scholarly journals Equitable Partition of Graphs into Independent Sets and Cliques

2019 ◽  
Author(s):  
Bruno Monteiro ◽  
Vinicius Dos Santos
Keyword(s):  
Polynomial Time ◽  
Independent Sets ◽  
Equitable Partition ◽  
Equitable Coloring ◽  
Vertex Set ◽  
Np Complete

A graph is (k, l) if its vertex set can be partitioned into k independent sets and l cliques. Deciding if a graph is (k, l) can be seen as a generalization of coloring, since deciding is a graph belongs to (k, 0) corresponds to deciding if a graph is k-colorable. A coloring is equitable if the cardinalities of the color classes differ by at most 1. In this paper, we generalize both the (k, l) and the equitable coloring problems, by showing that deciding whether a given graph can be equitably partitioned into k independent sets and l cliques is solvable in polynomial time if max(k, l) 2, and NP complete otherwise.

scholarly journals On 2-Colorability Problem for Hypergraphs with P_8-free Incidence Graphs

2019 ◽  
Vol 17 (2) ◽  
pp. 257-263
Author(s):  
Ruzayn Quaddoura
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Incidence Graph ◽  
Coloring Problem ◽  
Vertex Set ◽  
Np Complete

A 2-coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. The hypergraph 2- Coloring Problem is the question whether a given hypergraph is 2-colorable. It is known that deciding the 2-colorability of hypergraphs is NP-complete even for hypergraphs whose hyperedges have size at most 3. In this paper, we present a polynomial time algorithm for deciding if a hypergraph, whose incidence graph is P_8-free and has a dominating set isomorphic to C_8, is 2-colorable or not. This algorithm is semi generalization of the 2-colorability algorithm for hypergraph, whose incidence graph is P_7-free presented by Camby and Schaudt.

scholarly journals Complexity of conditional colouring with given template

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peter J. Dukes ◽  
Steve Lowdon ◽  
Gary Macgillivray
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Vertex Set ◽  
International Audience ◽  
Partitioning Problem ◽  
Np Complete ◽  
Graph Families ◽  
General Collection

Graph Theory International audience We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

scholarly journals MPF Problem over Modified Medial Semigroup Is NP-Complete

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 571 ◽  
Author(s):  
Eligijus Sakalauskas ◽  
Aleksejus Mihalkovich
Keyword(s):  
Power Function ◽  
Polynomial Time ◽  
Cryptographic Protocols ◽  
Binary Field ◽  
Dual Interpretation ◽  
Np Complete ◽  
Matrix Power ◽  
Necessary Constraints ◽  
One Way Function ◽  
Polynomial Time Reduction

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals The Complexity Landscape of Resource-Constrained Scheduling

2020 ◽  
Author(s):  
Robert Ganian ◽  
Thekla Hamm ◽  
Guillaume Mescoff
Keyword(s):  
Artificial Intelligence ◽  
Polynomial Time ◽  
Project Scheduling ◽  
Scheduling Problem ◽  
Comprehensive Understanding ◽  
Resource Constrained ◽  
Special Cases ◽  
Project Scheduling Problem ◽  
Np Complete ◽  
New Algorithms

The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

scholarly journals Hypergraph domination and strong independence

2009 ◽  
Vol 3 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Bibin Jose ◽  
Zsolt Tuza
Keyword(s):  
Dominating Set ◽  
Independent Sets ◽  
Dominating Sets ◽  
Tight Bound ◽  
Open Problems ◽  
Odd Cycle ◽  
Vertex Set

We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus-Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D X such that, for every vertex x? X\D there exists an edge E ? E with x ? E and E?D ??.) As an example, it is shown that the tight bound ??(H)+??(H) ? n+2 holds in hypergraphs H = (X, E) of order n ? 6, where H is defined as H = (X, E) with E = {X\E | E ? E}, and ?? is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S X is strongly independent if |S?E|? 1 for all E ? E.).

EFFICIENT APPROACH FOR WEB SERVICES SELECTION WITH MULTI-QoS CONSTRAINTS

2008 ◽  
Vol 17 (03) ◽  
pp. 349-371 ◽  
Author(s):  
TAO HUANG ◽  
LEI LI ◽  
JUN WEI
Keyword(s):  
Web Services ◽  
Polynomial Time ◽  
High Performance ◽  
Feasible Solution ◽  
Search Space ◽  
User Preference ◽  
Composite Service ◽  
Efficient Approach ◽  
Complete Problem ◽  
Np Complete

With the increasing number of Web Services with similar or identical functionality, the non-functional properties of a Web Service will become more and more important. Hence, a choice needs to be made to determine which services are to participate in a given composite service. In general, multi-QoS constrained Web Services composition, with or without optimization, is a NP-complete problem on computational complexity that cannot be exactly solved in polynomial time. A lot of heuristics and approximation algorithms with polynomial- and pseudo-polynomial-time complexities have been designed to deal with this problem. However, these approaches suffer from excessive computational complexities that cannot be used for service composition in runtime. In this paper, we propose a efficient approach for multi-QoS constrained Web Services selection. Firstly, a user preference model was proposed to collect the user's preference. And then, a correlation model of candidate services are established in order to reduce the search space. Based on these two model, a heuristic algorithm is then proposed to find a feasible solution for multi-QoS constrained Web Services selection with high performance and high precision. The experimental results show that the proposed approach can achieve the expecting goal.

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