# Changing Bases: Multistage Optimization for Matroids and Matchings

@inproceedings{Gupta2014ChangingBM,
title={Changing Bases: Multistage Optimization for Matroids and Matchings},
author={Anupam Gupta and Kunal Talwar and Udi Wieder},
booktitle={ICALP},
year={2014}
}
• Published in ICALP 2014
• Computer Science, Mathematics
This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge is to continually maintain near-optimal solutions to an underlying optimization problem, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an… Expand
Multistage Matchings
• Computer Science
• SWAT
• 2018
14 We consider a multistage version of the Perfect Matching problem which models the 15 scenario where the costs of edges change over time and we seek to obtain a solution that achieves 16 low totalExpand
Approximating Multistage Matching Problems
• Computer Science
• IWOCA
• 2021
It is shown that multistage perfect matching problems are NP-hard even in very restricted scenarios, and new approximation algorithms and methods are proposed to transfer results between different problem variants without loosing approximation guarantees. Expand
Parameterized Algorithms for Diverse Multistage Problems
• Computer Science
• ESA
• 2021
This work introduces a framework allowing it to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. Expand
LP-based algorithms for multistage minimization problems
• Computer Science, Mathematics
• WAOA
• 2020
A new two-threshold rounding scheme, tailored for multistage problems, is introduced and it is shown that this rounding scheme gives a 2$f$-approximation algorithm for the multistages variant of the f-Set Cover problem, where each element belongs to at most f sets. Expand
Multistage Knapsack
• Computer Science, Mathematics
• MFCS
• 2019
A PTAS is proposed for the Multistage Knapsack problem and it is proved that there is no FPTAS for the problem even in the case where T=2, unless P=NP, and a pseudopolynomial time algorithm is given for the cases where the number of steps is bounded by a fixed constant. Expand
Multistage Problems on a Global Budget
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 2021
This work studies the different (time) layers of a temporal graph, and finds that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally easier than the ones of polynomial-time solvable problemssuch as Matching. Expand
Multistage Vertex Cover
• Computer Science, Mathematics
• IPEC
• 2019
The goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that two vertex cover sets of every two consecutive layers differ not too much (specified by a given parameter). Expand
A Multistage View on 2-Satisfiability
It is proved that Multistage 2-SAT is NP-hard even in quite restricted cases and parameterized algorithms for Multistages are presented and proved to be asymptotically optimal. Expand
A Combinatorial Metrical Task System Problem Under the Uniform Metric
• Mathematics, Computer Science
• ALT
• 2016
It is shown that Weighted Marking algorithm still keeps $$O(\log n)$$ competitive ratio for the standard MTS problem with n states, and combining with known sampling techniques for combinatorial sets, WeightedMarking algorithm works efficiently for various classes of combinatorsial sets. Expand
A General Approach to Approximate Multistage Subgraph Problems
• Computer Science
• ArXiv
• 2021
This work presents a framework that provides a (1/ √ 2χ)-approximation algorithm for the 2-stage restriction of an MSP if the similarity of subsequent solutions is measured as the intersection cardinality and said MSP is preficient, i.e., the authors can efficiently find a single-stage solution that prefers some given subset. Expand

#### References

SHOWING 1-10 OF 30 REFERENCES
Infrastructure Leasing Problems
• Computer Science, Mathematics
• IPCO
• 2007
A general approach to solving a wide class of Steiner Tree leasing problems by showing a close connection between deterministic leasing problems and problems in multistage stochastic optimization. Expand
The power of deferral: maintaining a constant-competitive steiner tree online
• Computer Science, Mathematics
• STOC '13
• 2013
Showing that these dual radii cannot change too rapidly is the technical heart of the paper, and allows us to give a hard bound on the number of swaps per arrival, while maintaining a constant-competitive tree at all times. Expand
The Power of Recourse for Online MST and TSP
• Mathematics, Computer Science
• ICALP
• 2012
An algorithm is introduced that maintains a nearly optimal tree when given constant amortized budget and a promising proof technique is specified and conjecture a structural property of optimal solutions that would prove a constant competitive ratio with a single recourse action. Expand
A Theory and Algorithms for Combinatorial Reoptimization
• Mathematics, Computer Science
• LATIN
• 2012
A general model for combinatorial reoptimization is developed, encompassing classical objective functions as well as the goal of minimizing the transition cost from one solution to the other, and distinguishes here for the first time between classes of re Optimization problems, by their hardness status with respect to minimizing transition costs while guaranteeing a good approximation for the underlying optimization problem. Expand
A Greedy Heuristic for the Set-Covering Problem
• V. Chvátal
• Mathematics, Computer Science
• Math. Oper. Res.
• 1979
It turns out that the ratio between the two grows at most logarithmically in the largest column sum of A when all the components of cT are the same, which reduces to a theorem established previously by Johnson and Lovasz. Expand
Online Primal-Dual Algorithms for Covering and Packing
• Mathematics, Computer Science
• Math. Oper. Res.
• 2009
This work provides general deterministic primal-dual algorithms for online fractional covering and packing problems and also provides deterministic algorithms for several integral online covering andpacking problems. Expand
Offline and online facility leasing
• Computer Science, Mathematics
• Discret. Optim.
• 2013
This work modifications an O ( log n ) -competitive algorithm of Fotakis (2007) for the online facility location problem and reanalyzes his algorithm via the dual-fitting technique to prove that it achieves the O (log n ) competitive ratio. Expand
Dynamic Steiner Tree Problem
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 1991
This paper proposes a new problem called the dynamic Steiner tree problem, and it is shown that the worst-case performance for any algorithm is at least $\frac{1}{2}\lg n$ times the cost of an optimum solution with complete rearrangement. Expand
Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes
• Mathematics
• 2014
We consider the problem of maximizing a nonnegative submodular set function $f:2^N \rightarrow {\mathbb R}_+$ over a ground set $N$ subject to a variety of packing-type constraints includingExpand
Metrical Task Systems and the k-Server Problem on HSTs
• Mathematics, Computer Science
• ICALP
• 2010
The main technical contribution here is to extend many of these techniques to work directly on HSTs to obtain a refined guarantee for the unfair metrical task systems problem on an HST. Expand