License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.3
URN: urn:nbn:de:0030-drops-188284
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18828/
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Chandrasekaran, Karthekeyan ; Wang, Weihang

Approximating Submodular k-Partition via Principal Partition Sequence

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LIPIcs-APPROX3.pdf (0.8 MB)

Abstract

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).

BibTeX - Entry

@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/opus/volltexte/2023/18828},
  URN =		{urn:nbn:de:0030-drops-188284},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.3},
  annote =	{Keywords: Approximation algorithms}
}

Keywords: Approximation algorithms
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Issue Date: 2023
Date of publication: 04.09.2023

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