scholarly journals Continuous facility location on graphs

2021 ◽  
Author(s):  
Tim A. Hartmann ◽  
Stefan Lendl ◽  
Gerhard J. Woeginger
Keyword(s):  
Facility Location ◽  
Unit Length ◽  
Facility Location Problem ◽  
Covering Problem ◽  
Fixed Parameter Tractable ◽  
Unit Fractions ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Polynomially Solvable ◽  
Interior Points

AbstractWe study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range $$\delta >0$$ δ > 0 . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most $$\delta $$ δ from one of these facilities. We investigate this covering problem in terms of the rational parameter $$\delta $$ δ . We prove that the problem is polynomially solvable whenever $$\delta $$ δ is a unit fraction, and that the problem is NP-hard for all non unit fractions $$\delta $$ δ . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for $$\delta <3/2$$ δ < 3 / 2 , and it is W[2]-hard for $$\delta \ge 3/2$$ δ ≥ 3 / 2 .

scholarly journals Dispersing Obnoxious Facilities on a Graph

Algorithmica ◽  
2021 ◽  
Author(s):  
Alexander Grigoriev ◽  
Tim A. Hartmann ◽  
Stefan Lendl ◽  
Gerhard J. Woeginger
Keyword(s):  
Facility Location ◽  
Location Problem ◽  
Unit Length ◽  
Facility Location Problem ◽  
Np Hard ◽  
Obnoxious Facilities ◽  
Rational Parameter ◽  
Polynomially Solvable

AbstractWe study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

scholarly journals Facility Location Problem with Capacity Constraints: Algorithmic and Mechanism Design Perspectives

2020 ◽  
Vol 34 (02) ◽  
pp. 1806-1813 ◽  
Author(s):  
Haris Aziz ◽  
Hau Chan ◽  
Barton Lee ◽  
Bo Li ◽  
Toby Walsh
Keyword(s):  
Mechanism Design ◽  
Facility Location ◽  
Location Problem ◽  
Optimal Solution ◽  
Facility Location Problem ◽  
Solution Quality ◽  
Maximum Cost ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
The One

We consider the facility location problem in the one-dimensional setting where each facility can serve a limited number of agents from the algorithmic and mechanism design perspectives. From the algorithmic perspective, we prove that the corresponding optimization problem, where the goal is to locate facilities to minimize either the total cost to all agents or the maximum cost of any agent is NP-hard. However, we show that the problem is fixed-parameter tractable, and the optimal solution can be computed in polynomial time whenever the number of facilities is bounded, or when all facilities have identical capacities. We then consider the problem from a mechanism design perspective where the agents are strategic and need not reveal their true locations. We show that several natural mechanisms studied in the uncapacitated setting either lose strategyproofness or a bound on the solution quality %on the returned solution for the total or maximum cost objective. We then propose new mechanisms that are strategyproof and achieve approximation guarantees that almost match the lower bounds.

APPROXIMATE BLOCK SORTING

2006 ◽  
Vol 17 (02) ◽  
pp. 337-355 ◽  
Author(s):  
MEENA MAHAJAN ◽  
RAGHAVAN RAMA ◽  
VENKATESH RAMAN ◽  
S. VIJAYKUMAR
Keyword(s):  
Approximation Algorithm ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fixed Parameter Tractable ◽  
Identity Permutation ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Increasing Sequences ◽  
Better Than

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

scholarly journals Krausz dimension and its generalizations in special graph classes

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Olga Glebova ◽  
Yury Metelsky ◽  
Pavel Skums
Keyword(s):  
Open Problem ◽  
Chordal Graphs ◽  
Np Hard ◽  
Induced Subgraphs ◽  
Fixed Parameter Tractable ◽  
Graph Classes ◽  
Fixed Parameter ◽  
Minimum Number ◽  
International Audience ◽  
Split Graphs

Graph Theory International audience A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

Bounds on the partition dimension of convex polytopes

2020 ◽  
Vol 23 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Faisal Nadeem ◽  
Mohammad Azeem
Keyword(s):  
Combinatorial Optimization ◽  
Computer Science ◽  
Facility Location ◽  
Robot Navigation ◽  
Convex Polytopes ◽  
Pharmaceutical Chemistry ◽  
Resolving Set ◽  
Minimum Number ◽  
Partition Dimension ◽  
Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

scholarly journals Facility location selection for the furniture industry of Bangladesh: Comparative AHP and FAHP analysis

2021 ◽  
Vol 13 ◽  
pp. 184797902110308
Author(s):  
Md Nazmul Hasan Suman ◽  
Nagib MD Sarfaraj ◽  
Fuad Ahmed Chyon ◽  
Md Rafiul Islsm Fahim
Keyword(s):  
Facility Location ◽  
Raw Materials ◽  
Fuzzy Ahp ◽  
Facility Location Problem ◽  
Decision Makers ◽  
Energy Availability ◽  
Furniture Industry ◽  
Data Set ◽  
Location Selection ◽  
Hierarchy Process

The furniture industry is growing to a great extent in Bangladesh. Many market researchers believe that the industry has enormous potentiality. However, the expansion of this industry may face complexities within a few years. Due to the wrong selection of facilities, many organizations failed to earn profit as expected. It also needs a large investment. Selecting a suitable place for a new facility is going to be the biggest question of upcoming years. This study aimed to analyze Bangladesh’s furniture industry, address the facility location problem, and provide a constructive solution to the decision-makers. In this study, seven criteria were considered: availability of raw materials, transportations, skilled labor, proximity to customers, energy availability, economic zone facility, and environmental impact, and five ideal locations or alternatives: Khulna, Chattogram, Bogura, Gazipur, and Manikganj. Thirty-four experts took part in the survey to analyze the significant criteria for selecting a furniture industry’s facility location and alternatives or potential locations for the facility. The Analytical Hierarchy Process (AHP) and Fuzzy AHP methods (FAHP), two MCDM techniques, were used to analyze the data set. A sensitivity analysis was done to determine the model’s robustness for any critical changes in the real world. The result showed that ‘energy availability is the most significant criterion to select a facility location for the furniture industry, where it got 35.1% criteria weight in AHP and 33.9% in FAHP. ‘Chattogram’ was selected as the most suitable place containing 33.74% normalized weight in AHP and 33.81% normalized weight in FAHP.

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