## Choice of parameter for DP-based FPT algorithms: four case studies

##### Abstract

This thesis studies dynamic programming algorithms and structural parameters used when solving computationally hard problems. In particular, we look at algorithms that make use of structural decompositions to overcome difficulties of solving a problem, and find alternative runtime parameterizations for some of these problems.

The algorithms we look at make use of branch decompositions to guide the algorithm when doing dynamic programming. Algorithms of this type comprise of two parts; the first part computes a decomposition of the input, and the second part solves the given problem by dynamic programming over the computed decomposition. By altering what properties of an input instance these decompositions should exploit, the runtime of the complete algorithm will change. We look at four cases where altering the structural properties of the decomposition (i.e., changing what width measure for the decomposition to minimize), is used to improve an algorithm.

The first case looks at using branch decompositions of low maximum matchingwidth (mm-width) instead of tree-decompositions of low treewidth when solving Dominating Set. The result of this is an algorithm that is faster than the treewidth-algorithms on instances where the treewidth is at least 1.55 times the mm-width.

In the second case, we look at using branch decompositions of low splitmatching- width (sm-width) for cases when using tree-decompositions or kexpressions will not do. This study leads to new tractability results for Hamiltonian Cycle, Edge Dominating Set, Chromatic Number, and MaxCut for a class of dense graphs.

For the third case, we look at using branch decompositions of low Q-rank-width as an alternative to using branch decompositions of low rank-width for solving a large class of problems definable as [σ,ρ]-partition problems. This class consists of many domination-type problems such as Dominating Set and Independent Set. One of the results of using such an alternative branch decompositions is that we get an improved worst case runtime for Dominating Set parameterized by the clique-width cw; namely O ∗((cw)O(cw)) over the previous best O ∗(2O((cw)2)).

The fourth case looks at using branch decompositions of low projectionsatisfiable- width (ps-width) for solving #SAT and MaxSAT on CNF formulas. We define the notion of having low ps-width and show that by using a dynamic programming algorithm that makes use of the ps-width of a branch decomposition, we get new tractability results for #SAT and MaxSAT, and a framework unifying many previous tractability results.

We also show that deciding boolean-width of a graph is NP-hard and deciding mim-width of a graph is W[1]-hard. In fact, under the assumption NP =ZPP, we show that we cannot approximate mim-width to within a constant factor in polynomial time.