On some subclasses of circular-arc catch digraphs

Catch digraphs was introduced by Hiroshi Maehera in 1984 as an analog of intersection graphs where a family of pointed sets represents a digraph. After that Prisner continued his research particularly on interval catch digraphs by characterizing them diasteroidal triple free. It has numerous applications in the field of real world problems like network technology and telecommunication operations. In this article we introduce a new class of catch digraphs, namely circular-arc catch digraphs. The definition is same as interval catch digraph, only the intervals are replaced by circular-arcs here. We present the characterization of proper circular-arc catch digraphs, which is a natural subclass of circular-arc catch digraphs where no circular-arc is contained in other properly. We do the characterization by introducing a concept monotone circular ordering for the vertices of the augmented adjacency matrices of it. Next we find that underlying graph of a proper oriented circular-arc catch digraph is a proper circular-arc graph. Also we characterize proper oriented circular-arc catch digraphs by defining a certain kind of circular vertex ordering of its vertices. Another interesting result is to characterize oriented circular-arc catch digraphs which are tournaments in terms of forbidden subdigraphs. Further we study some properties of an oriented circular-arc catch digraph. In conclusion we discuss the relations between these subclasses of circular-arc catch digraphs

Toward a Dichotomy for Approximation of H-Coloring

Given two (di)graphs G, H and a cost function c:V(G) x V(H) -> Q_{>= 0} cup {+infty}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f:V(G)-> V(H) (a.k.a H-coloring) that minimizes sum limits_{v in V(G)}c(v,f(v)). The complexity of exact minimization of this problem is well understood [Pavol Hell and Arash Rafiey, 2012], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor. In terms of digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph (i.e. symmetric digraph). For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: - Dichotomy for Graphs: MinHOM(H) has a 2|V(H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a bi-arc digraph; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of |V(H)|

Parametric classification of directed acyclic graphs, A

2017 Summer.Includes bibliographical references.We consider four NP-hard optimization problems on directed acyclic graphs (DAGs), namely, max clique, min coloring, max independent set and min clique cover. It is well-known that these four problems can be solved in polynomial time on transitive DAGs. It is also known that there can be no polynomial O(n1-ϵ)-approximation algorithms for these problems on the general class of DAGs unless P = NP. We propose a new parameter, β, as a measure of departure from transitivity for DAGs. We define β to be the number of vertices in a longest path in a DAG such that there is no edge from the first to the last vertex of the path, and 2 if the graph is transitive. Different values of β define a hierarchy of classes of DAGs, starting with the class of transitive DAGs. We give a polynomial time algorithm for finding a max clique when β is bounded by a fixed constant. The algorithm is exponential in β, but we also give a polynomial β-approximation algorithm. We prove that the other three decision problems are NP-hard even for β ≥ 4 and give polynomial algorithms with approximation bounds of β or better in each case. Furthermore, generalizing the definition of quasi-transitivity introduced by Ghouilà-Houri, we define β-quasi-transitivity and prove a more generalized version their theorem relating quasi-transitive orientation and transitive orientation

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for $k=7$ colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

Generalizations of comparability graphs

2022 Summer.Includes bibliographical references.In rational decision-making models, transitivity of preferences is an important principle. In a transitive preference, one who prefers x to y and y to z must prefer x to z. Many preference relations, including total order, weak order, partial order, and semiorder, are transitive. As a preference which is transitive yet not all pairs of elements are comparable, partial orders have been studied extensively. In graph theory, a comparability graph is an undirected graph which connects all comparable elements in a partial order. A transitive orientation is an assignment of direction to every edge so that the resulting directed graph is transitive. A graph is transitive if there is such an assignment. Comparability graphs are a class of graphs where clique, coloring, and many other optimization problems are solved by polynomial algorithms. It also has close connections with other classes of graphs, such as interval graphs, permutation graphs, and perfect graphs. In this dissertation, we define new measures for transitivity to generalize comparability graphs. We introduce the concept of double threshold digraphs together with a parameter λ which we define as our degree of transitivity. We also define another measure of transitivity, β, as the longest directed path such that there is no edge from the first vertex to the last vertex. We present approximation algorithms and parameterized algorithms for optimization problems and demonstrate that they are efficient for "almost-transitive" preferences

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

Twin-width I: tractable FO model checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|\phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|\phi|) \cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarsk\'y et al. [FOCS '15].Comment: 49 pages, 9 figure