The complexity of the max word problem and the power of one-way interactive proof systems

We study the complexity of the max word problem for matrices, a variation of the well-known word problem for matrices. We show that the problem is NP-complete, and cannot be approximated within any constant factor, unless P=NP. We describe applications of this result to probabilistic finite state automata, rational series andk-regular sequences. Our proof is novel in that it employs the theory of interactive proof systems, rather than a standard reduction argument. As another consequence of our results, we characterize NP exactly in terms ofone-way interactive proof systems.

[1]  J. Rotman.  The theory of groups : an introduction , 1966 .

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

[3]  Richard E. Ladner,et al.  Interactive Proof Systems with Polynomially Bounded Strategies , 1995, J. Comput. Syst. Sci..

[4]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[5]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[6]  Richard E. Ladner,et al.  Interactive proof systems with polynomially bounded strategies , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[7]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 1992, computational complexity.

[8]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

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

[10]  Mariëlle Stoelinga,et al.  An Introduction to Probabilistic Automata , 2002, Bull. EATCS.

[11]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[12]  Arto Salomaa,et al.  Automata-Theoretic Aspects of Formal Power Series , 1978, Texts and Monographs in Computer Science.

[13]  Anne Condon.  The Complexity of the Max Word Problem , 1991, STACS.

[14]  M. Hall.  The Theory Of Groups , 1959 .

[15]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[16]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[17]  Richard J. Lipton,et al.  Word Problems Solvable in Logspace , 1977, JACM.

[18]  Richard J. Lipton,et al.  On the complexity of space bounded interactive proofs , 1989, 30th Annual Symposium on Foundations of Computer Science.

[19]  Anne Condon.  Space-bounded probabilistic game automata , 1991, JACM.

[20]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.