# Equal-Subset-Sum Faster Than the Meet-in-the-Middle

## TL;DR

This paper presents a faster randomized algorithm for the NP-complete Equal-Subset-Sum problem, improving the best known worst-case time complexity from 1.7321^n to 1.7088^n, and also offers a polynomial space algorithm with improved running time.

## Contribution

It introduces a new Monte Carlo algorithm with improved exponential time complexity and analyzes a randomized polynomial space algorithm for Equal-Subset-Sum.

## Key findings

- New Monte Carlo algorithm with O*(1.7088^n) time complexity.
- Polynomial space randomized algorithm running in O*(2.6817^n) time.
- Improved bounds answer an open problem from Woeginger's survey.

## Abstract

In the Equal-Subset-Sum problem, we are given a set $S$ of $n$ integers and the problem is to decide if there exist two disjoint nonempty subsets $A,B \subseteq S$, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in $O^{*}(3^{n/2}) \le O^{*}(1.7321^n)$ time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give $O^{*}(1.7088^n)$ worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in $O^{*}(3^n)$ time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in $O^{*}(2.6817^n)$ time and polynomial space.