Dieter van Melkebeek

Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses

Traditional hardness versus randomness results focus on time-efficient randomized decision procedures. We generalize these trade-offs to a much wider class of randomized processes. We work out various applications, most notably to derandomizing Arthur-Merlin games. We show that every language with a bounded round Arthur-Merlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomial-time hierarchy (and hence the polynomial-time hierarchy collapses). Since the graph nonisomorphism problem has a bounded round Arthur-Merlin game, this provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We also establish hardness versus randomness trade-offs for space bounded computation.

Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ε we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd-ε) then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ε = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on n-vertex d-uniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d=2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k2-ε) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k^2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

Time-space lower bounds for satisfiability

We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant <i>c</i> less than the golden ratio there exists a positive constant <i>d</i> such that no deterministic random-access Turing machine can solve satisfiability in time <i>n</i>^<i>c</i> and space <i>n</i>^<i>d</i>, where <i>d</i> approaches 1 when <i>c</i> does. On conondeterministic instead of deterministic machines, we prove the same for any constant <i>c</i> less than &2radic;.Our lower bounds apply to nondeterministic linear time and almost all natural NP-complete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space <i>n</i>^<i>1/c</i>.Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove time-space lower bounds for languages higher up in the polynomial-time hierarchy.

Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses

We establish hardness versus randomness trade-offs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round Arthur-Merlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round Arthur-Merlin game has subexponential size membership proofs for infinitely many input lengths unless the polynomial-time hierarchy collapses. This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given random bit sequence is not too high. We then apply our derandomization technique to four fundamental complexity theoretic constructions: The Valiant-Vazirani random hashing technique which prunes the number of satisfying assignments of a Boolean formula to one, and related procedures like computing satisfying assignments to Boolean formulas non-adaptively given access to an oracle for satisfiability. The algorithm of Bshouty et al. for learning Boolean circuits. Constructing matrices with high rigidity. Constructing polynomial-size universal traversal sequences. We also show that if linear space requires exponential size circuits, then space bounded randomized computations can be simulated deterministically with only a constant factor overhead in space.

Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

Consider the following two-player communication process to decide a language <i>L</i>: The first player holds the entire input <i>x</i> but is polynomially bounded; the second player is computationally unbounded but does not know any part of <i>x</i>; their goal is to decide cooperatively whether <i>x</i> belongs to <i>L</i> at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer <i>d</i> ≥ 3 and positive real <i>ε</i>, we show that, if satisfiability for <i>n</i>-variable <i>d</i>-CNF formulas has a protocol of cost <i>O</i>(<i>nd</i> − <i>ε</i>), then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for <i>ε</i> = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on <i>n</i>-vertex <i>d</i>-uniform hypergraphs, this statement holds for any integer <i>d</i> ≥ 2. The case <i>d</i> = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of <i>O</i>(<i>k</i>2 − <i>ε</i>) edges unless coNP is in NP/poly, where <i>k</i> denotes the size of the deletion set. Kernels consisting of <i>O</i>(<i>k</i>2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

A survey of lower bounds for satisfiability and related problems

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving non-trivial lower bounds on the computational complexity of satisfiability. In this article, we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the state-of-the-art results and present the underlying arguments in a unified framework.

Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasi-polynomial time in general, and polynomial time for constant depth. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four circuits.

Separating Complexity Classes Using Autoreducibility

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Posts program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines

We establish the first polynomial-strength time-space lower bounds for problems in the linear-time hierarchy on randomized machines with two-sided error. We show that for any integer

A time lower bound for satisfiability

\ell > 1