scholarly journals Improving Vertex Cover as a Graph Parameter

2015 ◽  
Vol Vol. 17 no.2 (Discrete Algorithms) ◽  
Author(s):  
Robert Ganian
Keyword(s):  
Vertex Cover ◽  
Graph Structure ◽  
Fpt Algorithms ◽  
Graph Problems ◽  
Graph Classes ◽  
Wide Range ◽  
Dense Graphs ◽  
International Audience ◽  
Hard Problems ◽  
Tree Width

International audience Parameterized algorithms are often used to efficiently solve NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with graph problems which are hard to solve even when parameterized by tree-width; however, the drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a generalization of vertex cover called twin-cover and show that FPT algorithms exist for a wide range of difficult problems when parameterized by twin-cover. The advantage of twin-cover over vertex cover is that it imposes a lesser restriction on the graph structure and attains low values even on dense graphs. Apart from introducing the parameter itself, this article provides a number of new FPT algorithms parameterized by twin-cover with a special emphasis on solving problems which are not in FPT even when parameterized by tree-width. It also shows that MS1 model checking can be done in elementary FPT time parameterized by twin-cover and discusses the field of kernelization.

scholarly journals Smaller Cuts, Higher Lower Bounds

2021 ◽  
Vol 17 (4) ◽  
pp. 1-40
Author(s):  
Amir Abboud ◽  
Keren Censor-Hillel ◽  
Seri Khoury ◽  
Ami Paz
Keyword(s):  
Lower Bounds ◽  
Vertex Cover ◽  
Shortest Paths ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Graph Problems ◽  
All Pairs Shortest Paths ◽  
Dense Graphs ◽  
Hard Problems ◽  
A New Technique

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

scholarly journals Tropical Graph Parameters

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Nadia Labai ◽  
Johann Makowsky
Keyword(s):  
Polynomial Time ◽  
Finite Rank ◽  
Bounded Linear ◽  
Graph Classes ◽  
Connection Matrices ◽  
International Audience ◽  
Tree Width ◽  
Graph Parameters ◽  
Valued Graph ◽  
Second Order Logic

International audience Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matricesof fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption. Les matrices de connexion pour des fonctions sur les graphes à valeurs dans un corps ont été introduites par M. Freedman, L. Lovász and A. Schrijver (2007). Une fonctions sur les graphes ayant des matrices de connexion de rang fini peut être calculée en temps polynomial sur toute famille de graphes de largeur arborescente (”tree-width”) bornée. Nous introduisons des matrices de jointure (”join matrices”) qui généralisent les matrices deconnexion, et nous permettons aux fonctions sur les graphes de prendre leurs valeurs dans des semianneaux tropicaux réels. Nous montrons qu’une fonction sur les graphes ayant des matrices de jointure de rang fini peut être calculée en temps polynomial sur des graphes de largeur de clique (”clique-width”) bornée. Dans le cas des semi-anneaux commutatifs, cela reste vrai pour les graphes de largeur de clique linéaire bornée. B. Godlin, T. Kotek and J.A. Makowsky (2008) ont montré que certaines hypothèses de definissabilité en Logique du Second Ordre Monadique concernant desopérations sur les graphes entraine la finitude des rangs. Nous exhibons un ensemble non dénombrable d’opérations ayant une matrice de connexion et des matrices de jointure de rang fini. Cela démontre que l’hypothèse de rang fini est beaucoup plus faible que l’hypothèse de definissabilité.

scholarly journals Combinatorial Dominance Guarantees for Heuristic Algorithms

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Daniel Berend ◽  
Steven S. Skiena ◽  
Yochai Twitto
Keyword(s):  
Polynomial Time ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Approximation Schemes ◽  
Set Cover ◽  
Time Approximation ◽  
Analysis Techniques ◽  
Wide Range ◽  
International Audience

International audience An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals On edge-intersection graphs of k-bend paths in grids

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Therese Biedl ◽  
Michal Stern
Keyword(s):  
Conflict Resolution ◽  
Planar Graph ◽  
Bipartite Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Intersection Graphs ◽  
Grid Networks ◽  
Graph Classes ◽  
International Audience ◽  
Number Of Bends

International audience Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.

Faster FPT Algorithms for Deletion to Pairs of Graph Classes

2021 ◽  
pp. 314-326
Author(s):  
Ashwin Jacob ◽  
Diptapriyo Majumdar ◽  
Venkatesh Raman
Keyword(s):  
Fpt Algorithms ◽  
Graph Classes

scholarly journals Solving NP-hard problems in ‘almost trees’: Vertex cover

1985 ◽  
Vol 10 (1) ◽  
pp. 27-45 ◽  
Author(s):  
Don Coppersmith ◽  
Uzi Vishkin
Keyword(s):  
Vertex Cover ◽  
Np Hard ◽  
Hard Problems ◽  
Np Hard Problems

scholarly journals Convex Partitions of Graphs induced by Paths of Order Three

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
C. C. Centeno ◽  
S. Dantas ◽  
M. C. Dourado ◽  
Dieter Rautenbach ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Convex Sets ◽  
Graph Classes ◽  
Convex Partitions ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.

scholarly journals Disimplicial arcs, transitive vertices, and disimplicial eliminations

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Martiniano Eguia ◽  
Francisco Soulignac
Keyword(s):  
Efficient Algorithm ◽  
Bipartite Graphs ◽  
Space Complexity ◽  
Time And Space ◽  
Graph Classes ◽  
International Audience ◽  
Time And Space Complexity ◽  
Finite Posets

International audience In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A <i>diclique</i> of a digraph is a pair $V$ &rarr; $W$ of sets of vertices such that $v$ &rarr; $w$ is an arc for every $v$ &isin; $V$ and $w$ &isin; $W$. An arc $v$ &rarr; $w$ is <i>disimplicial</i> when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.

scholarly journals Finite Controllability for Ontology-Mediated Query Answering of CRPQ

2020 ◽  
Author(s):  
Diego Figueira ◽  
Santiago Figueira ◽  
Edwin Pin Baque
Keyword(s):  
Query Answering ◽  
Graph Structure ◽  
First Order ◽  
Underlying Graph ◽  
Graph Classes ◽  
Regular Path ◽  
Order Constraints ◽  
Simple Recursion ◽  
Regular Path Queries ◽  
Path Queries

Finite ontology mediated query answering (FOMQA) is the variant of ontology mediated query answering (OMQA) where the represented world is assumed to be finite, and thus only finite models of the ontology are considered. We study the property of finite-controllability, that is, whether FOMQA and OMQA are equivalent, for fragments of C2RPQ. C2RPQ is the language of conjunctive two-way regular path queries, which can be regarded as the result of adding simple recursion to Conjunctive Queries. For graph classes S, we consider fragments C2RPQ(S) of C2RPQ as the queries whose underlying graph structure is in S. We completely classify the finitely controllable and non-finitely controllable fragments under: inclusion dependencies, (frontier-)guarded rules, frontier-one rules (either with or without constants), and more generally under guarded-negation first-order constraints. For the finitely controllable fragments, we show a reduction to the satisfiability problem for guarded-negation first-order logic, yielding a 2EXPTIME algorithm (in combined complexity) for the corresponding (F)OMQA problem.

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