Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution

Abstract

Abstract.In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk 2 3 k ) time and the problem to count the number of perfect matchings in O * (2 k ) time. Using a generalisation of fast subset convolution, we obtain faster algorithms for all [ρ, σ]-domination problems with finite or cofinite ρ and σ on tree decompositions. These include many well known graph problems. We give additional results on many more graph covering and partitioning problems.

Graph Minors and Parameterized Algorithm Design

“…There is a quite extended bibliography on how to do fast dynamic programming on graphs of bounded treewidth; as a sample of this, we just mention [5,6,9,11,13,16,22,35,35,36,37,37,38,114,120,120]. t: Bounds are much better for the function t. For most natural graph parameters, it holds that t(k) = O(k) while for some of them, including tw and pw, it holds that t(k) = Θ(k).…”

On the Boolean-Width of a Graph: Structure and Applications

“…tree-width branch-width clique-width rank-width boolean-width MDS O * (2 1.58tw ) [21] O * (2 2bw ) [6] O * (2 4cw ) [16] O * (2 0.75rw 2 +O(rw) ) [2,9] O * (2 3boolw ) [3] Figure 1. Upper bounds tying parameters tw =tree-width, bw =branch-width, cw =clique-width, rw =rank-width and boolw =boolean-width, and runtimes achievable for Minimum Dominating Set using various parameters.…”

“…These are related to domination, independence and homomorphism, including Max or Min Perfect Code, Max or Min Independent Dominating Set, Min k -Dominating Set, Max Induced k -Regular Subgraph, Max Induced k -Bounded Degree Subgraph, H -Coloring, H -Homomorphism, H -Covering, H -Partial Covering. Algorithms parameterized by either the tree-width of the input graph, or by its clique-width, have already been given for this class of problems [22,10], recently improved by van Rooij et al [21]. The runtime achieved by these algorithms are O * (2 O(tw) ) and O * (2 2 poly(cw) ).…”

“…This results in several papers aiming at the design of algorithms having the lowest dependency on the parameters [22,10,21], originally done for tree-width and clique-width, but having an implication to the other parameters as well, e.g., by using the relationships in Figure 1 in a straightforward manner. Also, these are well-behaved problems when parameterized by tree-width tw , namely there are O * (2 O(tw) ) algorithms solving the problem if a tree decomposition of width tw is given [22], with the fastest known for (σ, ρ)-problems [21], where d(σ) and d(ρ) are problem specific constants (see below). This is not the same situation for clique-width, where until now the best runtime contains a O * (2 2 poly(cw) ) double exponential factor [10].…”

On the parameterized complexity of monotone and antimonotone weighted circuit satisfiability

“…The total running time of the dynamic programming algorithm outlined above is O(9 ω N O(1) ). It may be possible to obtain improvement on the running time of the dynamic programming algorithm above using the generalized subset convolution technique (see [27]). …”