Some polynomially solvable cases of the inverse ordered 1-median problem on trees

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3651-3664 ◽  
Author(s):  
Kien Nguyen
Keyword(s):  
Minimum Cost ◽  
Np Hard ◽  
Median Problem ◽  
Star Graphs ◽  
Polynomially Solvable ◽  
Np Hardness

We consider the problem of modifying the edge lengths of a tree at minimum cost such that a prespecified vertex become an ordered 1-median of the perturbed tree. We call this problem the inverse ordered 1-median problem on trees. Gassner showed in 2012 that the inverse ordered 1-median problem on trees is, in general, NP-hard. We, however, address some situations, where the corresponding inverse 1-median problem is polynomially solvable. For the problem on paths with n vertices, we develop an O(n3) algorithm based on a greedy technique. Furthermore, we prove the NP-hardness of the inverse ordered 1- median problem on star graphs and propose a quadratic algorithm that solves the inverse ordered 1-median problem on unweighted stars.

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.

Two-Machine Ordered Flow Shop Scheduling with Generalized Due Dates

2020 ◽  
Vol 37 (01) ◽  
pp. 1950032
Author(s):  
Myoung-Ju Park ◽  
Byung-Cheon Choi ◽  
Yunhong Min ◽  
Kyung Min Kim
Keyword(s):  
Flow Shop ◽  
Flow Shop Scheduling ◽  
Due Dates ◽  
Shop Scheduling ◽  
Processing Times ◽  
Specific Position ◽  
Polynomially Solvable ◽  
Definition Of ◽  
Tardy Jobs ◽  
Np Hardness

We consider a two-machine flow shop scheduling with two properties. The first is that each due date is assigned for a specific position different from the traditional definition of due dates, and the second is that a consistent pattern exists in the processing times within each job and each machine. The objective is to minimize maximum tardiness, total tardiness, or total number of tardy jobs. We prove the strong NP-hardness and inapproximability, and investigate some polynomially solvable cases. Finally, we develop heuristics and verify their performances through numerical experiments.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals Quasi-Popular Matchings, Optimality, and Extended Formulations

2021 ◽  
Author(s):  
Yuri Faenza ◽  
Telikepalli Kavitha
Keyword(s):  
Minimum Cost ◽  
Stable Marriage ◽  
Np Hard ◽  
Stable Marriage Problem ◽  
Polynomial Size ◽  
The Stable Marriage Problem ◽  
Matching Polytopes ◽  
The Cost ◽  
Popular Matching ◽  
Positive Results

Let [Formula: see text] be an instance of the stable marriage problem in which every vertex ranks its neighbors in a strict order of preference. A matching [Formula: see text] in [Formula: see text] is popular if [Formula: see text] does not lose a head-to-head election against any matching. Popular matchings generalize stable matchings. Unfortunately, when there are edge costs, to find or even approximate up to any factor a popular matching of minimum cost is NP-hard. Let [Formula: see text] be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most [Formula: see text] by paying the price of mildly relaxing popularity. Our main positive results are two bicriteria algorithms that find in polynomial time a “quasi-popular” matching of cost at most [Formula: see text]. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than [Formula: see text]. Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

Line-Constrained k-Median, k-Means, and k-Center Problems in the Plane

2016 ◽  
Vol 26 (03n04) ◽  
pp. 185-210
Author(s):  
Haitao Wang ◽  
Jingru Zhang
Keyword(s):  
Time Algorithm ◽  
Efficient Algorithms ◽  
Additional Constraint ◽  
Distance Measures ◽  
Np Hard ◽  
Center Problem ◽  
Median Problem ◽  
One Dimensional ◽  
The Given

The (weighted) [Formula: see text]-median, [Formula: see text]-means, and [Formula: see text]-center problems in the plane are known to be NP-hard. In this paper, we study these problems with an additional constraint that requires the sought [Formula: see text] facilities to be on a given line. We present efficient algorithms for various distance measures such as [Formula: see text]. We assume that all [Formula: see text] weighted points are given sorted by their projections on the given line. For [Formula: see text]-median, our algorithms for [Formula: see text] and [Formula: see text] metrics run in [Formula: see text] time and [Formula: see text] time, respectively. For [Formula: see text]-means, which is defined only on the squared [Formula: see text] distance, we give an [Formula: see text] time algorithm. For [Formula: see text]-center, our algorithms run in [Formula: see text] time for all three metrics, and in [Formula: see text] time for the unweighted version under [Formula: see text] and [Formula: see text] metrics. While our results for the [Formula: see text]-center problem are optimal, the results for the [Formula: see text]-median problem almost match the best algorithms for the corresponding one-dimensional problems.

