Fine-grained reductions from approximate counting to decision

Abstract

In this paper, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead, and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ( √ log n))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF(m) d for constant m can be solved in time n · poly(d) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time.We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O(c n )-time algorithm for k-SAT. Then we prove * We thank Rahul Santhanam and Ryan Williams for some valuable discussions. that for all k, there is an O((c + o(1)) n )-time algorithm for approximate #k-SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1 + ε in time 2 o(n) /ε 2 (taking ε > 0 as part of the input).

Approximately counting and sampling small witnesses using a colourful decision oracle

“…For example, Valiant and Vazirani [43] proved that any polynomial-time algorithm to decide SAT can be bootstrapped into a polynomial-time εapproximation algorithm for #SAT, or, more formally, that a size-n instance of any problem in #P can be ε-approximated in time poly(n, ε −1 ) using an NP-oracle. A similar result holds in the parameterised setting, where Müller [40] proved that a size-n instance of any problem in #W[i] with parameter k can be ε-approximated in time g(k) · poly(n, ε −1 ) using a W[i]-oracle for some computable function g : N → N. Another such result holds in the subexponential setting, where Dell and Lapinskas [14] proved that the (randomised) Exponential Time Hypothesis is equivalent to the statement: There is no ε-approximation algorithm for #3-SAT which runs on an n-variable instance in time ε −2 2 o(n) .…”

“…Thus if the dependence on ε were subpolynomial, Theorem 1 would essentially imply a fine-grained reduction from exact counting to decision. This is impossible under SETH in our setting; see [14,Theorem 3] for a more detailed discussion.…”

“…The above reductions all introduce significant overhead, so they are not fine-grained. Here only one general result is known, again due to Dell and Lapinskas [14]. Informally, if the decision problem reduces "naturally" to deciding whether an n-vertex bipartite graph contains an edge, then any algorithm for the decision version can be bootstrapped into an ε-approximation algorithm for the counting version with only O(ε −2 polylog(n)) overhead.…”

“…The reduction of [14] is general enough to cover core problems in fine-grained complexity such as OR-THOGONAL VECTORS, 3SUM and NEGATIVE-WEIGHT TRIANGLE, but it is not universal. In this paper, we substantially generalise it to cover any problem which can be "naturally" formulated as deciding whether a k-partite k-hypergraph contains an edge; thus we essentially recover the original result on taking k = 2.…”

“…For any problem which satisfies this property, our result implies that any new decision algorithm will automatically lead to a new approximate counting algorithm whose running time is at most a factor of log O(k) n larger. Our framework covers several reduction targets in fine-grained complexity not covered by [14], including k-ORTHOGONAL VECTORS, k-SUM and EXACT-WEIGHT k-CLIQUE, as well as some key problems in parameterised complexity including weight-k CSPs and size-k induced subgraph problems. (Note that the overhead of log O(k) n can be re-expressed as k 2k n o(1) using a standard trick, so an FPT decision algorithm is transformed into an FPT approximate counting algorithm; see Section 1.3.…”