We describe a simple and efficient heuristic algorithm for the graph coloring problem and show that for all k 1, it finds an optimal coloring for almost all k-colorable graphs. We also show that an algorithm proposed by Brelaz and justified on experimental grounds optimally colors almost all k-colorable graphs. Efficient implementations of both algorithms are given. The first one runs in O(n+m log k) time where n is the number of vertices and m the number of edges. The new implementation of Brelaz's algorithm runs in O(m log n) time. We observe that the popular greedy heuristic works poorly on k-colorable graphs.

In this paper, we present improved inapproximability re-sults for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chro-matic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*astad's celebrated result [13] shows that the maximumclique size in a graph with n vertices is inapproximable inpolynomial time within a factor n1-ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's re-sult. We prove that clique size is inapproximable within a factor n2(log n)1-fl (corresponding to ffl = 1(log n)fl) for some constant fl> 0 unless NP ` ZPTIME(2(log n) O(1)). This improves the previous best inapproximability factor of

Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a new permutation of the vertices is chosen which respects the independent sets of a previous coloring, then applying the greedy algorithm will result in a new coloring in which the number of colors used does not increase, yet may decrease. We introduce several heuristics for generating new permutations that are fast when implemented and effective in reducing the coloring number. The resulting Iterated Greedy algorithm(IG) can obtain colorings in the range 100 to 103 on graphs in G 1000; 1 2 . More interestingly, it can optimally color k-colorable graphs with k up to 60 and n = 1000, exceeding results of anything in the literature for these graphs. We couple this algorithm with several other c...

In the past few years, there has been significant progress in our understanding of the extent to which near-optimal solutions can be efficiently computed for NP-hard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations.

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Live range splitting techniques improve global register allocation by splitting the live ranges of variables into segments that are individually allocated registers. Load/store range analysis is a new technique for live range splitting that is based on reaching definition and live variable analyses. Our analysis localizes the profits and the register requirements of every access to every variable to provide a fine granularity of candidates for register allocation. Experiments on a suite of C and FORTRAN benchmark programs show that a graph coloring register allocator operating on load/store ranges often provides better allocations than the same allocator operating on live ranges. Experimental results also show that the computational cost of using load/store ranges for register allocation is moderately more than the cost of using live ranges. 1 Introduction Register allocation maps variables in an intermediate language program to either registers or memory locations in order to minimiz...

We give a new proof showing that it is NP-hard to color a 3-colorable graph using just four colors. This result is already known [19], but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3].

Many problems in combinatorial optimization are NP-hard (see [60]). This has forced researchers to explore techniques for dealing with NP-completeness. Some have considered algorithms that solve “typical”

Finding the minimum column multiplicity for a bound set of variables is an important problem in Curtis decomposition. To investigate this problem, we compared two graph coloring programs: one exact, and another one based on heuristics which can give, however, provably exact results on some types of graphs. These programs were incorporated into the multi-valued decomposer MVGUD. We proved that the exact graph coloring is not necessary for high-quality functional decomposers. Thus we improved by orders of magnitude the speed of the column multiplicity problem, with very little or no sacrifice of decomposition quality. Comparison of our experimental results with competing decomposers shows that for nearly all benchmarks our solutions are best and time is usually not too high.

This thesis investigates various combinations of Constraint Satisfaction Strategies with Genetic Algorithms (GA) for Solving Examination timetabling Problems (ETTPs). Since Timetabling Problems (TTPs) in their simplest form can be mapped onto GraphColouring Problems (GCP), strategies for solving these problems are also included in the research. The main GA-related issues addressed in this thesis involve the fitness function, the crossover operator and the representation. The primary contributions this investigation presents can be summarised as follows: (1) in relation to the fitness function, the Hardness Theory (HT) which intends to measure how hard it is to solve a Constraint Satisfaction Problem (CSP) has been applied with the aim of improving the quality of solutions produced by the GA with the standard penalty function which has been criticised for exhibiting a series of defects. The key idea is that the fitness value for each individual in the population at a given generation, is the measure of difficulty of solving the remaining unsolved problem, consisting of the events yet to be scheduled and the edges that connect them. Despite the fact