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

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 $O^{\ast }({1.7088}^n)$ 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 $O^{\ast }(3^n)$ time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in $O^{\ast }({2.6817}^n)$ time and polynomial space.