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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known.
The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters.
We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A.
Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth.

Tanmay Inamdar, Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability). In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{inamdar_et_al:LIPIcs.ESA.2023.63, author = {Inamdar, Tanmay and Lokshtanov, Daniel and Saurabh, Saket and Surianarayanan, Vaishali}, title = {{Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {63:1--63:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.63}, URN = {urn:nbn:de:0030-drops-187167}, doi = {10.4230/LIPIcs.ESA.2023.63}, annote = {Keywords: FPT Approximation, Minimum Bisection, Unbreakable Tree Decomposition, Treewidth} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In the Min k-Cut problem, the input is a graph G and an integer k. The task is to find a partition of the vertex set of G into k parts, while minimizing the number of edges that go between different parts of the partition. The problem is NP-complete, and admits a simple 3ⁿ⋅n^𝒪(1) time dynamic programming algorithm, which can be improved to a 2ⁿ⋅n^𝒪(1) time algorithm using the fast subset convolution framework by Björklund et al. [STOC'07]. In this paper we give an algorithm for Min k-Cut with running time 𝒪((2-ε)ⁿ), for ε > 10^{-50}. This is the first algorithm for Min k-Cut with running time 𝒪(cⁿ) for c < 2.

Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. Breaking the All Subsets Barrier for Min k-Cut. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 90:1-90:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2023.90, author = {Lokshtanov, Daniel and Saurabh, Saket and Surianarayanan, Vaishali}, title = {{Breaking the All Subsets Barrier for Min k-Cut}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {90:1--90:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.90}, URN = {urn:nbn:de:0030-drops-181422}, doi = {10.4230/LIPIcs.ICALP.2023.90}, annote = {Keywords: Exact algorithms, min k-cut, exponential algorithms, graph algorithms, k-way cut} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ℝ^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ∩ P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ≥ 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension.

Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, and Vaishali Surianarayanan. Anonymity-Preserving Space Partitions. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hebertjohnson_et_al:LIPIcs.ISAAC.2021.32, author = {H\'{e}bert-Johnson, \'{U}rsula and Sonar, Chinmay and Suri, Subhash and Surianarayanan, Vaishali}, title = {{Anonymity-Preserving Space Partitions}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.32}, URN = {urn:nbn:de:0030-drops-154654}, doi = {10.4230/LIPIcs.ISAAC.2021.32}, annote = {Keywords: Anonymity, Hitting Set, LP, Constant Approximation, Fixed-Parameter Tractable, Space Partitions, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

In the Dominating Set problem the input is a graph G and an integer k, the task is to determine whether there exists a vertex set S of size at most k so that every vertex not in S has at least one neighbor in S. We consider the parameterized complexity of the Dominating Set problem, parameterized by the solution size k, and the weak closure of the input graph G. Weak closure of graphs was recently introduced by Fox et al. [SIAM J. Comp. 2020 ] and captures sparseness and triadic closure properties found in real world graphs. A graph G is weakly c-closed if for every induced subgraph G' of G, there exists a vertex v ∈ V(G') such that every vertex u in V(G') which is non-adjacent to v has less than c common neighbors with v. The weak closure of G is the smallest integer γ such that G is weakly γ-closed. We give an algorithm for Dominating Set with running time k^O(γ² k³) n^O(1), resolving an open problem of Koana et al. [ISAAC 2020].
One of the ingredients of our algorithm is a proof that the VC-dimension of (the set system defined by the closed neighborhoods of the vertices of) a weakly γ-closed graph is upper bounded by 6γ. This result may find further applications in the study of weakly closed graphs.

Daniel Lokshtanov and Vaishali Surianarayanan. Dominating Set in Weakly Closed Graphs is Fixed Parameter Tractable. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lokshtanov_et_al:LIPIcs.FSTTCS.2021.29, author = {Lokshtanov, Daniel and Surianarayanan, Vaishali}, title = {{Dominating Set in Weakly Closed Graphs is Fixed Parameter Tractable}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.29}, URN = {urn:nbn:de:0030-drops-155404}, doi = {10.4230/LIPIcs.FSTTCS.2021.29}, annote = {Keywords: Dominating Set, Weakly Closed Graphs, FPT, Domination Cores, VC-dimension} }

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