Search Results

Documents authored by Könemann, Jochen

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)

Copy BibTex To Clipboard

@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.ISAAC.2016.33
Fast Approximation Algorithms for the Generalized Survivable Network Design Problem

Authors: Andreas Emil Feldmann, Jochen Könemann, Kanstantsin Pashkovich, and Laura Sanità

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
In a standard f-connectivity network design problem, we are given an undirected graph G = (V, E), a cut-requirement function f : 2^V to N, and non-negative costs c(e) for all e in E. We are then asked to find a minimum-cost vector x in N^E such that x(delta(S)) geq f (S) for all S subseteq V. We focus on the class of such problems where f is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the f-connectivity problem with general proper functions f. Implementing Jain’s algorithm, this yields a strongly polynomial time (2 + epsilon)-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).
Cite as

Andreas Emil Feldmann, Jochen Könemann, Kanstantsin Pashkovich, and Laura Sanità. Fast Approximation Algorithms for the Generalized Survivable Network Design Problem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{feldmann_et_al:LIPIcs.ISAAC.2016.33,
  author =	{Feldmann, Andreas Emil and K\"{o}nemann, Jochen and Pashkovich, Kanstantsin and Sanit\`{a}, Laura},
  title =	{{Fast Approximation Algorithms for the Generalized Survivable Network Design Problem}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{33:1--33:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.33},
  URN =		{urn:nbn:de:0030-drops-68035},
  doi =		{10.4230/LIPIcs.ISAAC.2016.33},
  annote =	{Keywords: strongly polynomial runtime, generalized survivable network design, primal-dual method}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2016.3
A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs

Authors: Zachary Friggstad, Jochen Könemann, and Mohammad Shadravan

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract
We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is O(log k) in quasi-bipartite graphs with k terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor. A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs.
Cite as

Zachary Friggstad, Jochen Könemann, and Mohammad Shadravan. A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 3:1-3:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{friggstad_et_al:LIPIcs.SWAT.2016.3,
  author =	{Friggstad, Zachary and K\"{o}nemann, Jochen and Shadravan, Mohammad},
  title =	{{A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{3:1--3:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.3},
  URN =		{urn:nbn:de:0030-drops-60323},
  doi =		{10.4230/LIPIcs.SWAT.2016.3},
  annote =	{Keywords: Approximation algorithm, Primal-Dual algorithm, Directed Steiner tree}
}
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)

Copy BibTex To Clipboard

@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}
}
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact