scholarly journals Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

2021 ◽  
pp. 1-32
Author(s):  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Optimization Problem ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Np Hard ◽  
Polynomial Time Algorithms ◽  
Split Graphs ◽  
Np Complete ◽  
Cycle Path

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

scholarly journals Internet shopping optimization problem

2010 ◽  
Vol 20 (2) ◽  
pp. 385-390 ◽  
Author(s):  
JACEK B£A ZÿEWICZ ◽  
Mikhail Kovalyov ◽  
Jędrzej Musiał ◽  
Andrzej Urbanski ◽  
Adam Wojciechowski
Keyword(s):  
Polynomial Time ◽  
Optimization Problem ◽  
Partial Solution ◽  
The Internet ◽  
Internet Shopping ◽  
Single Product ◽  
Polynomial Time Algorithms ◽  
Special Cases ◽  
Multiple Item ◽  
Np Hardness

Internet shopping optimization problemA high number of Internet shops makes it difficult for a customer to review manually all the available offers and select optimal outlets for shopping. A partial solution to the problem is brought by price comparators which produce price rankings from collected offers. However, their possibilities are limited to a comparison of offers for a single product requested by the customer. The issue we investigate in this paper is a multiple-item multiple-shop optimization problem, in which total expenses of a customer to buy a given set of items should be minimized over all available offers. In this paper, the Internet Shopping Optimization Problem (ISOP) is defined in a formal way and a proof of its strong NP-hardness is provided. We also describe polynomial time algorithms for special cases of the problem.

scholarly journals Are Stable Instances Easy?

2012 ◽  
Vol 21 (5) ◽  
pp. 643-660 ◽  
Author(s):  
YONATAN BILU ◽  
NATHAN LINIAL
Keyword(s):  
Polynomial Time ◽  
Discrete Optimization ◽  
Optimization Problem ◽  
Discrete Optimization Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Hard Problems ◽  
Np Hard Problems

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

scholarly journals Equitable Coloring of Graphs. Recent Theoretical Results and New Practical Algorithms

2016 ◽  
Vol 26 (3) ◽  
pp. 281-295 ◽  
Author(s):  
Hanna Furmańczyk ◽  
Andrzej Jastrzębski ◽  
Marek Kubale
Keyword(s):  
Polynomial Time ◽  
Np Hard ◽  
Coloring Problem ◽  
Equitable Coloring ◽  
Practical Algorithms ◽  
Polynomial Time Algorithms ◽  
Sequencing And Scheduling ◽  
General Graphs ◽  
Theoretical Results

AbstractIn many applications in sequencing and scheduling it is desirable to have an underlaying graph as equitably colored as possible. In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.

scholarly journals On the Computation of Fully Proportional Representation

2013 ◽  
Vol 47 ◽  
pp. 475-519 ◽  
Author(s):  
N. Betzler ◽  
A. Slinko ◽  
J. Uhlmann
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Proportional Representation ◽  
Fixed Parameter Tractability ◽  
Np Hard ◽  
Winner Determination ◽  
Negative Results ◽  
Polynomial Time Algorithms ◽  
Fixed Parameter

We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

scholarly journals The resolving number of a graph Delia

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Clique Number ◽  
Metric Dimension ◽  
Np Hard ◽  
Arbitrary Graph ◽  
International Audience ◽  
Graph Parameters ◽  
Two Parameters

Graph Theory International audience We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

scholarly journals Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

2021 ◽  
Author(s):  
Yasaman KalantarMotamedi
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Hamiltonian Cycle ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rigorous Proof ◽  
Complete Problem ◽  
Time Solution ◽  
Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

scholarly journals Polynomial Time Algorithms for Variants of Graph Matching on Partial k-Trees

2016 ◽  
Vol 41 (3) ◽  
pp. 163-181
Author(s):  
Takayuki Nagoya
Keyword(s):  
Polynomial Time ◽  
Graph Matching ◽  
Graph Isomorphism ◽  
Exact Algorithms ◽  
Polynomial Time Algorithms ◽  
Np Complete

AbstractIn this paper, we deal with two variants of graph matching, the graph isomorphism with restriction and the prefix set of graph isomorphism. The former problem is known to be NP-complete, whereas the latter problem is known to be GI-complete. We propose polynomial time exact algorithms for these problems on partial k-trees.

scholarly journals Acyclic colourings of graphs with bounded degree

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Mieczyslaw Borowiecki ◽  
Anna Fiedorowicz ◽  
Katarzyna Jesse-Jozefczyk ◽  
Elzbieta Sidorowicz
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Cubic Graph ◽  
Bounded Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

RANK PROBLEMS FOR COMPOSITE TRANSFORMATIONS

1995 ◽  
Vol 05 (03) ◽  
pp. 309-316 ◽  
Author(s):  
PAVEL GORALČÍK ◽  
VÁCLAV KOUBEK
Keyword(s):  
Polynomial Time ◽  
Np Hard ◽  
Finite Set ◽  
Np Complete

Let (X, F) be a pair consisting of a finite set X and a set F of transformations of X, and, let <F> and F(l) denote, respectively, the semigroup generated by F and the part of <F> consisting of the transformations determined by a generator sequence of length no more than a given integer l. We show the following: • The problem whether or not, for a given pair (X, F) and a given integer r, there is an idempotent transformation of rank r in <F> is PSPACE-complete. • For each fixed r≥1, it is decidable in a polynomial time, for a given pair (X, F), whether or not <F> contains an idempotent transformation of rank r, and, if yes then a generator sequence of polynomial length composing to an idempotent transformation of rank r can be obtained in a polynomial time. • For each fixed r≥1, the problem whether or not, for a given (X, F) and l, there is an idempotent transformation of rank r in F(l) is NP-complete. • For each fixed r≥2, to decide, for a given (X, F), whether or not <F> contains a transformation of rank r is NP-hard.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

Sign in / Sign up

Export Citation Format

Share Document