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DOI: 10.4230/LIPIcs.SoCG.2023.54

New Approximation Algorithms for Touring Regions

Authors: Benjamin Qi and Richard Qi

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract

We analyze the touring regions problem: find a (1+ε)-approximate Euclidean shortest path in d-dimensional space that starts at a given starting point, ends at a given ending point, and visits given regions R₁, R₂, R₃, … , R_n in that order. Our main result is an O (n/√ε log{1/ε} + 1/ε)-time algorithm for touring disjoint disks. We also give an O(min(n/ε, n²/√ε))-time algorithm for touring disjoint two-dimensional convex fat bodies. Both of these results naturally generalize to larger dimensions; we obtain O(n/{ε^{d-1}} log²1/ε + 1/ε^{2d-2}) and O(n/ε^{2d-2})-time algorithms for touring disjoint d-dimensional balls and convex fat bodies, respectively.

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Benjamin Qi and Richard Qi. New Approximation Algorithms for Touring Regions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{qi_et_al:LIPIcs.SoCG.2023.54,
  author =	{Qi, Benjamin and Qi, Richard},
  title =	{{New Approximation Algorithms for Touring Regions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{54:1--54:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.54},
  URN =		{urn:nbn:de:0030-drops-179047},
  doi =		{10.4230/LIPIcs.SoCG.2023.54},
  annote =	{Keywords: shortest paths, convex bodies, fat objects, disks}
}

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DOI: 10.4230/LIPIcs.ISAAC.2022.41

On Maximizing Sums of Non-Monotone Submodular and Linear Functions

Authors: Benjamin Qi

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract

We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by [Bodek and Feldman '22]. In this problem, we are given query access to a non-negative submodular function f: 2^N → ℝ_{≥ 0} and a linear function 𝓁: 2^N → ℝ over the same ground set N, and the objective is to output a set T ⊆ N approximately maximizing the sum f(T)+𝓁(T). Specifically, an algorithm is said to provide an (α,β)-approximation for RegularizedUSM if it outputs a set T such that E[f(T)+𝓁(T)] ≥ max_{S ⊆ N}[α ⋅ f(S)+β⋅ 𝓁(S)]. We also study the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied [Sviridenko et al. '17, Feldman '18, Harshaw et al. '19]. On the other hand, we are aware of only one prior work that studies RegularizedCSM with non-monotone f [Lu et al. '21], and that work constrains 𝓁 to be non-positive. In this work, we provide improved (α,β)-approximation algorithms for both {RegularizedUSM} and {RegularizedCSM} with non-monotone f. In particular, we are the first to provide nontrivial (α,β)-approximations for RegularizedCSM where the sign of 𝓁 is unconstrained, and the α we obtain for RegularizedUSM improves over [Bodek and Feldman '22] for all β ∈ (0,1). In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the α our algorithm obtains for {RegularizedCSM} with unconstrained 𝓁 is essentially tight for β ≥ e/(e+1). Using similar ideas, we are also able to show 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to [Oveis Gharan and Vondrak '10].

Cite as

Benjamin Qi. On Maximizing Sums of Non-Monotone Submodular and Linear Functions. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{qi:LIPIcs.ISAAC.2022.41,
  author =	{Qi, Benjamin},
  title =	{{On Maximizing Sums of Non-Monotone Submodular and Linear Functions}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.41},
  URN =		{urn:nbn:de:0030-drops-173263},
  doi =		{10.4230/LIPIcs.ISAAC.2022.41},
  annote =	{Keywords: submodular maximization, regularization, continuous greedy, inapproximability}
}

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Documents authored by Qi, Benjamin

New Approximation Algorithms for Touring Regions

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On Maximizing Sums of Non-Monotone Submodular and Linear Functions

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