2 Search Results for "Linhares, André"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.50
New Results on a General Class of Minimum Norm Optimization Problems

Authors: Kuowen Chen, Jian Li, Yuval Rabani, and Yiran Zhang

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the general norm optimization for combinatorial problems, initiated by Chakrabarty and Swamy (STOC 2019). We propose a general formulation that captures a large class of combinatorial structures: we are given a set 𝒰 of n weighted elements and a family of feasible subsets ℱ. Each subset S ∈ ℱ is called a feasible solution/set of the problem. We denote the value vector by v = {v_i}_{i ∈ [n]}, where v_i ≥ 0 is the value of element i. For any subset S ⊆ 𝒰, we use v[S] to denote the n-dimensional vector {v_e⋅ 𝟏[e ∈ S]}_{e ∈ 𝒰} (i.e., we zero out all entries that are not in S). Let f: ℝⁿ → ℝ_+ be a symmetric monotone norm function. Our goal is to minimize the norm objective f(v[S]) over feasible subset S ∈ ℱ. The problem significantly generalizes the corresponding min-sum and min-max problems. We present a general equivalent reduction of the norm minimization problem to a multi-criteria optimization problem with logarithmic budget constraints, up to a constant approximation factor. Leveraging this reduction, we obtain constant factor approximation algorithms for the norm minimization versions of several covering problems, such as interval cover, multi-dimensional knapsack cover, and logarithmic factor approximation for set cover. We also study the norm minimization versions for perfect matching, s-t path and s-t cut. We show the natural linear programming relaxations for these problems have a large integrality gap. To complement the negative result, we show that, for perfect matching, it is possible to obtain a bi-criteria result: for any constant ε,δ > 0, we can find in polynomial time a nearly perfect matching (i.e., a matching that matches at least 1-ε proportion of vertices) and its cost is at most (8+δ) times of the optimum for perfect matching. Moreover, we establish the existence of a polynomial-time O(log log n)-approximation algorithm for the norm minimization variant of the s-t path problem. Specifically, our algorithm achieves an α-approximation with a time complexity of n^{O(log log n / α)}, where 9 ≤ α ≤ log log n.
Cite as

Kuowen Chen, Jian Li, Yuval Rabani, and Yiran Zhang. New Results on a General Class of Minimum Norm Optimization Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2025.50,
  author =	{Chen, Kuowen and Li, Jian and Rabani, Yuval and Zhang, Yiran},
  title =	{{New Results on a General Class of Minimum Norm Optimization Problems}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.50},
  URN =		{urn:nbn:de:0030-drops-234276},
  doi =		{10.4230/LIPIcs.ICALP.2025.50},
  annote =	{Keywords: Approximation Algorithms, Minimum Norm Optimization, Linear Programming}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.32
Improved Algorithms for MST and Metric-TSP Interdiction

Authors: André Linhares and Chaitanya Swamy

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

Abstract
We consider the MST-interdiction problem: given a multigraph G = (V, E), edge weights {w_e >= 0}_{e in E}, interdiction costs {c_e >= 0}_{e in E}, and an interdiction budget B >= 0, the goal is to remove a subset R of edges of total interdiction cost at most B so as to maximize the w-weight of an MST of G-R:=(V,E-R). Our main result is a 4-approximation algorithm for this problem. This improves upon the previous-best 14-approximation [Zenklusen, FOCS 2015]. Notably, our analysis is also significantly simpler and cleaner than the one in [Zenklusen, FOCS 2015]. Whereas Zenklusen uses a greedy algorithm with an involved analysis to extract a good interdiction set from an over-budget set, we utilize a generalization of knapsack called the tree knapsack problem that nicely captures the key combinatorial aspects of this "extraction problem." We prove a simple, yet strong, LP-relative approximation bound for tree knapsack, which leads to our improved guarantees for MST interdiction. Our algorithm and analysis are nearly tight, as we show that one cannot achieve an approximation ratio better than 3 relative to the upper bound used in our analysis (and the one in [Zenklusen, FOCS 2015]). Our guarantee for MST-interdiction yields an 8-approximation for metric-TSP interdiction (improving over the 28-approximation in [Zenklusen, FOCS 2015]). We also show that maximum-spanning-tree interdiction is at least as hard to approximate as the minimization version of densest-k-subgraph.
Cite as

André Linhares and Chaitanya Swamy. Improved Algorithms for MST and Metric-TSP Interdiction. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{linhares_et_al:LIPIcs.ICALP.2017.32,
  author =	{Linhares, Andr\'{e} and Swamy, Chaitanya},
  title =	{{Improved Algorithms for MST and Metric-TSP Interdiction}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{32:1--32:14},
  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.2017.32},
  URN =		{urn:nbn:de:0030-drops-74552},
  doi =		{10.4230/LIPIcs.ICALP.2017.32},
  annote =	{Keywords: Approximation algorithms, interdiction problems, LP-rounding algorithms, iterative rounding, tree-knapsack problem, supermodular functions}
}
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