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58
Almost All kColorable Graphs Are Easy to Color
"... 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 kcolorable graphs. We also show that an algorithm proposed by Brelaz and justified on experimental grounds optimally colors almost all kc ..."
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Cited by 68 (0 self)
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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 kcolorable graphs. We also show that an algorithm proposed by Brelaz and justified on experimental grounds optimally colors almost all kcolorable 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 kcolorable graphs.
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and acce ..."
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Cited by 66 (9 self)
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The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
"... In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*ast ..."
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Cited by 59 (8 self)
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In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic 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 n1ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's result. We prove that clique size is inapproximable within a factor n2(log n)1fl (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
Iterated Greedy Graph Coloring and the Difficulty Landscape
, 1992
"... 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 ne ..."
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Cited by 34 (2 self)
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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 kcolorable 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...
Computing NearOptimal Solutions to Combinatorial Optimization Problems
 IN COMBINATORIAL OPTIMIZATION, DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1995
"... In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the ..."
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Cited by 32 (0 self)
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In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard 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.
Inapproximability of combinatorial optimization problems. The Computing Research Repository
, 2004
"... ..."
On the Hardness of 4coloring a 3colorable Graph
 In Proceedings of the 15th Annual IEEE Conference on Computational Complexity
, 2000
"... We give a new proof showing that it is NPhard to color a 3colorable 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 res ..."
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Cited by 21 (2 self)
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We give a new proof showing that it is NPhard to color a 3colorable 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 3colorable 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].
Load/Store Range Analysis for Global Register Allocation
 Proc. of the SIGPLAN Conference on Programming Language Design and Implementation
, 1994
"... 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. ..."
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Cited by 20 (0 self)
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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...
The approximability of NPhard problems
 In Proceedings of the Annual ACM Symposium on Theory of Computing
, 1998
"... Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical” ..."
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Cited by 16 (0 self)
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Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical”