TY - GEN

T1 - LP-Based Algorithms for Multistage Minimization Problems

AU - Bampis, Evripidis

AU - Escoffier, Bruno

AU - Kononov, Alexander

N1 - Funding Information: Acknowledgement. For the first two authors, this work was partially funded by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR). The research of the third author was partially supported by RFBR grant 20-07-00458. Publisher Copyright: © 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - We consider a multistage framework introduced recently (Eisenstat et al., Gupta et al., both in ICALP 2014), where given a time horizon t= 1, 2, …, T, the input is a sequence of instances of a (static) combinatorial optimization problem I1, I2, …, IT, (one for each time step), and the goal is to find a sequence of solutions S1, S2, …, ST (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. For several polynomial-time solvable problems, such as Minimum Cost Perfect Matching, the multistage variant becomes hard to approximate (even for two time steps for Minimum Cost Perfect Matching). In this paper, we study multistage variants of some important discrete minimization problems (including Minimum Cut, Vertex Cover, Prize-Collecting Steiner Tree, Prize-Collecting Traveling Salesman). We focus on the natural question of whether linear-programming-based methods may help in developing good approximation algorithms in this framework. We first introduce a new two-threshold rounding scheme, tailored for multistage problems. Using this rounding scheme we propose a 3.53-approximation algorithm for a multistage variant of the Prize-Collecting Steiner Tree problem, and a 3.034-approximation algorithm for a multistage variant of the Prize-Collecting Metric Traveling Salesman problem. Then, we show that Min Cut remains polytime solvable in its multistage variant, and Vertex Cover remains 2-approximable, as particular cases of a more general statement which easily follows from the work of (Hochbaum, EJOR 2002) on monotone and IP2 problems.

AB - We consider a multistage framework introduced recently (Eisenstat et al., Gupta et al., both in ICALP 2014), where given a time horizon t= 1, 2, …, T, the input is a sequence of instances of a (static) combinatorial optimization problem I1, I2, …, IT, (one for each time step), and the goal is to find a sequence of solutions S1, S2, …, ST (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. For several polynomial-time solvable problems, such as Minimum Cost Perfect Matching, the multistage variant becomes hard to approximate (even for two time steps for Minimum Cost Perfect Matching). In this paper, we study multistage variants of some important discrete minimization problems (including Minimum Cut, Vertex Cover, Prize-Collecting Steiner Tree, Prize-Collecting Traveling Salesman). We focus on the natural question of whether linear-programming-based methods may help in developing good approximation algorithms in this framework. We first introduce a new two-threshold rounding scheme, tailored for multistage problems. Using this rounding scheme we propose a 3.53-approximation algorithm for a multistage variant of the Prize-Collecting Steiner Tree problem, and a 3.034-approximation algorithm for a multistage variant of the Prize-Collecting Metric Traveling Salesman problem. Then, we show that Min Cut remains polytime solvable in its multistage variant, and Vertex Cover remains 2-approximable, as particular cases of a more general statement which easily follows from the work of (Hochbaum, EJOR 2002) on monotone and IP2 problems.

KW - Approximation algorithms

KW - LP-rounding

KW - Multistage optimization

UR - http://www.scopus.com/inward/record.url?scp=85112297864&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-80879-2_1

DO - 10.1007/978-3-030-80879-2_1

M3 - Conference contribution

AN - SCOPUS:85112297864

SN - 9783030808785

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 15

BT - Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers

A2 - Kaklamanis, Christos

A2 - Levin, Asaf

PB - Springer Science and Business Media Deutschland GmbH

T2 - 18th International Workshop on Approximation and Online Algorithms, WAOA 2019

Y2 - 9 September 2020 through 10 September 2020

ER -