimization problems are much more difficult than their counterparts in classical graph theory. In particular standard relationship such as the Max Flow -Min Cut equality do not hold any longer. In this article we identify cases where these problems are polynomial, for example when the edges of a given color form a connected subgraph, and otherwise give hardness and non approximability results for these problems.

In this paper, we deal with the compact routing

In this paper, we address the problem of gathering information in a specific node (or sink) of a radio network, where interference constraints are present. We take into account the fact that, when a node transmits, it produces interference in an area bigger than the area in which its message can actually be received. The network is modeled by a graph; a node is able to transmit one unit of information to the set of vertices at distance at most dT in the graph, but when doing so it generates interference that does not allow nodes at distance up to dI (dI ≥ dT) to listen to other transmissions. Time is synchronous and divided into time-steps in each of which a round (set of non-interfering radio transmissions) is performed. We give general lower bounds on the number of rounds required to gather into a sink of a general graph, and present an algorithm working on any graph, with an approximation factor of 4. We also show that the problem of finding an optimal strategy for gathering is NP-HARD, for any values of dI and dT. If dI > dT, we show that the problem remains hard when restricted to the uniform case where each vertex in the network has exactly one piece of information to communicate to the sink.

No abstract

International audienceA proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper coloring of G (1997). If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. Max Coloring Problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes, in particular, there exists a PTAS for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The Exponential Time Hypothesis (ETH) states that 3-SAT cannot be solved in sub-exponen-tial time. We show that, assuming ETH, the best algorithm to compute the weighted chromatic number of n-node trees has time-complexity n Θ(log n) . Our result mainly relies on proving that, when computing an optimal proper weighted coloring of a graph G, it is hard to combine colorings of its connected components

Abstract. In this paper we present a new heuristic called Adaptive Broadcast Consumption (ABC for short) for the Minimum-Energy Broadcast Routing (MEBR) problem. We first investigate the problern trying to understand which are the main properties not taken into account by the dassie and well-studied MST and BIP heuristics, then we propose a new algorithm proving that it computes the MEBR with an approximation ratio less than or equal to MST, for which we prove an approximation ratio of at most 12.15 instead of the well-known 12 [10]. Finally we present experimental results supporting our intuitive ideas, comparing ABC with other heuristics presented in the literature and showing its good performance on random instances even compared to the optimum.

In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d ≥ 2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, for any fixed instance, as a limit for α going to infinity, the ratio tends to the lower bound of Clementi et al. (Proceedings of the 18th annual symposium on theoretical aspects of computer science (STACS), pp. 121-131, 2001), Wan et al. (Wirel. Netw. 8(6):607-617, 2002) given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.45-approximation ratio for Algorithmica (2007) 49: 318-336 319 the 2-dimensional case, thus significantly decreasing the previously known 12 upper bound (Wan et al. in Wirel. Netw. 8(6):607-617, 2002) (actually corrected to 12.15 in Klasing et al. (Proceedings of the 3rd IFIP-TC6 international networking conference, pp. 866-877, 2004)). Finally, we extend our analysis to any number of dimensions d ≥ 2 and any α ≥ d, obtaining a general approximation ratio of 3 d − 1, again independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d.