A Generic Convolution Algorithm for Join Operations on Tree Decompositions

Abstract

The fast subset convolution algorithm by Björklund et al. (STOC 2007) has made quite an impact on parameterised complexity. Amongst the many applications are dynamic programming algorithms on tree decompositions, where the computations in so-called join nodes are a recurring example of where convolution-like operations are used. As such, several generalisations of the original fast subset convolution algorithm have been proposed, all based on concepts that strongly relate to either Möbius transforms or to Fourier transforms.

We present a new convolution generalisation that uses both Möbius transforms and Fourier transforms on the same transformation domain. This results in new faster algorithms on tree decompositions for a broad class of vertex subset problems known as the $[\sigma ,\rho ]$-domination problems. We solve them in $\mathcal {O}(s^{t+2} t n^2 (t\log (s)+\log (n)))$ arithmetic operations, where t is the treewidth, s is the (fixed) number of states required to represent partial solutions of the specific problem, and n is the number of vertices in the graph. This improves the previous best bound of $\mathcal {O}( s^{t+2} (st)^{2(s-2)} n^3 )$ arithmetic operations (van Rooij, Bodlaender, Rossmanith, ESA 2009). Specifically, this removes the dependence of the degree of the polynomial on s.