Martin Vatshelle

Boolean-width of graphs

Abstract We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with a decomposition tree of boolean-width k , we solve Maximum Weight Independent Set in time O ( n 2 k 2 2 k ) and Minimum Weight Dominating Set in time O ( n 2 + n k 2 3 k ) . With an additional n 2 factor in the runtime, we can also count all independent sets and dominating sets of each cardinality. Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of G F ( 2 ) -sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ ( n 2 ) vertices has boolean-width Θ ( log n ) and rank-width, clique-width, tree-width, and branch-width Θ ( n ) .

Graph classes with structured neighborhoods and algorithmic applications

Given a graph in any of the following graph classes: trapezoid graphs, circular permutation graphs, convex graphs, Dilworth k graphs, k-polygon graphs, circular arc graphs and complements of k-degenerate graphs, we show how to compute decompositions with the number of d-neighborhoods bounded by a polynomial of the input size. Combined with results of Bui-Xuan, Telle and Vatshelle (2013) [1] this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes. The boolean-width of a graph is related to the number of 1-neighborhoods and our results imply that any of these graph classes have boolean-width O(logn).

Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems

Given a graph G we provide dynamic programming algorithms for many locally checkable vertex subset and vertex partitioning problems. Their runtime is polynomial in the number of equivalence classes of problem-specific equivalence relations on subsets of vertices, defined on a given decomposition tree of G. Using these algorithms all these problems become solvable in polynomial time for many well-known graph classes like interval graphs and permutation graphs (Belmonte and Vatshelle (2013) [1]). Given a decomposition of boolean-width k we show that the algorithms will have runtime O(n^42^O^(^k^^^2^)), providing the first large class of problems solvable in fixed-parameter single-exponential time in boolean-width.

Finding good decompositions for dynamic programming on dense graphs

It is well-known that for graphs with high edge density the tree-width is always high while the clique-width can be low. Boolean-width is a new parameter that is never higher than tree-width or clique-width and can in fact be as small as logarithmic in clique-width. Boolean-width is defined using a decomposition tree by evaluating the number of neighborhoods across the resulting cuts of the graph. Several NP-hard problems can be solved efficiently by dynamic programming when given a decomposition of boolean-width k, e.g. Max Weight Independent Set in time O(n2k22k) and Min Weight Dominating Set in time O(n2+nk23k). Finding decompositions of low boolean-width is therefore of practical interest. There is evidence that computing boolean-width is hard, while the existence of a useful approximation algorithm is still open. In this paper we introduce and study a heuristic algorithm that finds a reasonably good decomposition to be used for dynamic programming based on boolean-width. On a set of graphs of practical relevance, specifically graphs in TreewidthLIB, the best known upper bound on their tree-width is compared to the upper bound on their boolean-width given by our heuristic. For the large majority of the graphs on which we made the tests, the tree-width bound is at least twice as big as the boolean-width bound, and boolean-width compares better the higher the edge density. This means that, for problems like Dominating Set, using boolean-width should outperform dynamic programming by tree-width, at least for graphs of edge density above a certain bound. In view of the amount of previous work on heuristics for tree-width these results indicate that boolean-width could in the future outperform tree-width in practice for a large class of graphs and problems.

Boolean-Width of Graphs

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divide-and-conquer approach. Boolean-width is similar to rank-width, which is related to the number of GF(2)-sums (1+1=0) of neighborhoods instead of the Boolean-sums (1+1=1) used for boolean-width. For an n-vertex graph G given with a decomposition tree of boolean-width k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n + 23k k )). We show for any graph that its boolean-width is never more than the square of its rank-width. We also exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on ?(n 2) vertices has boolean-width ?(logn) and tree-width, branch-width, clique-width and rank-width ?(n). Moreover, any optimal rank-decomposition of such a graph will have boolean-width ?(n), i.e. exponential in the optimal boolean-width.

On the Boolean-width of a graph: structure and applications

Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of boolean-width k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O*(2O(k2)). We relate the boolean-width of a graph to its branch-width and to the boolean-width of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width in [S. Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3):239-244, 2008]. For an n-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is Θ(log2 n) - setting boolean-width apart from other graph invariants - and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time O*(2O(log4 n)).

Faster algorithms on branch and clique decompositions

We combine two techniques recently introduced to obtain faster dynamic programming algorithms for optimization problems on graph decompositions. The unification of generalized fast subset convolution and fast matrix multiplication yields significant improvements to the running time of previous algorithms for several optimization problems. As an example, we give an O*(3ω/2k) time algorithm for Minimum Dominating Set on graphs of branchwidth k, improving on the previous O*(4k) algorithm. Here ω is the exponent in the running time of the best matrix multiplication algorithm (currently ω < 2.376). For graphs of cliquewidth k, we improve from O*(8k) to O*(4k). We also obtain an algorithm for counting the number of perfect matchings of a graph, given a branch decomposition of width k, that runs in time O*(2ω/2k). Generalizing these approaches, we obtain faster algorithms for all so-called [ρ, σ]-domination problems on branch decompositions if ρ and ρ are finite or cofinite. The algorithms presented in this paper either attain or are very close to natural lower bounds for these problems.

Feedback Vertex Set on Graphs of Low Cliquewidth

The Feedback Vertex Set problem asks whether a graph contains q vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if q vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for problems based on non-local properties are usually more complicated. In this paper, given a graph G of cliquewidth cw and a cw-expression of G, we solve the Minimum Feedback Vertex Set problem in time

An algorithm for the maximum weight independent set problem on outerstring graphs

O(n^22^{2cw^2 \log cw})

Solving #SAT and MaxSAT by dynamic programming

. Our algorithm applies a non-standard dynamic programming on a so-called k-module decomposition of a graph, as defined by Rao [26], which is easily derivable from a k-expression of the graph. The related notion of module-width of a graph is tightly linked to both cliquewidth and nlc-width, and in this paper we give an alternative equivalent characterization of module-width.