Graph coloring with cardinality constraints on the neighborhoods

AbstractExtensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph G a k-coloring, i.e., a partition V1,…,Vk of the vertex set of G such that, for some specified neighborhood Ñ(v) of each vertex v, the number of vertices in Ñ(v)∩Vi is (at most) a given integer hvi. The complexity of some variations is discussed according to Ñ(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when G is a special tree)

Graph coloring with cardinality constraints on the neighborhoods

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph a k-coloring, i.e., a partition (V_1,\cdots,V_k) of the vertex set of G such that, for some specified neighborhood \tilde|{N}(v) of each vertex v, the number of vertices in \tilde|{N}(v)\cap V_i is (at most) a given integer h_i^v. The complexity of some variations is discussed according to \tilde|{N}(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when is a special tree)

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances

A study on exponential-size neighborhoods for the bin packing problem with conflicts

We propose an iterated local search based on several classes of local and large neighborhoods for the bin packing problem with conflicts. This problem, which combines the characteristics of both bin packing and vertex coloring, arises in various application contexts such as logistics and transportation, timetabling, and resource allocation for cloud computing. We introduce $O(1)$ evaluation procedures for classical local-search moves, polynomial variants of ejection chains and assignment neighborhoods, an adaptive set covering-based neighborhood, and finally a controlled use of 0-cost moves to further diversify the search. The overall method produces solutions of good quality on the classical benchmark instances and scales very well with an increase of problem size. Extensive computational experiments are conducted to measure the respective contribution of each proposed neighborhood. In particular, the 0-cost moves and the large neighborhood based on set covering contribute very significantly to the search. Several research perspectives are open in relation to possible hybridizations with other state-of-the-art mathematical programming heuristics for this problem.Comment: 26 pages, 8 figure