Clique Is Hard on Average for Regular Resolution Albert Atserias Ilario Bonacina Susanna F. de Rezende Universitat Politècnica de Catalunya Universitat Politècnica de Catalunya KTH Royal Institute of Technology Department of Computer Science Department of Computer Science School of Electrical Engineering and Barcelona, Spain Barcelona, Spain Computer Science [email protected] [email protected] Stockholm, Sweden [email protected] Massimo Lauria Jakob Nordström Alexander Razborov Sapienza Università di Roma KTH Royal Institute of Technology University of Chicago Department of Statistical Sciences School of Electrical Engineering and Chicago, USA Rome, Italy Computer Science [email protected] [email protected] Stockholm, Sweden Steklov Mathematical Institute [email protected] Moscow, Russia [email protected] ABSTRACT there does not even exist any polynomial-time algorithm for ap- p 1−ϵ We prove that for k ≪ 4 n regular resolution requires length nΩ¹kº proximating the maximal size of a clique to within a factor n to establish that an Erdős–Rényi graph with appropriately cho- for any constant ϵ > 0, where n is the number of vertices in the sen edge density does not contain a k-clique. This lower bound graph [13, 34]. Furthermore, the problem appears to be hard not is optimal up to the multiplicative constant in the exponent, and only in the worst case but also on average in the Erdős-Rényi ran- also implies unconditional nΩ¹kº lower bounds on running time for dom graph model—we know of no efficient algorithms for finding several state-of-the-art algorithms for finding maximum cliques in cliques of maximum size asymptotically almost surely on random graphs. graphs with appropriate edge densities [16, 31]. In terms of upper bounds, the k-clique problem can clearly be k n CCS CONCEPTS solved in time roughly n simply by checking if any of the k many sets of vertices of size k forms a clique, which is polynomial • Theory of computation → Proof complexity; • Mathemat- k ¹nωk/3º ics of computing → Random graphs; if is constant. This can be improved slightly to O using algebraic techniques [26], where ω ≤ 2:373 is the matrix multipli- KEYWORDS cation exponent, although in practice such algebraic algorithms are outperformed by combinatorial ones [33]. k Proof complexity, regular resolution, -clique, Erdős-Rényi random The motivating problem behind this work is to determine the graphs exact time complexity of the clique problem when k is given as a ACM Reference Format: parameter. As noted above, all known algorithms require time nΩ¹kº. Albert Atserias, Ilario Bonacina, Susanna F. de Rezende, Massimo Lauria, It appears quite likely that some dependence on k is needed in the Jakob Nordström, and Alexander Razborov. 2018. Clique Is Hard on Aver- exponent, since otherwise we have the parameterized complexity Proceedings of 50th Annual ACM SIGACT age for Regular Resolution . In collapse FPT = W[1] [11]. Even more can be said if we are willing Symposium on the Theory of Computing (STOC’18). ACM, New York, NY, to believe the Exponential Time Hypothesis (ETH) [14]—then the USA, 12 pages. https://doi.org/10.1145/3188745.3188856 exponent has to depend linearly on k [8], so that the trivial upper 1 INTRODUCTION bound is essentially tight. Obtaining such a lower bound unconditionally would, in par- Deciding whether a graph has a k-clique is one of the most basic ticular, imply P , NP, and so currently seems completely out of computational problems on graphs, and has been extensively stud- reach. But is it possible to prove nΩ¹kº lower bounds in restricted ied in computational complexity theory ever since it appeared in but nontrivial models of computation? For circuit complexity, this Karp’s list of 21 NP-complete problems [15]. Not only is this prob- challenge has been met for circuits that are of bounded depth [30] lem widely believed to be infeasible to solve exactly—unless P = NP or are monotone [32]. In this paper we focus on computational Permission to make digital or hard copies of all or part of this work for personal or models that are powerful enough to capture algorithms that are classroom use is granted without fee provided that copies are not made or distributed used in practice. for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM When analysing such algorithms, it is convenient to view the must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, execution trace as a proof establishing the maximal clique size to post on servers or to redistribute to lists, requires prior specific permission and/or a for the input graph. In particular, if this graph does not have a fee. Request permissions from [email protected]. STOC’18, June 25–29, 2018, Los Angeles, CA, USA k-clique, then the trace provides an efficiently verifiable proof of © 2018 Association for Computing Machinery. the statement that the graph is k-clique-free. If one can establish a ACM ISBN 978-1-4503-5559-9/18/06...