scholarly journals Object Reachability via Swaps along a Line

2019 ◽  
Vol 33 ◽  
pp. 2037-2044 ◽  
Author(s):  
Sen Huang ◽  
Mingyu Xiao
Keyword(s):  
Social Network ◽  
Open Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Resources Allocation ◽  
Weak Ties ◽  
Initial Endowment ◽  
Single Object ◽  
Star Network ◽  
Polynomially Solvable

The HOUSING MARKET problem is a widely studied resources allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We consider the problem whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie is allowed). This problem is polynomially solvable in a star-network and NPcomplete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Furthermore, we show that the problem on a path will become NP-hard when the preferences of the agents are weak (ties are allowed).

THE PROBLEM OF PARTITIONING WITH DUPLICATIONS AND ITS APPLICATIONS

1994 ◽  
Vol 03 (03) ◽  
pp. 395-405
Author(s):  
J. HARALAMBIDES ◽  
S. TRAGOUDAS
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vlsi Layout ◽  
Optimal Polynomial ◽  
Partitioning Problem ◽  
Polynomially Solvable ◽  
The Cost ◽  
Parallel Graph ◽  
Special Case

The problem of partitioning the elements of a graph G=(V, E) into two equal size sets A and B that share at most d elements such that the total number of edges (u, v), u∈A−B, v∈B−A is minimized, arises in the areas of Hypermedia Organization, Network Integrity, and VLSI Layout. We formulate the problem in terms of element duplication, where each element c∈A∩B is substituted by two copies c′∈A and c″∈B As a result, edges incident to c′ or c″ need not count in the cost of the partition. We show that this partitioning problem is NP-hard in general, and we present a solution which utilizes an optimal polynomial time algorithm for the special case where G is a series-parallel graph. We also discuss special other cases where the partitioning problem or variations are polynomially solvable.

scholarly journals Finding a Shortest Odd Hole

2021 ◽  
Vol 17 (2) ◽  
pp. 1-21
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t , there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t . In this article, we give an algorithm that finds a shortest odd hole, if one exists.

A Pseudo-Polynomial Time Algorithm for a Special Multiobjective Order Picking Problem

2015 ◽  
Vol 14 (05) ◽  
pp. 1111-1128 ◽  
Author(s):  
Özgür Özpeynirci ◽  
Cansu Kandemir
Keyword(s):  
Polynomial Time ◽  
Exact Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Picking ◽  
Special Design ◽  
Nondominated Points ◽  
Polynomially Solvable ◽  
Nondominated Point ◽  
Special Case

In this study, we work on the order picking problem (OPP) in a specially designed warehouse with a single picker. Ratliff and Rosenthal [Operations Research31(3) (1983) 507–521] show that the special design of the warehouse and use of one picker lead to a polynomially solvable case. We address the multiobjective version of this special case and investigate the properties of the nondominated points. We develop an exact algorithm that finds any nondominated point and present an illustrative example. Finally we conduct a computational test and report the results.

scholarly journals Partitioning a split graph into induced subgraphs isomorphic to the path of order 3.

2019 ◽  
Vol 55 (1) ◽  
pp. 32-49
Author(s):  
O. I. Duginov
Keyword(s):  
Computational Complexity ◽  
Efficient Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Team Formation ◽  
Urgent Problem ◽  
Induced Subgraphs ◽  
Split Graph ◽  
Vertex Set ◽  
Polynomially Solvable

The study of the computational complexity of problems on graphs is an urgent problem. We show that the problem of deciding whether the vertex set of a given split graph of order 3n can be partitioned into induced subgraphs isomorphic to P3 is a polynomially solvable problem. We develop a polynomial-time algorithm based on the method of augmenting graphs. The developed efficient algorithm can be used for solving team formation problems.

scholarly journals The graph isomorphism problem on geometric graphs

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Ryuhei Uehara
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Polynomial Hierarchy ◽  
Intersection Graphs ◽  
Graph Isomorphism Problem ◽  
Graph Classes

Special issue PRIMA 2013 International audience The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM

2009 ◽  
Vol 20 (03) ◽  
pp. 479-499
Author(s):  
SANGUTHEVAR RAJASEKARAN ◽  
VAMSI KUNDETI
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Cospectral Graphs ◽  
Graph Isomorphism Problem ◽  
Spectra Of Graphs ◽  
New Construction

The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prove that our algorithms will work on all possible instances. In this paper, we also give a new construction for cospectral graphs.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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