Results 1  10
of
62
Finding branchdecompositions and rankdecompositions
, 2007
"... Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm w ..."
Abstract

Cited by 28 (1 self)
 Add to MetaCart
Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixedparameter tractable, that is, they run in time O(n 3) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branchdecomposition or a rankdecomposition of optimal width due to Oum and Seymour [Testing branchwidth. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixedparameter tractable.)
Digraph measures: Kelly decompositions, games and orderings
"... We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We i ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kellywidth, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting a number of potential applications including polynomialtime algorithms for NPcomplete problems on graphs of bounded Kellywidth, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connecti ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
The recognizability of sets of graphs is a robust property
"... Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorith ..."
Abstract

Cited by 13 (9 self)
 Add to MetaCart
Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of contextfree sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) contextfree graph grammars, to Vertex Replacement (VR) contextfree graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifierfree formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HRrecognizability and VRrecognizability coincide. The same combinatorial condition equates HRcontextfree and VRcontextfree sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures. 1
Solving #SAT Using Vertex Covers
, 2006
"... We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of form ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph ” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clusteringwidth. Our algorithm runs in uniform polynomial time on formulas with bounded clusteringwidth. It is known that the number of models of formulas with bounded cliquewidth, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clusteringwidth and the other parameters mentioned are incomparable: there are formulas with bounded clusteringwidth and arbitrarily large cliquewidth, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clusteringwidth and bounded cliquewidth, treewidth, and branchwidth.
Graphs of bounded rankwidth
 Princeton University
, 2005
"... We define rankwidth of graphs to investigate cliquewidth. Rankwidth is a complexity measure of decomposing a graph in a kind of treestructure, called a rankdecomposition. We show that graphs have bounded rankwidth if and only if they have bounded cliquewidth. It is unknown how to recognize g ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
We define rankwidth of graphs to investigate cliquewidth. Rankwidth is a complexity measure of decomposing a graph in a kind of treestructure, called a rankdecomposition. We show that graphs have bounded rankwidth if and only if they have bounded cliquewidth. It is unknown how to recognize graphs of cliquewidth at most k for fixed k> 3 in polynomial time. However, we find an algorithm recognizing graphs of rankwidth at most k, by combining following three ingredients. First, we construct a polynomialtime algorithm, for fixed k, that confirms rankwidth is larger than k or outputs a rankdecomposition of width at most f(k) for some function f. It was known that many hard graph problems have polynomialtime algorithms for graphs of bounded cliquewidth, however, requiring a given decomposition corresponding to cliquewidth (kexpression); we remove this requirement. Second, we define graph vertexminors which generalizes matroid minors, and prove that if {G1, G2,...} is an infinite sequence of graphs of bounded rankwidth,
Computing graph polynomials on graphs of bounded cliquewidth
 GraphTheoretic Concepts in Computer Science, 32nd International Workshop, WG 2006
"... Abstract. We discuss the complexity of computing various graph polynomials of graphs of fixed cliquewidth. We show that the chromatic polynomial, the matching polynomial and the twovariable interlace polynomial of a graph G of cliquewidth at most k with n vertices can be computed in time O(n f(k) ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
Abstract. We discuss the complexity of computing various graph polynomials of graphs of fixed cliquewidth. We show that the chromatic polynomial, the matching polynomial and the twovariable interlace polynomial of a graph G of cliquewidth at most k with n vertices can be computed in time O(n f(k)), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 k+2 for the chromatic polynomial. 1