Sigve Hortemo Sæther

Solving #SAT and MaxSAT by dynamic programming

We look at dynamic programming algorithms for propositional model counting, also called #SAT, and MaxSAT. Tools from graph structure theory, in particular treewidth, have been used to successfully identify tractable cases in many subfields of AI, including SAT, Constraint Satisfaction Problems (CSP), Bayesian reasoning, and planning. In this paper we attack #SAT and MaxSAT using similar, but more modern, graph structure tools. The tractable cases will include formulas whose class of incidence graphs have not only unbounded treewidth but also unbounded clique-width. We show that our algorithms extend all previous results for MaxSAT and #SAT achieved by dynamic programming along structural decompositions of the incidence graph of the input formula. We present some limited experimental results, comparing implementations of our algorithms to state-of-the-art #SAT and MaxSAT solvers, as a proof of concept that warrants further research.

Faster algorithms for vertex partitioning problems parameterized by clique-width

Abstract 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 3 k − 1 ) -expression via rank-width and then solving Dominating Set using the ( 2 3 k − 1 ) -expression, the above algorithm will only give a runtime of 4 2 3 k 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 et al. (2013) [7] . 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.

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 projection satisfiable if there is some complete assignment satisfying these clauses only. Let the ps-value of the formula be the number of projection 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 \({\mathcal O}(m^2(m + n)s)\) algorithms for formulas F of m clauses and n variables and total 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.

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal

On Satisfiability Problems with a Linear Structure

(\sigma ,\rho )

Solving Hamiltonian Cycle by an EPT Algorithm for a Non-sparse Parameter

(σ,ρ)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter ☆

Abstract A graph of treewidth k has a representation by subtrees of a ternary tree, with subtrees of adjacent vertices sharing a tree node, and any tree node sharing at most k + 1 subtrees. Likewise for branchwidth, but with a shift to the edges of the tree rather than the nodes. In this paper we show that the mm-width of a graph – maximum matching width – combines aspects of both these representations, targeting tree nodes for adjacency and tree edges for the parameter value. The proof of this new characterization of mm-width is based on a definition of canonical minimum vertex covers of bipartite graphs. We show that these behave in a monotone way along branch decompositions over the vertex set of a graph. We use these representations to compare mm-width with treewidth and branchwidth, and also to give another new characterization of mm-width, by subgraphs of chordal graphs. We prove that given a graph G and a branch decomposition of maximum matching width k we can solve the Minimum Dominating Set Problem in time O ∗ ( 8 k ) , thereby beating O ∗ ( 3 tw ( G ) ) whenever tw ( G ) > log 3 8 × k ≈ 1 . 893 k . Note that mmw ( G ) ≤ tw ( G ) + 1 ≤ 3 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 Minimum Dominating Set whenever tw ( G ) > 1 . 549 × mmw ( G ) .

Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set.

It was recently shown \cite{STV} that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within each partite set). Here we relax this condition in several directions: First, we show that it holds for