60,448 research outputs found
Local enumeration and majority lower bounds
Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art ÎŁ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, PudlĂĄk, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any ÎŁ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(â_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) â„ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a ÎŁ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^Ï(ân) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598âż, substantially improving on the trivial bound of 3^{n/2} â 1.732âż. This already improves ÎŁ^3_3 lower bounds for Majority function to 1.251âż. The previous bound was 1.154âż which follows from the work of HĂ„stad, Jukna, and PudlĂĄk (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph TurĂĄn problem. Therefore our techniques might be of independent interest in extremal combinatorics
On Oracles and Algorithmic Methods for Proving Lower Bounds
This paper studies the interaction of oracles with algorithmic approaches to proving circuit complexity lower bounds, establishing new results on two different kinds of questions.
1) We revisit some prominent open questions in circuit lower bounds, and provide a clean way of viewing them as circuit upper bound questions. Let Missing-String be the (total) search problem of producing a string that does not appear in a given list L containing M bit-strings of length N, where M < 2?. We show in a generic way how algorithms and uniform circuits (from restricted classes) for Missing-String imply complexity lower bounds (and in some cases, the converse holds as well).
We give a local algorithm for Missing-String, which can compute any desired output bit making very few probes into the input, when the number of strings M is small enough. We apply this to prove a new nearly-optimal (up to oracles) time hierarchy theorem with advice.
We show that the problem of constructing restricted uniform circuits for Missing-String is essentially equivalent to constructing functions without small non-uniform circuits, in a relativizing way. For example, we prove that small uniform depth-3 circuits for Missing-String would imply exponential circuit lower bounds for ?? EXP, and depth-3 lower bounds for Missing-String would imply non-trivial circuits (relative to an oracle) for ?? EXP problems. Both conclusions are longstanding open problems in circuit complexity.
2) It has been known since Impagliazzo, Kabanets, and Wigderson [JCSS 2002] that generic derandomizations improving subexponentially over exhaustive search would imply lower bounds such as NEXP ? ? ?/poly. Williams [SICOMP 2013] showed that Circuit-SAT algorithms running barely faster than exhaustive search would imply similar lower bounds. The known proofs of such results do not relativize (they use techniques from interactive proofs/PCPs). However, it has remained open whether there is an oracle under which the generic implications from circuit-analysis algorithms to circuit lower bounds fail.
Building on an oracle of Fortnow, we construct an oracle relative to which the circuit approximation probability problem (CAPP) is in ?, yet EXP^{NP} has polynomial-size circuits.
We construct an oracle relative to which SAT can be solved in "half-exponential" time, yet exponential time (EXP) has polynomial-size circuits. Improving EXP to NEXP would give an oracle relative to which ?? ? has "half-exponential" size circuits, which is open. (Recall it is known that ?? ? is not in "sub-half-exponential" size, and the proof relativizes.) Moreover, the running time of the SAT algorithm cannot be improved: relative to all oracles, if SAT is in "sub-half-exponential" time then EXP does not have polynomial-size circuits
Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT
We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of LovĂĄsz that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
On product, generic and random generic quantum satisfiability
We report a cluster of results on k-QSAT, the problem of quantum
satisfiability for k-qubit projectors which generalizes classical
satisfiability with k-bit clauses to the quantum setting. First we define the
NP-complete problem of product satisfiability and give a geometrical criterion
for deciding when a QSAT interaction graph is product satisfiable with positive
probability. We show that the same criterion suffices to establish quantum
satisfiability for all projectors. Second, we apply these results to the random
graph ensemble with generic projectors and obtain improved lower bounds on the
location of the SAT--unSAT transition. Third, we present numerical results on
random, generic satisfiability which provide estimates for the location of the
transition for k=3 and k=4 and mild evidence for the existence of a phase which
is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published
version. New proof in appendi
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