6 Search Results for "Natura, Bento"

Document
DOI: 10.4230/LIPIcs.STACS.2026.31
Lower Bounds for Ranking-Based Pivot Rules

Authors: Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland’s rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.
Cite as

Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis. Lower Bounds for Ranking-Based Pivot Rules. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{disser_et_al:LIPIcs.STACS.2026.31,
  author =	{Disser, Yann and Loho, Georg and Maat, Matthew and Mosis, Nils},
  title =	{{Lower Bounds for Ranking-Based Pivot Rules}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.31},
  URN =		{urn:nbn:de:0030-drops-255207},
  doi =		{10.4230/LIPIcs.STACS.2026.31},
  annote =	{Keywords: lower bounds, Markov decision processes, parity games, pivot rules, policy iteration, simplex method}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.IPEC.2025.1
A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk)

Authors: Martin Koutecký

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Integer Programming (IP) is a fundamental but computationally hard problem. Still, certain efficiently solvable subclasses have been identified over time, most notably totally unimodular IPs in the 1950s, and fixed-dimension IPs in the 1980s. Starting around the year 2000, a stream of research has identified block-structured IPs as yet another tractable subclass. In this paper, we give a brief and incomplete review of this history, with a focus on several of the author’s contributions.
Cite as

Martin Koutecký. A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk). In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{koutecky:LIPIcs.IPEC.2025.1,
  author =	{Kouteck\'{y}, Martin},
  title =	{{A Brief History of Parameterized Algorithms for Block-Structured Integer Programs}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.1},
  URN =		{urn:nbn:de:0030-drops-251338},
  doi =		{10.4230/LIPIcs.IPEC.2025.1},
  annote =	{Keywords: Integer Programming, Parameterized Algorithm, Graver Basis, Treedepth, n-fold, tree-fold, 2-stage stochastic, multistage stochastic, Mixed-Integer Programming}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2025.2
A Strongly Polynomial Algorithm for Linear Programs with at Most Two Non-Zero Entries per Row or Column (Invited Talk)

Authors: Daniel Dadush, Zhuan Khye Koh, Bento Natura, Neil Olver, and László A. Végh

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh (MOR '17) and Megiddo (SICOMP '83) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL '04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices and extreme rays. Our approach is black-box and can be applied to any log-barrier path following method.
Cite as

Daniel Dadush, Zhuan Khye Koh, Bento Natura, Neil Olver, and László A. Végh. A Strongly Polynomial Algorithm for Linear Programs with at Most Two Non-Zero Entries per Row or Column (Invited Talk). In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dadush_et_al:LIPIcs.STACS.2025.2,
  author =	{Dadush, Daniel and Koh, Zhuan Khye and Natura, Bento and Olver, Neil and V\'{e}gh, L\'{a}szl\'{o} A.},
  title =	{{A Strongly Polynomial Algorithm for Linear Programs with at Most Two Non-Zero Entries per Row or Column}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.2},
  URN =		{urn:nbn:de:0030-drops-228273},
  doi =		{10.4230/LIPIcs.STACS.2025.2},
  annote =	{Keywords: Linear Programming, Strongly Polynomial Algorithms, Interior Point Methods}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.80
The Pareto Cover Problem

Authors: Bento Natura, Meike Neuwohner, and Stefan Weltge

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We introduce the problem of finding a set B of k points in [0,1]ⁿ such that the expected cost of the cheapest point in B that dominates a random point from [0,1]ⁿ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for k = 2 when each coordinate is drawn from {0,1}, and obtain an FPTAS for general fixed k under mild assumptions on the distributions.
Cite as

Bento Natura, Meike Neuwohner, and Stefan Weltge. The Pareto Cover Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 80:1-80:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{natura_et_al:LIPIcs.ESA.2022.80,
  author =	{Natura, Bento and Neuwohner, Meike and Weltge, Stefan},
  title =	{{The Pareto Cover Problem}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{80:1--80:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.80},
  URN =		{urn:nbn:de:0030-drops-170186},
  doi =		{10.4230/LIPIcs.ESA.2022.80},
  annote =	{Keywords: Pareto, Covering, Optimization, Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.79
A Faster Interior-Point Method for Sum-Of-Squares Optimization

Authors: Shunhua Jiang, Bento Natura, and Omri Weinstein

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are a central tool in polynomial optimization and capture convex programming in the Lasserre hierarchy. Let p = ∑_i q²_i be an n-variate SOS polynomial of degree 2d. Denoting by L : = binom(n+d,d) and U : = binom(n+2d,2d) the dimensions of the vector spaces in which q_i’s and p live respectively, our algorithm runs in time Õ(LU^{1.87}). This is polynomially faster than state-of-art SOS and semidefinite programming solvers [Jiang et al., 2020; Huang et al., 2021; Papp and Yildiz, 2019], which achieve runtime Õ(L^{0.5} min{U^{2.37}, L^{4.24}}). The centerpiece of our algorithm is a dynamic data structure for maintaining the inverse of the Hessian of the SOS barrier function under the polynomial interpolant basis [Papp and Yildiz, 2019], which efficiently extends to multivariate SOS optimization, and requires maintaining spectral approximations to low-rank perturbations of elementwise (Hadamard) products. This is the main challenge and departure from recent IPM breakthroughs using inverse-maintenance, where low-rank updates to the slack matrix readily imply the same for the Hessian matrix.
Cite as

Shunhua Jiang, Bento Natura, and Omri Weinstein. A Faster Interior-Point Method for Sum-Of-Squares Optimization. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jiang_et_al:LIPIcs.ICALP.2022.79,
  author =	{Jiang, Shunhua and Natura, Bento and Weinstein, Omri},
  title =	{{A Faster Interior-Point Method for Sum-Of-Squares Optimization}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{79:1--79:20},
  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.2022.79},
  URN =		{urn:nbn:de:0030-drops-164205},
  doi =		{10.4230/LIPIcs.ICALP.2022.79},
  annote =	{Keywords: Interior Point Methods, Sum-of-squares Optimization, Dynamic Matrix Inverse}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.36
An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems

Authors: Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh

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

Abstract
We present an accelerated, or "look-ahead" version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of O(mlog m) iterations; the previous best bound was O(m²log m) by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems, and O(m + nlog n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton–Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017).
Cite as

Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh. An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dadush_et_al:LIPIcs.ESA.2021.36,
  author =	{Dadush, Daniel and Koh, Zhuan Khye and Natura, Bento and V\'{e}gh, L\'{a}szl\'{o} A.},
  title =	{{An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{36:1--36:15},
  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.2021.36},
  URN =		{urn:nbn:de:0030-drops-146172},
  doi =		{10.4230/LIPIcs.ESA.2021.36},
  annote =	{Keywords: Newton-Dinkelbach method, fractional optimization, parametric optimization, strongly polynomial algorithms, two variables per inequality systems, Markov decision processes, submodular function minimization}
}
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