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Documents authored by Olver, Neil

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.OPODIS.2024.38
Efficient Algorithms for Demand-Aware Networks and a Connection to Virtual Network Embedding

Authors: Aleksander Figiel, Janne H. Korhonen, Neil Olver, and Stefan Schmid

Published in: LIPIcs, Volume 324, 28th International Conference on Principles of Distributed Systems (OPODIS 2024)

Abstract
Emerging optical switching technologies enable demand-aware datacenter networks, whose topology can be flexibly optimized toward the traffic they serve. This paper revisits the bounded-degree network design problem underlying such demand-aware networks. Namely, given a distribution over communicating node pairs (represented has a demand graph), we want to design a network with bounded maximum degree (called host graph) that minimizes the expected communication distance. We improve the understanding of this problem domain by filling several gaps in prior work. First, we present the first practical algorithm for solving this problem on arbitrary instances without violating the degree bound. Our algorithm is based on novel insights obtained from studying a new Steiner node version of the problem, and we report on an extensive empirical evaluation, using several real-world traffic traces from datacenters, finding that our approach results in improved demand-aware network designs. Second, we shed light on the complexity and hardness of the bounded-degree network design problem by formally establishing its NP-completeness for any degree. We use our techniques to improve prior upper bounds for sparse instances. Finally, we study an intriguing connection between demand-aware network design and the virtual networking embedding problem, and show that the latter cannot be used to approximate the former: there is no universal host graph which can provide a constant approximation for our problem.
Cite as

Aleksander Figiel, Janne H. Korhonen, Neil Olver, and Stefan Schmid. Efficient Algorithms for Demand-Aware Networks and a Connection to Virtual Network Embedding. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 38:1-38:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{figiel_et_al:LIPIcs.OPODIS.2024.38,
  author =	{Figiel, Aleksander and Korhonen, Janne H. and Olver, Neil and Schmid, Stefan},
  title =	{{Efficient Algorithms for Demand-Aware Networks and a Connection to Virtual Network Embedding}},
  booktitle =	{28th International Conference on Principles of Distributed Systems (OPODIS 2024)},
  pages =	{38:1--38:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-360-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{324},
  editor =	{Bonomi, Silvia and Galletta, Letterio and Rivi\`{e}re, Etienne and Schiavoni, Valerio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2024.38},
  URN =		{urn:nbn:de:0030-drops-225742},
  doi =		{10.4230/LIPIcs.OPODIS.2024.38},
  annote =	{Keywords: demand-aware networks, algorithms, virtual network embedding}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.73
Majorizing Measures for the Optimizer

Authors: Sander Borst, Daniel Dadush, Neil Olver, and Makrand Sinha

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes. One of the crowning achievements of the theory is Talagrand’s tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum. In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof of the majorizing measures theorem based on two parts: - We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program. This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization. - We obtain tree-based upper and lower bound certificates by rounding, in a series of steps, the primal and dual solutions to this convex program. While duality has conceptually been part of the theory since its beginnings, as far as we are aware no explicit link to convex optimization has been previously made.
Cite as

Sander Borst, Daniel Dadush, Neil Olver, and Makrand Sinha. Majorizing Measures for the Optimizer. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{borst_et_al:LIPIcs.ITCS.2021.73,
  author =	{Borst, Sander and Dadush, Daniel and Olver, Neil and Sinha, Makrand},
  title =	{{Majorizing Measures for the Optimizer}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{73:1--73:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.73},
  URN =		{urn:nbn:de:0030-drops-136120},
  doi =		{10.4230/LIPIcs.ITCS.2021.73},
  annote =	{Keywords: Majorizing measures, Generic chaining, Gaussian processes, Convex optimization, Dimensionality Reduction}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.19
Exploring the Tractability of the Capped Hose Model

Authors: Thomas Bosman and Neil Olver

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
Robust network design concerns the design of networks to support uncertain or varying traffic patterns. An especially important case is the VPN problem, where the total traffic emanating from any node is bounded, but there are no further constraints on the traffic pattern. Recently, Fréchette et al. [INFOCOM, 2013] studied a generalization of the VPN problem where in addition to these so-called hose constraints, there are individual upper bounds on the demands between pairs of nodes. They motivate their model, give some theoretical results, and propose a heuristic algorithm that performs well on real-world instances. Our theoretical understanding of this model is limited; it is APX-hard in general, but tractable when either the hose constraints or the individual demand bounds are redundant. In this work, we uncover further tractable cases of this model; our main result concerns the case where each terminal needs to communicate only with two others. Our algorithms all involve optimally embedding a certain auxiliary graph into the network, and have a connection to a heuristic suggested by Fréchette et al. for the capped hose model in general.
Cite as

Thomas Bosman and Neil Olver. Exploring the Tractability of the Capped Hose Model. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 19:1-19:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bosman_et_al:LIPIcs.ESA.2017.19,
  author =	{Bosman, Thomas and Olver, Neil},
  title =	{{Exploring the Tractability of the Capped Hose Model}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{19:1--19:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.19},
  URN =		{urn:nbn:de:0030-drops-78663},
  doi =		{10.4230/LIPIcs.ESA.2017.19},
  annote =	{Keywords: robust network design, VPN problem}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.17
On the Integrality Gap of the Prize-Collecting Steiner Forest LP

Authors: Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, and Jens Vygen

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G=(V,E), nonnegative edge costs {c_e} for e in E, terminal pairs {(s_i,t_i)} for i=1,...,k, and penalties {pi_i} for i=1,...,k for each terminal pair; the goal is to find a forest F to minimize c(F) + sum{ pi_i: (s_i,t_i) is not connected in F }. The Steiner forest problem can be viewed as the special case where pi_i are infinite for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4 even if the input instance is planar. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP-approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.
Cite as

Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, and Jens Vygen. On the Integrality Gap of the Prize-Collecting Steiner Forest LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{konemann_et_al:LIPIcs.APPROX-RANDOM.2017.17,
  author =	{K\"{o}nemann, Jochen and Olver, Neil and Pashkovich, Kanstantsin and Ravi, R. and Swamy, Chaitanya and Vygen, Jens},
  title =	{{On the Integrality Gap of the Prize-Collecting Steiner Forest LP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{17:1--17:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.17},
  URN =		{urn:nbn:de:0030-drops-75665},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.17},
  annote =	{Keywords: Integrality gap, Steiner tree, Steiner forest, prize-collecting, Lagrangianmultiplier- preserving}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.176
On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree

Authors: Andreas Emil Feldmann, Jochen Könemann, Neil Olver, and Laura Sanità

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The bottleneck of the currently best (ln(4) + epsilon)-approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well. We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known. In this paper, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with 3 Steiner neighbors. This implies faster ln(4)-approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard.
Cite as

Andreas Emil Feldmann, Jochen Könemann, Neil Olver, and Laura Sanità. On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 176-191, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{feldmann_et_al:LIPIcs.APPROX-RANDOM.2014.176,
  author =	{Feldmann, Andreas Emil and K\"{o}nemann, Jochen and Olver, Neil and Sanit\`{a}, Laura},
  title =	{{On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{176--191},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.176},
  URN =		{urn:nbn:de:0030-drops-46962},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.176},
  annote =	{Keywords: Steiner tree, bidirected cut relaxation, hypergraphic relaxation, polyhedral equivalence, approximation algorithms}
}
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