10 Search Results for "Olver, Neil"

Document
DOI: 10.4230/LIPIcs.ESA.2024.14
Approximating Maximum-Size Properly Colored Forests

Authors: Yuhang Bai, Kristóf Bérczi, Gergely Csáji, and Tamás Schwarcz

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and (1,2)-Traveling Salesman problems. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.
Cite as

Yuhang Bai, Kristóf Bérczi, Gergely Csáji, and Tamás Schwarcz. Approximating Maximum-Size Properly Colored Forests. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bai_et_al:LIPIcs.ESA.2024.14,
  author =	{Bai, Yuhang and B\'{e}rczi, Krist\'{o}f and Cs\'{a}ji, Gergely and Schwarcz, Tam\'{a}s},
  title =	{{Approximating Maximum-Size Properly Colored Forests}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.14},
  URN =		{urn:nbn:de:0030-drops-210858},
  doi =		{10.4230/LIPIcs.ESA.2024.14},
  annote =	{Keywords: Approximation algorithm, (1,2)-traveling salesman problem, Degree bounded spanning tree, Properly colored forest}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.42
From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

Authors: Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. - We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. - We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k).
Cite as

Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chekuri_et_al:LIPIcs.ESA.2024.42,
  author =	{Chekuri, Chandra and Jain, Rhea and Kulkarni, Shubhang and Zheng, Da Wei and Zhu, Weihao},
  title =	{{From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{42:1--42:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.42},
  URN =		{urn:nbn:de:0030-drops-211134},
  doi =		{10.4230/LIPIcs.ESA.2024.42},
  annote =	{Keywords: Directed Planar Graphs, Submodular Functions, Steiner Tree, Network Design}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.62
Optimizing Throughput and Makespan of Queuing Systems by Information Design

Authors: Svenja M. Griesbach, Max Klimm, Philipp Warode, and Theresa Ziemke

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links (queues) is equipped with deterministic capacities and stochastic offsets where the latter depend on a realized state, and the number of states is assumed to be constant. A continuum of flow particles (agents) arrives at the system at a constant rate. A system operator knows the realization of the state and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized state, and decide on which link they use. Inflow into a link exceeding the link’s capacity builds up in a queue that increases the cost (total travel time) on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected cost along all links with positive inflow is equal at every point in time and not larger than the expected cost of any unused link. For a given time horizon T, the throughput induced by a signaling scheme is the total volume of flow that leaves the links in the interval [0,T]. The public signaling scheme maximizing the throughput may involve irrational numbers. We provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant ε > 0. The algorithm solves a Lagrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute the optimal signals. It uses a different cell decomposition technique together with a piecewise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.
Cite as

Svenja M. Griesbach, Max Klimm, Philipp Warode, and Theresa Ziemke. Optimizing Throughput and Makespan of Queuing Systems by Information Design. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{griesbach_et_al:LIPIcs.ESA.2024.62,
  author =	{Griesbach, Svenja M. and Klimm, Max and Warode, Philipp and Ziemke, Theresa},
  title =	{{Optimizing Throughput and Makespan of Queuing Systems by Information Design}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{62:1--62:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.62},
  URN =		{urn:nbn:de:0030-drops-211336},
  doi =		{10.4230/LIPIcs.ESA.2024.62},
  annote =	{Keywords: Information Design, Dynamic Flows, Public Signals, Convex Envelope}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.5
An O(loglog n)-Approximation for Submodular Facility Location

Authors: Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In the Submodular Facility Location problem (SFL) we are given a collection of n clients and m facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility f to which we assign the subset of clients S^f, one has to pay the opening cost g(S^f), where g() is a monotone submodular function with g(emptyset)=0. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [SODA'06] gave the current-best O(log n) approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an O(loglog n) approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the natural generalizations of SFL where the opening cost of each facility f is of the form p_f + g(S^f) or w_f * g(S^f), where p_f, w_f >= 0 are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility f can be modelled with a submodular function of the set of clients assigned to f.
Cite as

Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely. An O(loglog n)-Approximation for Submodular Facility Location. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abbasi_et_al:LIPIcs.ICALP.2024.5,
  author =	{Abbasi, Fateme and Adamczyk, Marek and Bosch-Calvo, Miguel and Byrka, Jaros{\l}aw and Grandoni, Fabrizio and Sornat, Krzysztof and Tinguely, Antoine},
  title =	{{An O(loglog n)-Approximation for Submodular Facility Location}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.5},
  URN =		{urn:nbn:de:0030-drops-201488},
  doi =		{10.4230/LIPIcs.ICALP.2024.5},
  annote =	{Keywords: approximation algorithms, facility location, submodular facility location, universal stochastic facility location}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.15
An Improved Approximation Algorithm for Dynamic Minimum Linear Arrangement

Authors: Marcin Bienkowski and Guy Even

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
The dynamic offline linear arrangement problem deals with reordering n elements subject to a sequence of edge requests. The input consists of a sequence of m edges (i.e., unordered pairs of elements). The output is a sequence of permutations (i.e., bijective mapping of the elements to n equidistant points). In step t, the order of the elements is changed to the t-th permutation, and then the t-th request is served. The cost of the output consists of two parts per step: request cost and rearrangement cost. The former is the current distance between the endpoints of the request, while the latter is proportional to the number of adjacent element swaps required to move from one permutation to the consecutive permutation. The goal is to find a minimum cost solution. We present a deterministic O(log n log log n)-approximation algorithm for this problem, improving over a randomized O(log² n)-approximation by Olver et al. [Neil Olver et al., 2018]. Our algorithm is based on first solving spreading-metric LP relaxation on a time-expanded graph, applying a tree decomposition on the basis of the LP solution, and finally converting the tree decomposition to a sequence of permutations. The techniques we employ are general and have the potential to be useful for other dynamic graph optimization problems.
Cite as

Marcin Bienkowski and Guy Even. An Improved Approximation Algorithm for Dynamic Minimum Linear Arrangement. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bienkowski_et_al:LIPIcs.STACS.2024.15,
  author =	{Bienkowski, Marcin and Even, Guy},
  title =	{{An Improved Approximation Algorithm for Dynamic Minimum Linear Arrangement}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.15},
  URN =		{urn:nbn:de:0030-drops-197252},
  doi =		{10.4230/LIPIcs.STACS.2024.15},
  annote =	{Keywords: Minimum Linear Arrangement, dynamic Variant, Optimization Problems, Graph Problems, approximation Algorithms}
}
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}
}
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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