On the second eigenvalue of random regular graphs
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
The greedy coloring is a bad probabilistic algorithm
Journal of Algorithms
Laplacian eigenvalues and the maximum cut problem
Mathematical Programming: Series A and B
Algorithmic theory of random graphs
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Gadgets, Approximation, and Linear Programming
SIAM Journal on Computing
Some optimal inapproximability results
Journal of the ACM (JACM)
An Upper Bound for the Maximum Cut Mean Value
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
Combinatorics, Probability and Computing
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
Journal of Computer and System Sciences - STOC 2001
The two possible values of the chromatic number of a random graph
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT
Theoretical Computer Science
Exact and approximative algorithms for coloring G(n,p)
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Random MAX SAT, random MAX CUT, and their phase transitions
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part II
The Lovász Number of Random Graphs
Combinatorics, Probability and Computing
Spectral techniques applied to sparse random graphs
Random Structures & Algorithms
The Lovász Number of Random Graphs
Combinatorics, Probability and Computing
Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time
Combinatorics, Probability and Computing
An improved algorithm for approximating the chromatic number of Gn,p
Information Processing Letters
Approximability Distance in the Space of H-Colourability Problems
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
The MAX-CUT of sparse random graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
| Hi-index | 0.00 |
We consider the MAX k-CUT problem on random graphs Gn,p. First, we bound the probable weight of a MAX k-CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k-CUT in expected polynomial time, with approximation ratio 1 + O((np)-1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)-relaxation of MAX k-CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of Gn,p, 1/n ≤ p ≤ 0.99, within a factor of O((np)1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP-hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006An extended abstract version of this paper appeared in Proc. ICALP 2003, Springer LNCS 2719, pp. 200–211.