Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems

TL;DR

This paper introduces space-efficient Monte Carlo algorithms that significantly improve the running time for solving Subset Sum, k-Sum, and related NP-hard problems, using minimal space and random access assumptions.

Contribution

It presents the first space-efficient algorithms with subexponential time for Subset Sum and k-Sum, resolving a long-standing open problem in exact algorithms for NP-hard problems.

Findings

Subset Sum and Knapsack solved in $O^*(2^{0.86n})$ time with polynomial space.

k-Sum instances for constant k solved in $O(n^{k-0.5}polylog(n))$ time and $O( ext{log } n)$ space.

Efficient algorithms for list intersection problems with bounded integers under random read-only access.

Abstract

We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with $n$ items using $O^*(2^{0.86n})$ time and polynomial space, where the $O^*(\cdot)$ notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on $n$ variables with few constraints in a similar running time. We also show that for any constant $k\geq 2$, random instances of $k$-Sum can be solved using $O(n^{k-0.5}polylog(n))$ time and $O(\log n)$ space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length $n$ with integers bounded by a polynomial in $n$ share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using $O(\log n)$ space significantly faster than the trivial $O(n^2)$ time algorithm if no value occurs too often in the same list.