An Ordered Flow Shop with Two Agents

2016 ◽  
Vol 33 (05) ◽  
pp. 1650037 ◽  
Author(s):  
Byung-Cheon Choi ◽  
Myoung-Ju Park
Keyword(s):  
Polynomial Time ◽  
Processing Time ◽  
Flow Shop ◽  
Fixed Number ◽  
Scheduling Problem ◽  
Np Hard ◽  
Two Agents ◽  
Agent Scheduling ◽  
Polynomially Solvable ◽  
Machine Problem

In this paper, we consider a two-agent scheduling problem in an [Formula: see text]-machine ordered flow shop where each agent is responsible for his own set of jobs and wishes to minimize the makespan. Since the problem is NP-hard, we develop a pseudo-polynomial time approach for the case with a fixed number of machines and investigate the conditions that make the problem polynomially solvable. Finally, we consider a three-machine problem with a special processing time structure, and demonstrate its polynomiality.

Complexity issues for the symmetric interval eigenvalue problem

2014 ◽  
Vol 13 (1) ◽  
Author(s):  
Milan Hladík
Keyword(s):  
Computational Complexity ◽  
Eigenvalue Problem ◽  
Relative Error ◽  
Symmetric Matrix ◽  
Approximation Error ◽  
Np Hard ◽  
Approximation Errors ◽  
The Matrix ◽  
Polynomially Solvable ◽  
Relative Approximation

AbstractWe study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.

Resource allocation problem under single resource assignment

2018 ◽  
Vol 52 (2) ◽  
pp. 371-382
Author(s):  
Sakib A. Mondal
Keyword(s):  
Resource Allocation ◽  
Minimum Cost ◽  
Allocation Problem ◽  
Np Hard ◽  
Resource Allocation Problem ◽  
Resource Assignment ◽  
Start Time ◽  
Finish Time ◽  
Time Period ◽  
Time Availability

We consider a NP-hard resource allocation problem of allocating a set of resources to meet demands over a time period at the minimum cost. Each resource has a start time, finish time, availability and cost. The objective of the problem is to assign resources to meet the demands so that the overall cost is minimum. It is necessary that only one resource contributes to the demand of a slot. This constraint will be referred to as single resource assignment (SRA) constraint. We would refer to the problem as the S_RA problem. So far, only 16-approximation to this problem is known. In this paper, we propose an algorithm with approximation ratio of 12.

scholarly journals The Inverse 1-Median Problem on Tree Networks with Variable Real Edge Lengths

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Longshu Wu ◽  
Joonwhoan Lee ◽  
Jianhua Zhang ◽  
Qin Wang
Keyword(s):  
Location Problem ◽  
Hamming Distance ◽  
Linear Time ◽  
Shopping Centers ◽  
Median Problem ◽  
Tree Networks ◽  
Total Cost ◽  
Negative Edge ◽  
Polynomial Time Algorithms ◽  
Np Hardness

Location problems exist in the real world and they mainly deal with finding optimal locations for facilities in a network, such as net servers, hospitals, and shopping centers. The inverse location problem is also often met in practice and has been intensively investigated in the literature. As a typical inverse location problem, the inverse 1-median problem on tree networks with variable real edge lengths is discussed in this paper, which is to modify the edge lengths at minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to the new edge lengths. First, this problem is shown to be solvable in linear time with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, the NP-hardness is proved under Hamming distance, and strongly polynomial time algorithms are presented underl1andl∞norms, respectively.

scholarly journals $k$-Colour Partitions of Acyclic Tournaments

10.37236/1902 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Paulo Barcia ◽  
J. Orestes Cerdeira
Keyword(s):  
Convex Hull ◽  
Minimum Cost ◽  
Hard Problem ◽  
Np Hard ◽  
Nonnegativity Constraints ◽  
Np Hard Problem ◽  
The Cost

Let $G_{1}$ be the acyclic tournament with the topological sort $0 < 1 < 2 < \dots < n < n+1$ defined on node set $N\cup \{0,n+1\}$, where $N=\{1,2,\dots,n\}$. For integer $k\geq 2$, let $G_{k}$ be the graph obtained by taking $k$ copies of every arc in $G_{1}$ and colouring every copy with one of $k$ different colours. A $k$-colour partition of $N$ is a set of $k$ paths from 0 to $n+1$ such that all arcs of each path have the same colour, different paths have different colours, and every node of $N$ is included in exactly one path. If there are costs associated with the arcs of $G_{k}$, the cost of a $k$-colour partition is the sum of the costs of its arcs. For determining minimum cost $k$-colour partitions we describe an $O(k^{2}n^{2k})$ algorithm, and show this is an NP-$hard$ problem. We also study the convex hull of the incidence vectors of $k$-colour partitions. We derive the dimension, and establish a minimal equality set. For $k>2$ we identify a class of facet inducing inequalities. For $k=2$ we show that these inequalities turn out to be equations, and that no other facet defining inequalities exists besides the trivial nonnegativity constraints.

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