Binh-Minh Bui-Xuan

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 ) .

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.

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)).

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

Feedback vertex set on graphs of low clique-width

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

Tree-representation of set families and applications to combinatorial decompositions

. 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.

Unifying two graph decompositions with modular decomposition

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 clique-width cw and a cw-expression of G, we solve the Minimum Feedback Vertex Set problem in time O(n^22^O^(^c^w^l^o^g^c^w^)). Our algorithm applies dynamic programming on a so-called k-module decomposition of a graph, as defined by Rao (2008) [29], which is easily derivable from ak-expression of the graph. The related notion of module-width of a graph is tightly linked to both clique-width and NLC-width, and in this paper we give an alternative equivalent characterization of module-width.

On Modular Decomposition Concepts: the case for Homogeneous Relations

The number of families over ground set V is 2^2^^^|^^^V^^^| and by this fact it is not possible to represent such a family using a number of bits polynomial in |V|. However, under some simple conditions, this becomes possible, like in the cases of a symmetric crossing family and a weakly partitive family, both representable using @Q(|V|) space. We give a general framework for representing any set family by a tree. It extends in a natural way the one used for symmetric crossing families in [W. Cunningham, J. Edmonds, A combinatorial decomposition theory, Canadian Journal of Mathematics 32 (1980) 734-765]. We show that it also captures the one used for weakly partitive families in [M. Chein, M. Habib, M.C. Maurer, Partitive hypergraphs, Discrete Mathematics 37 (1) (1981) 35-50]. We introduce two new classes of families: weakly partitive crossing families are those closed under the union, the intersection, and the difference of their crossing members, and union-difference families those closed under the union and the difference of their overlapping members. Each of the two cases encompasses symmetric crossing families and weakly partitive families. Applying our framework, we obtain a linear @Q(|V|) and a quadratic O(|V|^2) space representation based on a tree for them. We introduce the notion of a sesquimodule - one module and a half - in a digraph and in a generalization of digraphs called 2-structure. From our results on set families, we show for any digraph, resp. 2-structure, a unique decomposition tree using its sesquimodules. These decompositions generalize strictly the clan decomposition of a digraph and that of a 2-structure. We give polynomial time algorithms computing the decomposition tree for both cases of sesquimodular decomposition.

A generic approach to decomposition algorithms, with an application to digraph decomposition

We introduces the umodules, a generalization of the notion of graph module. The theory we develop captures among others undirected graphs, tournaments, digraphs, and 2-structures. We show that, under some axioms, a unique decomposition tree exists for umodules. Polynomial-time algorithms are provided for: non-trivial umodule test, maximal umodule computation, and decomposition tree computation when the tree exists. Our results unify many known decomposition like modular and bi-join decomposition of graphs, and a new decomposition of tournaments.