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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Let G be a directed multi-graph on n vertices and m edges with a designated source vertex s and a designated sink vertex t. We study the (s,t)-cuts of capacity minimum+1 and as an important application of them, we give a solution to the dual edge sensitivity for (s,t)-mincuts - reporting the (s,t)-mincut upon failure or addition of any pair of edges.
Picard and Queyranne [Mathematical Programming Studies, 13(1):8-16, 1980] showed that there exists a directed acyclic graph (DAG) that compactly stores all minimum (s,t)-cuts of G. This structure also acts as an oracle for the single edge sensitivity of minimum (s,t)-cut. Dinitz and Nutov [STOC, pages 509-518, 1995] showed that there exists an 𝒪(n) size 2-level cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (s,t)-cuts, no such compact structures exist till date. We present the following structural and algorithmic results on minimum+1 (s,t)-cuts.
1) There exists a pair of DAGs of size O(m) that compactly store all minimum+1 (s,t)-cuts of G. Each minimum+1 (s,t)-cut appears as a (s,t)-cut in one of the 2 DAGs and is 3-transversal - it intersects any path in the DAG at most thrice.
2) There exists an O(n²) size data structure that, given a pair of vertices {u,v} which are not separated by an (s,t)-mincut, can determine in 𝒪(1) time if there exists a minimum+1 (s,t)-cut, say (A,B), such that {s,u} ∈ A and {v,t} ∈ B; the corresponding cut can be reported in 𝒪(|B|) time.
3) There exists an O(n²) size data structure that solves the dual edge sensitivity problem for (s,t)-mincuts. It takes 𝒪(1) time to report the value of a resulting (s,t)-mincut (A,B) and 𝒪(|B|) time to report the cut.
4) For the data structure problems addressed in (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems – all-pairs directed reachability problem, the dual edge sensitivity problem for (s,t)-mincuts, and 2× 2 maximum flow. Assuming the directed reachability hypothesis, this leads to Ω(n²) lower bounds on the space for the latter two problems.

Surender Baswana, Koustav Bhanja, and Abhyuday Pandey. Minimum+1 (s,t)-cuts and Dual Edge Sensitivity Oracle. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baswana_et_al:LIPIcs.ICALP.2022.15, author = {Baswana, Surender and Bhanja, Koustav and Pandey, Abhyuday}, title = {{Minimum+1 (s,t)-cuts and Dual Edge Sensitivity Oracle}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {15:1--15:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.15}, URN = {urn:nbn:de:0030-drops-163566}, doi = {10.4230/LIPIcs.ICALP.2022.15}, annote = {Keywords: mincut, maxflow, fault tolerant} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

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.

Surender Baswana, Shiv Gupta, and Till Knollmann. Mincut Sensitivity Data Structures for the Insertion of an Edge. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baswana_et_al:LIPIcs.ESA.2020.12, author = {Baswana, Surender and Gupta, Shiv and Knollmann, Till}, title = {{Mincut Sensitivity Data Structures for the Insertion of an Edge}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.12}, URN = {urn:nbn:de:0030-drops-128781}, doi = {10.4230/LIPIcs.ESA.2020.12}, annote = {Keywords: Mincut, Sensitivity, Data Structure} }

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

We present an algorithm for a fault tolerant Depth First Search (DFS) Tree in an undirected graph. This algorithm is drastically simpler than the current state-of-the-art algorithms for this problem, uses optimal space and optimal preprocessing time, and still achieves better time complexity. This algorithm also leads to a better time complexity for maintaining a DFS tree in a fully dynamic environment.

Surender Baswana, Shiv Gupta, and Ayush Tulsyan. Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{baswana_et_al:LIPIcs.MFCS.2019.65, author = {Baswana, Surender and Gupta, Shiv and Tulsyan, Ayush}, title = {{Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {65:1--65:16}, 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.2019.65}, URN = {urn:nbn:de:0030-drops-110096}, doi = {10.4230/LIPIcs.MFCS.2019.65}, annote = {Keywords: Depth first search, DFS, Dynamic graph algorithms, Fault tolerant} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G=(V,E) with n=|V| and m=|E|, and an integer value k\geq 1, there is an algorithm that computes in O(2^{k}n log^2 n) time for any set F of size at most k the strongly connected components of the graph G\F. The running time of our algorithm is almost optimal since the time for outputting the SCCs of G\F is at least \Omega(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2^{k} n^2).
Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest.

Surender Baswana, Keerti Choudhary, and Liam Roditty. An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{baswana_et_al:LIPIcs.ICALP.2017.72, author = {Baswana, Surender and Choudhary, Keerti and Roditty, Liam}, title = {{An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {72:1--72:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.72}, URN = {urn:nbn:de:0030-drops-74168}, doi = {10.4230/LIPIcs.ICALP.2017.72}, annote = {Keywords: Fault tolerant, Directed graph, Strongly connected components} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We present a fully dynamic algorithm for maintaining approximate
maximum weight matching in general weighted graphs. The algorithm maintains a matching M whose weight is at least 1/8 M^{*} where M^{*} is the weight of the maximum weight matching. The algorithm achieves an expected amortized O(log n log C) time per edge insertion or deletion, where C is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph.

Abhash Anand, Surender Baswana, Manoj Gupta, and Sandeep Sen. Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 257-266, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{anand_et_al:LIPIcs.FSTTCS.2012.257, author = {Anand, Abhash and Baswana, Surender and Gupta, Manoj and Sen, Sandeep}, title = {{Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {257--266}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.257}, URN = {urn:nbn:de:0030-drops-38648}, doi = {10.4230/LIPIcs.FSTTCS.2012.257}, annote = {Keywords: Matching, Dynamic Algorithm, Graph Algorithm} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Let $G=(V,E)$ be any undirected graph on $V$ vertices and
$E$ edges. A path $\textbf{P}$ between any two vertices $u,v\in V$ is said to be $t$-approximate shortest path if its length is at most $t$ times the length of the shortest path between $u$ and $v$.
We consider the problem of building a compact data structure for a
given graph $G$ which is capable of answering the following query for
any $u,v,z\in V$ and $t>1$.
\centerline{\em report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails}
We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size {\em nearly} match the size of their best static counterparts.

Neelesh Khanna and Surender Baswana. Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 513-524, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{khanna_et_al:LIPIcs.STACS.2010.2481, author = {Khanna, Neelesh and Baswana, Surender}, title = {{Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {513--524}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2481}, URN = {urn:nbn:de:0030-drops-24812}, doi = {10.4230/LIPIcs.STACS.2010.2481}, annote = {Keywords: Shortest path, distance, distance queries, oracle} }

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