Unary Subset-Sum is in Logspace

In this paper we consider the Unary-Subset-Sum problem which is defined as follows: Given integers m1, . . . ,mn and B (written in unary), we define the subset sum problem to be that of determining whether or not there exists an S ⊆ [n] so that ∑ i∈Smi = B (note that for this problem the mi are often assumed to be non-negative). Let C = |B| + ∑n i=1 |xi| + 1. This problem can be solved using a standard dynamic program using space O(C) and time O(Cn). The dynamic program makes fundamental use of this large space and it is interesting to ask whether this requirement can be removed. Unary SubsetSum has been studied in small-space models of computation as early as 1980 in [4], where they showed that it was in NL. Since then the problem was studied in [2], where Cho and Huynh devised a complexity class between L and NL that contained Unary Subset-Sum as supporting evidence that it is not NL-complete. This problem was listed again in [1] claiming it to be an open problem as to whether or not it is in L. In 2010 it was recently shown in [3] that this problem was in Logspace as a consequence of a much more general algorithm. We provide a simple algorithm solving this problem in Logspace, which is also implementable in TC.