Solving MaxSAT and #SAT on structured CNF formulas

In this paper we propose a structural parameter of CNF formulas and use it to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time. Given a CNF formula we say that a set of clauses is precisely satisfiable if there is some complete assignment satisfying these clauses only. Let the ps-value of the formula be the number of precisely satisfiable sets of clauses. Applying the notion of branch decompositions to CNF formulas and using ps-value as cut function, we define the ps-width of a formula. For a formula given with a decomposition of polynomial ps-width we show dynamic programming algorithms solving weighted MaxSAT and #SAT in polynomial time. Combining with results of 'Belmonte and Vatshelle, Graph classes with structured neighborhoods and algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of structured CNF formulas. For example, we get $O(m^2(m + n)s)$ algorithms for formulas $F$ of $m$ clauses and $n$ variables and size $s$, if $F$ has a linear ordering of the variables and clauses such that for any variable $x$ occurring in clause $C$, if $x$ appears before $C$ then any variable between them also occurs in $C$, and if $C$ appears before $x$ then $x$ occurs also in any clause between them. Note that the class of incidence graphs of such formulas do not have bounded clique-width

Maximum matching width: new characterizations and a fast algorithm for dominating set

We give alternative definitions for maximum matching width, e.g. a graph $G$ has $\operatorname{mmw}(G) \leq k$ if and only if it is a subgraph of a chordal graph $H$ and for every maximal clique $X$ of $H$ there exists $A,B,C \subseteq X$ with $A \cup B \cup C=X$ and $|A|,|B|,|C| \leq k$ such that any subset of $X$ that is a minimal separator of $H$ is a subset of either $A, B$ or $C$. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph $G$ and a branch decomposition of mm-width $k$ we can solve Dominating Set in time $O^*({8^k})$, thereby beating $O^*(3^{\operatorname{tw}(G)})$ whenever $\operatorname{tw}(G) > \log_3{8} \times k \approx 1.893 k$. Note that $\operatorname{mmw}(G) \leq \operatorname{tw}(G)+1 \leq 3 \operatorname{mmw}(G)$ and these inequalities are tight. Given only the graph $G$ and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever $\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)$

Faster Algorithms For Vertex Partitioning Problems Parameterized by Clique-width

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for computing an optimal $k$-expression. For a graph of clique-width $k$, if we rely on known algorithms to compute a $(2^{3k}-1)$-expression via rank-width and then solving Dominating Set using the $(2^{3k}-1)$-expression, the above algorithm will only give a runtime of $4^{2^{3k}} n^{O(1)}$. There have been results which overcome this exponential jump; the best known algorithm can solve Dominating Set in time $2^{O(k^2)} n^{O(1)}$ by avoiding constructing a $k$-expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We improve this to $2^{O(k\log k)}n^{O(1)}$. Indeed, we show that for a graph of clique-width $k$, a large class of domination and partitioning problems (LC-VSP), including Dominating Set, can be solved in $2^{O(k\log{k})} n^{O(1)}$. Our main tool is a variant of rank-width using the rank of a $0$-$1$ matrix over the rational field instead of the binary field.Comment: 13 pages, 5 figure

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

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