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

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.

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-dev.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} }

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

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.

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-dev.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} }

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

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.

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-dev.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} }

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

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.

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-dev.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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