@inproceedings{20159,
abstract = {{Let G = (V,E) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows. Build a compact data structure for G and a given set S ⊆ V of vertices that, on receiving any edge (x,y) ∈ S×S of positive capacity as query input, can efficiently report the set of all pairs from S× S whose mincut value increases upon insertion of the edge (x,y) to G. The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, Mathematical Programming Study, 13 (1980), 8-16). We present the following results for the single source and the all-pairs versions of this problem.
1) Single source: Given any designated source vertex s, there exists a data structure of size 𝒪(|S|) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(|S|).
2) All-pairs: There exists an 𝒪(|S|²) size data structure that can output all those pairs of vertices from S× S whose mincut value gets increased upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(k), where k is the number of pairs of vertices whose mincut increases.
For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of 𝒪(n) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result by Hariharan et al. (STOC 2007) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in a tree of 𝒪(n) size.}},
author = {{Baswana, Surender and Gupta, Shiv and Knollmann, Till}},
booktitle = {{28th Annual European Symposium on Algorithms (ESA 2020)}},
editor = {{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}},
isbn = {{978-3-95977-162-7}},
issn = {{1868-8969}},
keywords = {{Mincut, Sensitivity, Data Structure}},
pages = {{12:1--12:14}},
publisher = {{Schloss Dagstuhl -- Leibniz-Zentrum für Informatik}},
title = {{{Mincut Sensitivity Data Structures for the Insertion of an Edge}}},
doi = {{10.4230/LIPIcs.ESA.2020.12}},
volume = {{173}},
year = {{2020}},
}
@inproceedings{2484,
abstract = {{We study the classic bin packing problem in a fully-dynamic setting, where new items can arrive and old items may depart. We want algorithms with low asymptotic competitive ratio while repacking items sparingly between updates. Formally, each item i has a movement cost c_i >= 0, and we want to use alpha * OPT bins and incur a movement cost gamma * c_i, either in the worst case, or in an amortized sense, for alpha, gamma as small as possible. We call gamma the recourse of the algorithm. This is motivated by cloud storage applications, where fully-dynamic bin packing models the problem of data backup to minimize the number of disks used, as well as communication incurred in moving file backups between disks. Since the set of files changes over time, we could recompute a solution periodically from scratch, but this would give a high number of disk rewrites, incurring a high energy cost and possible wear and tear of the disks. In this work, we present optimal tradeoffs between number of bins used and number of items repacked, as well as natural extensions of the latter measure.}},
author = {{Feldkord, Björn and Feldotto, Matthias and Gupta, Anupam and Guruganesh, Guru and Kumar, Amit and Riechers, Sören and Wajc, David}},
booktitle = {{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}},
editor = {{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, Dániel and Sannella, Donald}},
isbn = {{978-3-95977-076-7}},
issn = {{1868-8969}},
location = {{Prag}},
pages = {{51:1--51:24}},
publisher = {{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}},
title = {{{Fully-Dynamic Bin Packing with Little Repacking}}},
doi = {{10.4230/LIPIcs.ICALP.2018.51}},
volume = {{107}},
year = {{2018}},
}