Covering Minimal Separators and Potential Maximal Cliques in $P_t$-Free Graphs

A graph is called $P_t$-free if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geqslant 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. In both cases, the algorithms rely on deep structural insights into the considered graph classes. One of the main tools in the algorithms for MWIS in $P_5$-free graphs~[SODA 2014] and in $P_6$-free graphs~[SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to $P_7$-free graphs and is false in $P_8$-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.

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Approximation algorithms for maximum weight k-coverings of graphs by packings

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500993 ◽

2021 ◽

pp. 2150099

Author(s):

Fanica Gavril ◽

Mordechai Shalom ◽

Shmuel Zaks

Keyword(s):

Approximation Algorithms ◽

Independent Set ◽

Connected Component ◽

Maximum Weight ◽

Induced Subgraph ◽

Circle Graphs ◽

Vertex Set ◽

Induced Matchings ◽

Closed Neighborhood ◽

Vertex Disjoint

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

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From SAT to Maximum Independent Set: A New Approach to Characterize Tractable Classes

10.29007/c9f6 ◽

2018 ◽

Author(s):

Yazid Boumarafi ◽

Lakhdar Sais ◽

Yakoub Salhi

Keyword(s):

Independent Set ◽

Conjunctive Normal Form ◽

Propositional Satisfiability ◽

Maximum Independent Set ◽

Minimal Models ◽

New Approach ◽

Graph Classes ◽

Polynomial Time Algorithms ◽

Maximum Independent Set Problem

In this paper, we propose a new approach for defining tractable classes in propositional satisfiability problem (in short SAT). The basic idea consists in transforming SAT instances into instances of the problem of finding a maximum independent set. In this context, we only consider propositional formulæ in conjunctive normal form where each clause is either positive or binary negative. Tractable classes are obtained from existing polynomial time algorithms of the problem of finding a maximum independent set in the case of different graph classes, such as claw-free graphs and perfect graphs. We show, in particular, that the pigeonhole principle belongs to one of the defined tractable classes. Furthermore, we propose a characterization of the minimal models in the largest considered fragment based on the maximum independent set problem.

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Classes of graphs with restricted interval models

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.263 ◽

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.

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New Instances for Maximum Weight Independent Set From a Vehicle Routing Application

SN Operations Research Forum ◽

10.1007/s43069-021-00084-x ◽

2021 ◽

Vol 2 (4) ◽

Author(s):

Yuanyuan Dong ◽

Andrew V. Goldberg ◽

Alexander Noe ◽

Nikos Parotsidis ◽

Mauricio G. C. Resende ◽

...

Keyword(s):

Vehicle Routing ◽

Real World ◽

Vehicle Routing Problem ◽

Independent Set ◽

Maximum Weight ◽

Routing Problem ◽

Independent Set Problem ◽

Maximum Weight Independent Set ◽

Large Size

AbstractWe present a set of new instances of the maximum weight independent set problem. These instances are derived from a real-world vehicle routing problem and are challenging to solve in part because of their large size. We present instances with up to 881 thousand nodes and 383 million edges.

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