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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors.
We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future.
We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.

Evripidis Bampis, Bruno Escoffier, Kevin Schewior, and Alexandre Teiller. Online Multistage Subset Maximization Problems. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bampis_et_al:LIPIcs.ESA.2019.11, author = {Bampis, Evripidis and Escoffier, Bruno and Schewior, Kevin and Teiller, Alexandre}, title = {{Online Multistage Subset Maximization Problems}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.11}, URN = {urn:nbn:de:0030-drops-111320}, doi = {10.4230/LIPIcs.ESA.2019.11}, annote = {Keywords: Multistage optimization, Online algorithms} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incured for transitioning from one state to another. In order to model such situations, Gupta et al. (ICALP 2014) and Eisenstat et al. (ICALP 2014) introduced a multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that simultaneously (i) have good quality on the time steps and (ii) as stable as possible. We focus on the multistage version of the Knapsack problem where we are given a time horizon t=1,2,...,T, and a sequence of knapsack instances I_1,I_2,...,I_T, one for each time step, defined on a set of n objects. In every time step t we have to choose a feasible knapsack S_t of I_t, which gives a knapsack profit. To measure the stability/similarity of two consecutive solutions S_t and S_{t+1}, we identify the objects for which the decision, to be picked or not, remains the same in S_t and S_{t+1}, giving a transition profit. We are asked to produce a sequence of solutions S_1,S_2,...,S_T so that the total knapsack profit plus the overall transition profit is maximized.
We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case (e.g.multistage Spanning Tree, or multistage Bipartite Perfect Matching). Then, we prove that there is no FPTAS for the problem even in the case where T=2, unless P=NP. Furthermore, we give a pseudopolynomial time algorithm for the case where the number of steps is bounded by a fixed constant and we show that otherwise the problem remains NP-hard even in the case where all the weights, profits and capacities are 0 or 1.

Evripidis Bampis, Bruno Escoffier, and Alexandre Teiller. Multistage Knapsack. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bampis_et_al:LIPIcs.MFCS.2019.22, author = {Bampis, Evripidis and Escoffier, Bruno and Teiller, Alexandre}, title = {{Multistage Knapsack}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.22}, URN = {urn:nbn:de:0030-drops-109664}, doi = {10.4230/LIPIcs.MFCS.2019.22}, annote = {Keywords: Knapsack, Approximation Algorithms, Multistage Optimization} }

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