TY - GEN

T1 - On PTAS for the Geometric Maximum Connected k-Factor Problem

AU - Gimadi, Edward

AU - Rykov, Ivan

AU - Tsidulko, Oxana

N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2020.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We consider the Connected k-factor problem (k-CFP): given a complete edge-weighted n-vertex graph, the goal is to find a connected k-regular spanning subgraph of maximum or minimum total weight. The problem is called geometric, if the vertices of a graph correspond to a set of points in a normed space (formula presented) and the weight of an edge is the distance between its endpoints. The k-CFP is a natural generalization of the well-known Traveling Salesman Problem, which is equivalent to the 2-CFP. In this paper we complement the known (formula presented)-approximation algorithm for the maximum k-CFP from [Baburin et al., 2007] with an approximation algorithm for the geometric k-CFP, that guarantees a relative error (formula presented). Together these two algorithms form an asymptotically optimal algorithm for the geometric k-CFP with an arbitrary value of k in an arbitrary normed space of fixed dimension d. Finally, the asymptotically optimal algorithm can be easily transformed into a PTAS for the considered geometric problem.

AB - We consider the Connected k-factor problem (k-CFP): given a complete edge-weighted n-vertex graph, the goal is to find a connected k-regular spanning subgraph of maximum or minimum total weight. The problem is called geometric, if the vertices of a graph correspond to a set of points in a normed space (formula presented) and the weight of an edge is the distance between its endpoints. The k-CFP is a natural generalization of the well-known Traveling Salesman Problem, which is equivalent to the 2-CFP. In this paper we complement the known (formula presented)-approximation algorithm for the maximum k-CFP from [Baburin et al., 2007] with an approximation algorithm for the geometric k-CFP, that guarantees a relative error (formula presented). Together these two algorithms form an asymptotically optimal algorithm for the geometric k-CFP with an arbitrary value of k in an arbitrary normed space of fixed dimension d. Finally, the asymptotically optimal algorithm can be easily transformed into a PTAS for the considered geometric problem.

KW - Asymptotically optimal algorithm

KW - Connected k-factor problem

KW - Normed space

KW - NP-hard problem

KW - Polynomial time approximation scheme

UR - http://www.scopus.com/inward/record.url?scp=85078414202&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-38603-0_15

DO - 10.1007/978-3-030-38603-0_15

M3 - Conference contribution

AN - SCOPUS:85078414202

SN - 9783030386023

T3 - Communications in Computer and Information Science

SP - 194

EP - 205

BT - Optimization and Applications - 10th International Conference, OPTIMA 2019, Revised Selected Papers

A2 - Jaćimović, Milojica

A2 - Khachay, Michael

A2 - Malkova, Vlasta

A2 - Posypkin, Mikhail

PB - Springer Gabler

T2 - 10th International Conference on Optimization and Applications, OPTIMA 2019

Y2 - 30 September 2019 through 4 October 2019

ER -