$15.00 lower bound on the length of such proofs, then this implies a lower https://doi.org/10.1145/3188745.3188856 866 STOC’18, June 25–29, 2018, Los Angeles, CA, USA A. Atserias, I. Bonacina, S. de Rezende, M. Lauria, J. Nordström, and A. Razborov bound on the running time of the algorithm, and this lower bound resolution might be able to certify k-clique-freeness in polynomial holds even if the algorithm is a non-deterministic heuristic that length independent of the value of k. somehow magically gets to make all the right choices. This brings Our contribution. We prove optimal nΩ¹kº average-case lower us to the topic of proof complexity [9], which can be viewed as the bounds for regular resolution proofs of unsatisfiability for k-clique study of upper and lower bounds in restricted nondeterministic formulas on Erdős-Rényi random graphs. computational models. p Using a standard reduction from k-clique to SAT, we can trans- Theorem 1.1 (Informal). For any integer k ≪ 4 n, given an late the problem of k-cliques in graphs to that of satisfiability of n-vertex graph G sampled at random from the Erdős-Rényi model formulas in conjunctive normal form (CNF). If an algorithm for with the appropriate edge density, regular resolution asymptotically finding k-cliques is run on a graph G that is k-clique-free, then we almost surely requires length nΩ¹kº to certify that G does not contain can extract a proof of the unsatisfiability of the corresponding CNF a k-clique. formula—the k-clique formula on G—from the execution trace of In order to make this formal, we need to define how the prob- the algorithm. Is it possible to show any non-trivial lower bound lem is encoded: depending on the formula considered, the exact on the length of such proofs? Specifically, does the resolution proof statement of what we can prove differs. In this conference paper system—the method of reasoning underlying state-of-the-art SAT we consider the simpler encoding for which we can prove an nΩ¹kº Ω¹kº ω ¹1º p solvers [2, 23, 25]—require length n , or at least n k , to prove lower bound for k ≪ n. For a stronger encoding, which in par- the absence of k-cliques in a graph? This question was asked in, ticular captures this simpler one, we prove the above result in the e.g., [7] and remains open. full-length version of this paper. The hardness of k-clique formulas for resolution is also a problem At a high level, the proof is based on a bottleneck counting of intrinsic interest in proof complexity, since these formulas escape argument in the style of [12] with a slight twist that was introduced known methods of proving resolution lower bounds for a range in [29]. In its classical form, such a proof takes four steps. First, of interesting values of k including k = O¹1º. In particular, the one defines a distribution of random source-to-sink paths onthe interpolation technique [18, 28], the random restriction method [4], DAG representation of the proof. Second, a subset of the vertices and the size-width lower bound [5] all seem to fail. of the DAG is identified—the set of bottleneck nodes—such that To make this more precise, we should mention that some previ- any random path must necessarily pass through at least one such ous works do use the size-width method, but only for very large k. node. Third, for any fixed bottleneck node, one shows that it is very 5/6 It was shown in [3] that for n ≪ k ≤ n/3 resolution requires unlikely that a random path passes through this particular node. Ω¹1º length exp n to certify that a dense enough Erdős-Rényi ran- Given this, a final union bound argument yields the conclusion that dom graph is k-clique-free. The constant hidden in the Ω¹1º in- the DAG must have many bottleneck nodes, and so the resolution creases with the density of the graph and, in particular, for very proof must be long. Ω¹nº dense graphs and k = n/3 the length required is 2 . Also, for The twist in our argument is that, instead of single bottleneck a specially tailored CNF encoding, where the ith member of the nodes, we need to define bottleneck pairs of nodes. We then argue claimed k-clique is encoded in binary by logn variables, a lower that any random path passes through at least one such pair but ¹ º bound of nΩ k for k ≤ logn can be extracted from a careful read- that few random paths pass through any fixed pair; the latter part ing of [21].