Derandomized graph products

Berman and Schnitger gave a randomized reduction from approximating MAX-SNP problems within constant factors arbitrarily close to 1 to approximating clique within a factor ofnε (for some ε). This reduction was further studied by Blum, who gave it the namerandomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs. The main technical contribution of this paper is in proving a lower bound for the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounds for this probability.) This lower bound extends also to the case where different sets of vertices are specified for different time steps of the walk.

[1]  Nabil Kahale,et al.  Better expansion for Ramanujan graphs , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[2]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[3]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[4]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[5]  Piotr Berman,et al.  Approximating maximum independent set in bounded degree graphs , 1994, SODA '94.

[6]  John E. Hopcroft,et al.  Complexity of Computer Computations , 1974, IFIP Congress.

[7]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[8]  David Zuckerman.  Simulating BPP using a general weak random source , 2005, Algorithmica.

[9]  Edoardo Amaldi,et al.  The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations , 1995, Theor. Comput. Sci..

[10]  Jacques Stern,et al.  The hardness of approximate optima in lattices, codes, and systems of linear equations , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[11]  Mihir Bellare,et al.  Randomness in interactive proofs , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[12]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[13]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[14]  János Komlós,et al.  Deterministic simulation in LOGSPACE , 1987, STOC.

[15]  Aviad Cohen Avi Wigderson.  Dispersers , Deterministic Ampli cation , and Weak RandomSources , 1989 .

[16]  Jaikumar Radhakrishnan,et al.  Greed is good: Approximating independent sets in sparse and bounded-degree graphs , 1997, Algorithmica.

[17]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..

[18]  R. Bellman,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[19]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[20]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[21]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[22]  M. Murty.  Ramanujan Graphs , 1965 .

[23]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[24]  Nathan Linial,et al.  Graph products and chromatic numbers , 1989, 30th Annual Symposium on Foundations of Computer Science.

[25]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.