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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

In submodular k-partition, the input is a submodular function f:2^V → ℝ_{≥ 0} (given by an evaluation oracle) along with a positive integer k and the goal is to find a partition of the ground set V into k non-empty parts V_1, V_2, …, V_k in order to minimize ∑_{i=1}^k f(V_i). Narayanan, Roy, and Patkar [Narayanan et al., 1996] designed an algorithm for submodular k-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is 2 for the special case of graph cut functions (which was subsequently rediscovered by Ravi and Sinha [R. Ravi and A. Sinha, 2008]). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions - namely monotone, symmetric, and posimodular and show the following results:
1) The approximation factor of their algorithm for monotone submodular k-partition is 4/3. This result improves on the 2-factor that was known to be achievable for monotone submodular k-partition via other algorithms. Moreover, our upper bound of 4/3 matches the recently shown lower bound under polynomial number of function evaluation queries [Santiago, 2021]. Our upper bound of 4/3 is also the first improvement beyond 2 for a certain graph partitioning problem that is a special case of monotone submodular k-partition.
2) The approximation factor of their algorithm for symmetric submodular k-partition is 2. This result generalizes their approximation factor analysis beyond graph cut functions.
3) The approximation factor of their algorithm for posimodular submodular k-partition is 2. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is Ω(n/k).

Karthekeyan Chandrasekaran and Weihang Wang. Approximating Submodular k-Partition via Principal Partition Sequence. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chandrasekaran_et_al:LIPIcs.APPROX/RANDOM.2023.3, author = {Chandrasekaran, Karthekeyan and Wang, Weihang}, title = {{Approximating Submodular k-Partition via Principal Partition Sequence}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {3:1--3:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.3}, URN = {urn:nbn:de:0030-drops-188284}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.3}, annote = {Keywords: Approximation algorithms} }

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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)

We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems - namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph G = (V, E) with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k non-empty parts (V₁, V₂, …, V_k) - known as a k-partition - so as to minimize an objective of interest.
1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition.
2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut.
We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an n^{O(k)}p-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known n^{O(k²)}p-time deterministic algorithm, where n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).

Calvin Beideman, Karthekeyan Chandrasekaran, and Weihang Wang. Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{beideman_et_al:LIPIcs.ICALP.2022.16, author = {Beideman, Calvin and Chandrasekaran, Karthekeyan and Wang, Weihang}, title = {{Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {16:1--16:18}, 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.2022.16}, URN = {urn:nbn:de:0030-drops-163578}, doi = {10.4230/LIPIcs.ICALP.2022.16}, annote = {Keywords: hypergraphs, k-partitioning, counting, enumeration} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We introduce and study 𝓁_p-norm-multiway-cut: the input here is an undirected graph with non-negative edge weights along with k terminals and the goal is to find a partition of the vertex set into k parts each containing exactly one terminal so as to minimize the 𝓁_p-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when p = 1) and min-max multiway cut (when p = ∞), both of which are well-studied classic problems in the graph partitioning literature. We show that 𝓁_p-norm-multiway-cut is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an O(log² n)-approximation for all p ≥ 1. We also show an integrality gap of Ω(k^{1-1/p}) for a natural convex program and an O(k^{1-1/p-ε})-inapproximability for any constant ε > 0 assuming the small set expansion hypothesis.

Karthekeyan Chandrasekaran and Weihang Wang. 𝓁_p-Norm Multiway Cut. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chandrasekaran_et_al:LIPIcs.ESA.2021.29, author = {Chandrasekaran, Karthekeyan and Wang, Weihang}, title = {{𝓁\underlinep-Norm Multiway Cut}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.29}, URN = {urn:nbn:de:0030-drops-146103}, doi = {10.4230/LIPIcs.ESA.2021.29}, annote = {Keywords: multiway cut, approximation algorithms} }

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