The Randomized Coloring Procedure with Symmetry-Breaking

A basic randomized coloring procedurehas been used inprobabilistic proofs to obtain remarkably strong results on graphcoloring. These results include the asymptotic version of the ListColoring Conjecture due to Kahn, the extensions of Brooks' Theoremto sparse graphs due to Kim and Johansson, and Luby's fast paralleland distributed algorithms for graph coloring. The most challengingaspect of a typical probabilistic proof is showing adequateconcentration bounds for key random variables. In this paper, wepresent a simple symmetry-breaking augmentation to the randomizedcoloring procedure that works well in conjunction with Azuma'sMartingale Inequalityto easily yield the requisiteconcentration bounds. We use this approach to obtain a number ofresults in two areas: frugal coloringand weightedequitable coloring. A β-frugal coloringof agraph Gis a proper vertex-coloring of Gin whichno color appears more than βtimes in anyneighborhood. Let G= (V, E) be avertex-weighted graph with weight function w: V→[0, 1] and let W= ΣveVw(v). A weighted equitablecoloringof Gis a proper k-coloring suchthat the total weight of every color class is "large", i.e., "notmuch smaller" than W/k; this notion is useful inobtaining tail bounds for sums of dependent random variables.

[1]  Alexandr V. Kostochka,et al.  A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring , 2008, Combinatorics, Probability and Computing.

[2]  Sriram V. Pemmaraju,et al.  Equitable colorings extend Chernoff-Hoeffding bounds , 2001, SODA '01.

[3]  Jeong Han Kim.  On Brooks' Theorem for Sparse Graphs , 1995, Comb. Probab. Comput..

[4]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[5]  Jon Rigelsford,et al.  Scheduling Computer and Manufacturing Processes 2nd Edition , 2002 .

[6]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[7]  Michael Luby.  Removing Randomness in Parallel Computation without a Processor Penalty , 1993, J. Comput. Syst. Sci..

[8]  Jeff Kahn,et al.  Asymptotically Good List-Colorings , 1996, J. Comb. Theory, Ser. A.

[9]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[10]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[11]  Béla Bollobás,et al.  Equitable and proportional coloring of trees , 1983, J. Comb. Theory, Ser. B.

[12]  Bolyai János Matematikai Társulat,et al.  Combinatorial theory and its applications , 1970 .

[13]  Svante Janson,et al.  The infamous upper tail , 2002, Random Struct. Algorithms.

[14]  Öjvind Johansson.  Simple Distributed Delta+1-coloring of Graphs , 1999, Inf. Process. Lett..

[15]  Klaus H. Ecker,et al.  Scheduling Computer and Manufacturing Processes , 2001 .

[16]  Edward G. Coffman,et al.  Mutual Exclusion Scheduling , 1996, Theor. Comput. Sci..

[17]  Michael Molloy,et al.  Total Coloring With $\Delta + \mbox\lowercasepoly(\log \Delta)$ Colors , 1999 .

[18]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[19]  Alexandr V. Kostochka,et al.  On Equitable Coloring of d-Degenerate Graphs , 2005, SIAM J. Discret. Math..

[20]  Sandy Irani,et al.  Scheduling with conflicts, and applications to traffic signal control , 1996, SODA '96.

[21]  D. de Werra,et al.  Chromatic optimisation: Limitations, objectives, uses, references , 1982 .

[22]  P. Erdos-L Lovász.  Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .

[23]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[24]  M. Jacob.  A personal communication , 1989 .

[25]  Alessandro Panconesi,et al.  Fast distributed algorithms for Brooks-Vizing colourings , 2000, SODA '98.

[26]  Robert Krauthgamer,et al.  Bounded geometries, fractals, and low-distortion embeddings , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[27]  Robert Krauthgamer,et al.  Navigating nets: simple algorithms for proximity search , 2004, SODA '04.