Shortest Augmenting Paths for Online Matchings on Trees

Bosek, Bartłomiej; Leniowski, Dariusz; Sankowski, Piotr; Zych-Pawlewicz, Anna

doi:10.1007/s00224-017-9838-x

Cited by 3 publications

(3 citation statements)

References 17 publications

(19 reference statements)

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“…+ r τ,j+2κ−2 ). (7) Taking together Equations ( 4)- (7) and Subclaim E (where a = r τ,l , b = r τ,l+1 + . .…”

Section: Appendix D Maintaining the Invariantsmentioning

confidence: 99%

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Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

Bosek¹,

Zych-Pawlewicz²

2022

Preprint

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In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals is allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of O(k 7 log n) while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [9] in terms of both k and n. We complement this result by showing the lower bound of Ω(n) on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented.As a byproduct of independent interest we include a new result on coloring proper circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs A. We show that if there is a set A of non-intersecting unit arcs of size L 2 − 1 such that A ∪ A does not contain L + 1 arcs intersecting in one point, then it is possible to color A with L colors. This complements the work on unit circular arc coloring [2,28,29], which specifies sufficient conditions needed to color A with L + 1 colors or more.

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“…+ r τ,j+2κ−2 ). (7) Taking together Equations ( 4)- (7) and Subclaim E (where a = r τ,l , b = r τ,l+1 + . .…”

Section: Appendix D Maintaining the Invariantsmentioning

confidence: 99%

“…This question has been recenly receiving increasing attention, and the positive answer has been obtained for a large number of problems [4,6,7,8,15,23,21,1,27,9]. In particular Barba et al applies the recourse budget model to coloring general graphs [1].…”

Section: Introductionmentioning

confidence: 99%

Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

Bosek¹,

Zych-Pawlewicz²

2022

Preprint

Self Cite

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“…They also show that if the bipartite graph remains a forest, there exists an algorithm (not SAP) with O(n log n) replacements, and a matching lower bound. Bosek, Leniowski, Sankowski and Zych later analyzed the SAP protocol for forests, giving an upper bound of O(n log 2 n) replacements [6], later improved to the optimal O(n log n) total replacements [7]. For general bipartite graphs, no analysis of SAP is known that shows better than the trivial O(n 2 ) total replacements.…”

Section: Previous Workmentioning

confidence: 99%

Online Bipartite Matching with Amortized Replacements

Bernstein¹,

Holm²,

Rotenberg³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω(log n) lower bound. The previous best strategy known achieved amortized O( √ n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log 2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices.We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O(min{L log 2 n, √ n log n}) reassignments. We also show that this is close to tight because Ω(min{L, √ n}) reassignments can be necessary.

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An acceptability index based approach for solving shortest path problem on a network with interval weights

Ebrahimnejad

2021

RAIRO-Oper. Res.

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Based on the acceptability index for comparison of any two imprecise values, efficient algorithms have been proposed in the literature for solving shortest path (SP) problem when the weights of connected arcs in a transportation network are represented as interval numbers. In this study, a generalized Dijkstra algorithm is proposed to handle the SP problem with interval weights. Here it is shown that once the acceptability index is chosen, the interval SP problem is converted into crisp one, which is easily solved by the standard SP algorithms. The main contribution here is the reduction of the computational complexity of the existing algorithm for solving interval SP problem. To show the advantages of the proposed algorithm over existing algorithm the numerical example presented in literature is solved using the proposed algorithm and the obtained results are discussed. Moreover, an small sized telecommunication network is provided to illustrate the potential application of the proposed method. Finally, the practical relevance of the proposed algorithm is evaluated by means of a large scale pilot case where a pharmaceutical shipment between the cities in Iran should be transported.

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Shortest Augmenting Paths for Online Matchings on Trees

Cited by 3 publications

References 17 publications

Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

Online Bipartite Matching with Amortized Replacements

An acceptability index based approach for solving shortest path problem on a network with interval weights

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