a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NP-hard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Be-and timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�-colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3-colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first non-trivial approximation result as a function of the maximum degree. This result can be generalized to�-colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�-function. 1

The problem of computing good graph colorings arises in many diverse applications, such as in the estimation of sparse Jacobians and in the development of efficient, parallel iterative methods for solving sparse linear systems. In this paper we present an asynchronous graph coloring heuristic well suited to distributed memory parallel computers. We present experimental results obtained on an Intel iPSC/860 which demonstrate that, for graphs arising from finite element applications, the heuristic exhibits scalable performance and generates colorings usually within three or four colors of the best-known linear time sequential heuristics. For bounded degree graphs, we show that the expected running time of the heuristic under the P-RAM computation model is bounded by EO(log(n)= log log(n)). This bound is an improvement over the previously known best upper bound for the expected running time of a random heuristic for the graph coloring problem. Key words: distributed memory computers, grap...

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.

We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branch-and-bound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...

... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/3 for approximating inde-pendent sets in graphs with degree bounded by A. The analysis directs us towards a simple parallel and dis-tributed algorithm with identical performance, which on constant-degree graphs runs in O(log ” n) time us-ing linear number of processors. We also analyze the Greedy algorithm when run in combination with a frac-tional relaxation technique of Nemhauser and Trotter, and obtain an improved (2Z + 3)/5 performance ratio on graphs with average degree ~. Finally, we introduce a generally applicable technique for improving the approximation ratios of independent set algorithms, and illustrate it by improving the per-formance ratio of Greedy for large ∆.

Authentication using a path of trusted intermediaries, each able to authenticate the next in the path, is a well-known technique for authenticating channels in a large distributed system. In this paper, we explore the use of multiple paths to redundantly authenticate a channel and focus on two notions of path independence---disjoint paths and connective paths---that seem to increase assurance in the authentication. We give evidence that there are no efficient algorithms for locating maximum sets of paths with these independence properties and propose several approximation algorithms for these problems. We also describe a service we have deployed, called PathServer, that makes use of our algorithms to find such sets of paths to support authentication in PGP applications.

The efficiency of a parallel implementation of the conjugate gradient method preconditioned by an incomplete Cholesky factorization can vary dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we compare the performance of the preconditioned conjugate gradient method using these coloring orderings with a number of standard orderings on matrices arising from finite element models. Because optimal colorings for these systems may not be known a priori, we employ a graph coloring heuristic to obtain consistent colorings. Based on lower bounds obtained from the local structure of these systems, we find that the colorings determined by the heuristic are nearly optimal. For these problems, we find that the increase in parallelism afforded by the coloring-based orderings more than offsets any i...

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...

Authenticating the source of a message in a large distributed system can be difficult due to the lack of a single authority that can tell for whom a channel speaks. This has lead many to propose the use of a path of authorities, each able to authenticate the next, such that the first authority in the path can be authenticated by the message recipient and the last authority in the path can authenticate the message source. In this paper we suggest the use of multiple such paths, no two of which share a common authority, to provide independent confirmation of the message source. We formalize the problem of finding a maximum set of such paths of bounded length in a graph-theoretic framework, in which the problem can be seen to be NP-hard. We propose approximation algorithms for this problem and evaluate their accuracy on a graph induced by a database of PGP certificates. We also introduce the PathServer for PGP, a service for finding maximum sets of such paths to support authentication in ...

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.