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O’Donnell Electronic mail: [email protected] [ii] Chicago Journal of Theoretical Computer Science 1997-1 On Limited versus Polynomial Nondeterminism Uriel Feige Joe Kilian 12 March, 1997 Abstract Abstract-1 In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simula- tion of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic time f(n) is contained in deterministic time 2O(√f(n) polylog f(n)). 1 Introduction 1-1 A major open question in computational complexity is whether P = NP. In other words, is it true that if a language can be recognized in time f(n) by a nondeterministic Turing machineL (where n denotes the input length), then can be recognized in time (f(n))c by a deterministic Tur- ing machine (forL some constant c)? Let DTIME(f(n)) denote the class of languages accepted by a deterministic Turing machine in time O(f(n)), and let NTIME(f(n)) denote the class of languages accepted by a non- deterministic Turing machine in time O(f(n)). The trivial relations be- tween these classes are DTIME(f(n)) NTIME(f(n)) (by definition), and ⊂ NTIME(f(n)) DTIME(2O(f(n))) (by exhaustive search). Perhaps the only ⊂ nontrivial relation known is that DTIME(n) = NTIME(n) [PPST83]. 6 1-2 A natural question to ask is whether there is a general methodology that improves over exhaustive search, and applies to a wide range of NP prob- lems. For some specific NP problems, improvements over exhaustive search that involve the constant in the exponent were obtained [TT77, SS79, BE95]. 1 Chicago Journal of Theoretical Computer Science 1997-1 Feige and Kilian Limited vs. Polynomial Nondeterminism 1 § However, we seek a more dramatic improvement, with more general applica- bility. This leads to the following question: Question 1 Is there a subexponential deterministic simulation of nondeter- ministic computation? More explicitly, is there some constant δ<1such that NTIME(f(n)) DTIME(2O(f(n)δ))? ⊂ 1-3 Certain NP problems require only a limited amount of nondeterminism, in the sense that their (natural) NP-witness is at most polylogarithmic in the input length. For such problems, improvements over exhaustive search are sometimes dramatic. Examples of problems that require limited nonde- terminism can be constructed by considering parameterized versions of NP optimization problems, in which the parameter k is restricted to be small (typically constant, or O(log n)). In the problem definitions below, k is a positive-integer input parameter. Path Instance: A graph G of order n. Question: Does G contain a simple path of length at least k? Vertex Cover Instance: A graph G of order n. Question: Does G contain a set of vertices of cardinality at most k that is incident with every edge of G? Clique Instance: A graph G of order n. Question: Does G contain a complete subgraph of order at least k? Monotone Circuit Satisfiability Instance: A monotone circuit C on n Boolean inputs. Question: Does C accept an input vector of Hamming weight at most k? 2 Chicago Journal of Theoretical Computer Science 1997-1 Feige and Kilian Limited vs. Polynomial Nondeterminism 1 § Tournament Dominating Set Instance: A digraph G =(V;E) of order n, in which for every pair of vertices u; v V , either (u; v) E or (v; u) E. ∈ ∈ ∈ Question: Find a set D V of minimum cardinality such that for each vertex v V , there is⊂ an edge (u; v) E for some vertex u D. ∈ ∈ ∈ VC Dimension Instance: A set U of cardinality n and a family of n subsets of U. F Question: Find a set S U of maximum cardinality that is shattered by . Set S is shattered⊂ by if for every subset T there is a set FA such that A S = TF. ⊂F ∈F ∩ We remark that the last two problems above are search problems, rather than decision problems, and also that there is no explicit parameter k involved. However, a parameter of k log n is implicit, since the cardinality of the optimal solution is at most log' n +1. 1-4 For the problems above, with parameter k = log n, an exhaustive search requires time of roughly nlog n. However, the first two of these problems have algorithms of time complexity O(nc2k), for some constant c, which is polyno- mial even for k = log n [PY96, DF95, AYZ95]. (Note that these two problems are NP-hard for a general k, since the Vertex Cover and Hamiltonian Path problems are.) Is this phenomenon of dramatically improving over exhaus- tive search a general one when problems that require a limited amount of nondeterminism are concerned? Does the same apply for the k-Clique prob- lem when k = O(log n)? These types of questions are treated by the theory of fixed parameter intractability [DF92], and motivate the introduction of the complexity class LOGSNP [PY96]. We shall return to discuss these notions in Section 4.2.1. For now, we shall concentrate on the log-Clique problem in which one must find a clique of size k log n in the input graph. ' Question 2 Is the log-Clique problem solvable in polynomial time? 1-5 Our main result is a connection between Question 2, that deals with a case of limited nondeterminism, and Question 1, that deals with polynomial nondeterminism. In Section 2 we show that if the answer to Question 2 is positive, then so is the answer to Question 1. In Section 3 we present 3 Chicago Journal of Theoretical Computer Science 1997-1 Feige and Kilian Limited vs. Polynomial Nondeterminism 2 § extensions that deal with the complexity of finding approximate solutions to the k-Clique problem, for small values of k. In Section 4 we discuss our results and their implications. In particular, we discuss related work showing (for the k-Monotone Circuit Satisfiability problem [ADF95]) or implying in conjunction with our work (for the Tournament Dominating Set and VC- Dimension problems [PY96]) additional problems of limited nondeterminism whose complexity is related with subexponential simulation of NP. 2 On the Complexity of the log-Clique Problem 2-1 The results of this section were first proven as corollaries of the results of Section 3. The elementary proofs given here were noticed by Noam Nisan, who permitted us to use them. 2-2 Lemma 1 Suppose there is a polynomial time algorithm for the k-Clique problem when k log n. Then the k-Clique problem is solvable in time O(nO(√n)), for every≤ k. c Prove Lemma 1-1 Proof of Lemma 1 Let be an algorithm that in time O(n ) solves the k-Clique problem for k Alog n. Let (G; k) be an input instance for the k- Clique problem (with arbitrarily≤ large k). Reduce (G; k) to a new instance (G;ˆ kˆ)ofthek-Clique problem as follows. Without loss of generality, assume that k is a perfect square. (Otherwise, let k0 be the least perfect square greater than k. Add k0 k vertices to G, connected to every other vertex.) ˆ −n ˆ Graph G contains N = √k vertices, where each vertex of G corresponds to a √k-subset of vertices of G. Two vertices of Gˆ are connected by an edge if the 2√k vertices that they correspond to are distinct and form a clique of size 2√k in G. Set kˆ = √k.