Fast Subset Convolution

Years and Authors of Summarized Original Work

• 2007; Björklund, Husfeldt, Kaski, Koivisto

Problem Definition

A basic strategy to solve hard problems by dynamic programming is to express the partial solutions using a recurrence over the 2ⁿ subsets of an n-element set U. Our interest here is in recurrences that have the following structure:

For each subset $S \subseteq U$, in order to obtain the partial solution at S, we consider all possible ways to partition S into two disjoint parts, T and $S\setminus T$, with $T \subseteq S$.

Fast subset convolution [1] is a technique to speed up the evaluation of such recurrences, assuming the recurrence can be reduced to a suitable algebraic form. In more precise terms, let R be an algebraic ring, such as the integers equipped with the usual arithmetic operations (addition, negation, multiplication). We seek a fast solution to:

Problem (Subset Convolution)

INPUT: Two functions$f : 2^{U} \rightarrow R$and$g : 2^{U} \rightarrow R$.

OUTPUT: Th...