On classes of graphs with logarithmic boolean-width (original) (raw)
Related papers
On the Boolean-Width of a Graph: Structure and Applications
Lecture Notes in Computer Science, 2010
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 booleanwidth k , we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O * (2 O(k 2)). We relate the boolean-width of a graph to its branch-width and to the booleanwidth of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width by Oum (JGT 2008). For a random graph on n vertices we show that almost surely its boolean-width is Θ(log 2 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 * (2 O(log 4 n)) .
Lecture Notes in Computer Science, 2009
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 + 2 3k k)). We show that for any graph the square root of its boolean-width is never more than its rank-width. We also exhibit a class of graphs, the Hsugrids, having the property that a Hsu-grid on Θ(n 2) vertices has boolean-width Θ(log n) and tree-width, branch-width, clique-width and rank-width Θ(n). Moreover, any optimal rank-decomposition of such a graph will have boolean-width Θ(n) , i.e. exponential in the optimal boolean-width.
Fast algorithms for vertex subset and vertex partitioning problems on graphs of low boolean-width
We consider the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. Boolean-width is similar to rankwidth, 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. It compares well to the other four well-known width parameters tree-width, branch-width, clique-width, and rank-width: for many graph classes boolean-width is bounded whereas tree-width and branch-width are unbounded; for some graph classes boolean-width has been shown to be exponentially smaller than any of the other four; for arbitrary graphs, boolean-width is never larger than branchwidth (except for extreme values of zero and one), nor tree-width plus one, nor clique-width, and has been shown to be at least smaller than the square of rank-width. Boolean-width has been shown to be a very natural parameter to consider when solving Maximum Independent Set and Minimum Dominating Set using a divide-and-conquer approach. In this paper we investigate which are the graph problems having the same behaviour, and extend them to a large class of NP-hard vertex subset and vertex partitioning problems by giving algorithms that are FPT when parameterized by either boolean-width, rank-width or clique-width, with runtime single exponential in either parameter if given the pertinent optimal decomposition.
Discrete Applied Mathematics, 2016
dedicated to the second workshop in the series, held in 2005 in Prague, Czech Republic; 145-2 (2005) dedicated to the first workshop, held in 2001 in Barcelona, Spain; and 54-2/3 (1994) dedicated to a workshop held in 1989 in Eugene, which in retrospect we view as workshop number zero in what has evolved to become the successful biannual GROW workshop series. This issue comprises 14 papers authored mainly, but not exclusively, by participants of the workshop. All submissions have been carefully refereed, and we thank all the referees for their hard work. True to the name of the workshop, the papers in the current special issue report on investigations in three areas of research: Graph classes, Optimization, and Width parameters. Due to the close interconnections among these areas, most of the papers fit into more than one of them. Based on their main focus, we introduce the papers in this issue in the corresponding three groups. The area of Graph Classes is represented by papers proving new structural properties of various graph classes and exploring algorithmic consequences of these properties. Bonomo, Grippo, Milanič, and Safe initiate the study of graph classes of power-bounded clique-width and give sufficient and necessary conditions for this property. Brignall, Lozin, and Stacho study bichain graphs that are a bipartite analog of split permutation graphs. They show that these graphs admit a simple geometric representation and have a universal element of quadratic order. Golovach, Heggernes, Kanté, Kratsch, and Villanger show that all minimal dominating sets of a chordal bipartite graph can be generated in incremental polynomial, hence output polynomial, time. Konagaya, Otachi, and Uehara present a polynomial-time algorithm for the subgraph isomorphism problem on several subclasses of perfect graphs. Mertzios and Zaks study a conjecture by Golumbic, Monma, and Trotter that states that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. They prove the conjecture for every graph that admits a tolerance representation with exactly one unbounded vertex. The Optimization section consists of papers that study computational complexity and algorithmic issues of various optimization problems on graphs. Corneil, Dusart, Habib, Mamcarz, and de Montgolfier consider the problem of the recognition of various kinds of orderings produced by graph searches. To this aim, they introduce a new framework in order to handle a broad variety of
Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications
Algorithmica
A Not-All-Equal (NAE) decomposition of a graph G is a decomposition of the vertices of G into two parts such that each vertex in G has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph G is a decomposition of the vertices of G into two parts A and B such that each vertex in the graph G has exactly one neighbor in part A. Among our results, we show that for a given graph G, if G does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether G has a 1-in-Degree decomposition. In sharp contrast, we prove that for every r, r ≥ 3, for a given r-regular bipartite graph G determining whether G has a 1-in-Degree decomposition is NP-complete. These complexity results have been especially useful in proving NP-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph G determining whether there is a vector in the null-space of the 0,1-adjacency matrix of G such that its entries belong to {±1, ±2} is NP-complete. Among other results, we introduce a new version of Planar 1-in-3 SAT and we prove that this version is also NP-complete. In consequence of this result, we show that for a given planar (3, 4)-semiregular graph G determining whether there is a vector in the null-space of the 0,1-incidence matrix of G such that its entries belong to {±1, ±2} is NP-complete.
Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem
2008
In this paper we study the problem of finding an induced subgraph of size at most k with minimum degree at least d for a given graph G, from the parameterized complexity perspective. We call this problem Minimum Subgraph of Minimum Degree ≥d (MSMD d ). For d = 2 it corresponds to finding a shortest cycle of the graph. Our main motivation to study this problem is its strong relation to Dense k-Subgraph and Traffic Grooming problems.
On the complexity of some subgraph problems
Discrete Applied Mathematics, 2009
We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NPcompleteness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(n log n)) time algorithm for such problems in graphs with vertex degree bounded by 3.
Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width
Theory of Computing Systems, 2000
Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of tree-width at most k, i.e., that have tree decompositions of width at most k, where k is xed, every decision or optimization problem expressible in monadic secondorder logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quanti cations. We develop applications to several classes of graphs that include cographs and are, like cographs, de ned by forbidding subgraphs with \too many" induced paths with four vertices.
Overview of some solved NP-complete problems in graph theory
In the theory of complexity, NP (nondeterministic polynomial time) is a set of decision problems in polynomial time to be resolved in the nondeterministic Turing machine. Equivalently, it is a set of problems whose solutions can be verified on a deterministic Turing machine in polynomial time. The importance of this class of decision problems is that it contains many interesting problems of search and optimization, where we want to know if there is a solution to the problem. The contents of this paper are now handled NP-complete problems in graph theory.
Polynomial-time recognition of clique-width ≤3 graphs
Discrete Applied Mathematics, 2012
Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of cliquewidth at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n 2 m)) to recognize graphs of clique-width at most 3.