Samik Sengupta

Disjoint NP-Pairs

We study the question of whether the class DisjNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair; an oracle relative to which optimal proof systems exist, and hence complete pairs exist, but no pair is NP-hard; and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.

Reductions between disjoint NP-pairs

Razborov (1994) proved that existence of an optimal proof system implies existence of a many-one complete disjoint NP-pair. Kobler, Messner, and Toran (2003) defined a stronger form of many-one reduction and claimed to improve Razborovs result by showing under the same assumption that there is a strongly many-one complete disjoint NP-pair. Here we show that the two results are equivalent. More generally, we prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A,B) and (C,D) is a Turing reduction with the additional property that if the input belongs to A /spl cup/ B, then all queries belong to C /spl cup/ D. We prove under the reasonable assumption UP /spl cap/ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A,B) and (C,D) such that (A,B) is truth-table reducible to (C,D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DisjNP has a truth-table-complete disjoint NP-pair, but has no many-one- complete disjoint NP-pair.

Disjoint NP-pairs

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provide additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair, an oracle relative to which optimal proof systems exist, hence complete pairs exist, but no pair is NP-hard, and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.

A Protocol for Serializing Unique Strategies

We devise an efficient protocol by which a series of two-person games G i with unique winning strategies can be combined into a single game G with unique winning strategy, even when the result of G is a non-monotone function of the results of the G i that is unknown to the players. In computational complexity terms, we show that the class UAP of Niedermeier and Rossmanith [10] of languages accepted by unambiguous polynomial-time alternating TMs is self-low, i.e., \({\rm UAP}^{\ rm UAP} = {\rm UAP}\). It follows that UAP contains the Graph Isomorphism problem, nominally improving the problem’s classification into SPP by Arvind and Kurur [2] since UAP is a subclass of SPP [10]. We give some other applications, oracle separations, and results on problems related to unique-alternation formulas.

Properties of NP-complete sets

We study several properties of sets that are complete for NP. We prove that if L is an NP-complete set and S /spl nsupe/ L is a p-selective sparse set, then L -S is /spl les//sub m//sup p/-hard for NP. We demonstrate existence of a sparse set S /spl isin/ DTIME(2/sup 2n/) such that for every L /spl isin/ NP - P, L - S is not /spl les//sub m//sup p/-hard for NP. Moreover, we prove for every L /spl isin/ NP - P, that there exists a sparse S /spl isin/ EXP such that L - S is not /spl les//sub m//sup p/-hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one-way permutations to derive consequences for long-standing open questions about whether NP-complete sets are immune. For example, assuming that pseudorandom generators and secure one-way permutations exist, it follows easily that NP-complete sets are not p-immune. Assuming only that secure one-way permutations exist, we prove that no NP-complete set is DTIME(2/sup ne/)-immune. Also, using these hypotheses we show that no NP-complete set is quasipolynomial-close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing-complete sets for NP are not closed under union. Our hypothesis asserts existence of a UP-machine M that accepts 0* such that for some 0 < /spl epsi/ < 1, no 2/sup ne/ time-bounded machine can correctly compute infinitely many accepting computations of M, We show that if UP /spl cap/ coUP contains DTIME(2/sup ne/)-bi-immune sets, then this hypothesis is true.

Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

If every language in coNP has a constant-round interactive proof system, then the polynomial-time hierarchy collapses [R.B. Boppana, J. Hastad, S. Zachos, Does co-NP have short interactive proofs? Information Processing Letters 25 (2) (1987) 127-132]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [C. Lund, L. Fortnow, H. Karloff, N. Nisan, Algebraic methods for interactive proof systems, Journal of the Association for Computing Machinery 39 (4) (1992) 859-868]. We consider the question of whether it is possible for coNP to have interactive proof systems with polylogarithmic-round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system, then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yaps theorem.

ON HIGHER ARTHUR-MERLIN CLASSES

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

On Higher Arthur-Merlin Classes

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP,R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.