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Documents authored by Schäfer, Guido

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APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.20
Round and Bipartize for Vertex Cover Approximation

Authors: Danish Kashaev and Guido Schäfer

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (𝒢, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph 𝒢̃ : = 𝒢/S and satisfies ρ ∈ [2,∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/ρ) (1 - α) + 2 α, where α ∈ [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph 𝒢̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph.
Cite as

Danish Kashaev and Guido Schäfer. Round and Bipartize for Vertex Cover Approximation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kashaev_et_al:LIPIcs.APPROX/RANDOM.2023.20,
  author =	{Kashaev, Danish and Sch\"{a}fer, Guido},
  title =	{{Round and Bipartize for Vertex Cover Approximation}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.20},
  URN =		{urn:nbn:de:0030-drops-188459},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.20},
  annote =	{Keywords: Combinatorial optimization, approximation algorithms, rounding algorithms, beyond the worst-case analysis}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.13
Approximate Pricing in Networks: How to Boost the Betweenness and Revenue of a Node

Authors: Ruben Brokkelkamp, Sven Polak, Guido Schäfer, and Yllka Velaj

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
We introduce and study two new pricing problems in networks: Suppose we are given a directed graph G = (V, E) with non-negative edge costs (c_e)_{e in E}, k commodities (s_i, t_i, w_i)_{i in [k]} and a designated node u in V. Each commodity i in [k] is represented by a source-target pair (s_i, t_i) in V x V and a demand w_i>0, specifying that w_i units of flow are sent from s_i to t_i along shortest s_i, t_i-paths (with respect to (c_e)_{e in E}). The demand of each commodity is split evenly over all shortest paths. Assume we can change the edge costs of some of the outgoing edges of u, while the costs of all other edges remain fixed; we also say that we price (or tax) the edges of u. We study the problem of pricing the edges of u with respect to the following two natural objectives: (i) max-flow: maximize the total flow passing through u, and (ii) max-revenue: maximize the total revenue (flow times tax) through u. Both variants have various applications in practice. For example, the max flow objective is equivalent to maximizing the betweenness centrality of u, which is one of the most popular measures for the influence of a node in a (social) network. We prove that (except for some special cases) both problems are NP-hard and inapproximable in general and therefore resort to approximation algorithms. We derive approximation algorithms for both variants and show that the derived approximation guarantees are best possible.
Cite as

Ruben Brokkelkamp, Sven Polak, Guido Schäfer, and Yllka Velaj. Approximate Pricing in Networks: How to Boost the Betweenness and Revenue of a Node. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brokkelkamp_et_al:LIPIcs.ISAAC.2019.13,
  author =	{Brokkelkamp, Ruben and Polak, Sven and Sch\"{a}fer, Guido and Velaj, Yllka},
  title =	{{Approximate Pricing in Networks: How to Boost the Betweenness and Revenue of a Node}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.13},
  URN =		{urn:nbn:de:0030-drops-115091},
  doi =		{10.4230/LIPIcs.ISAAC.2019.13},
  annote =	{Keywords: Network pricing, Stackelberg network pricing, betweenness centrality, revenue maximization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.20
Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond

Authors: Georgios Birmpas, Evangelos Markakis, and Guido Schäfer

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study mechanism design for combinatorial cost sharing models. Imagine that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost sharing domains until recently [S. Dobzinski and S. Ovadia, 2017]. Still, many questions about the interplay between strategyproofness, cost recovery and economic efficiency remain unanswered. The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions which capture the behavior of their average cost-shares. Based on our trace-monotonicity property, we design a scheme of ascending cost sharing mechanisms which is applicable to the combinatorial cost sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced and O(H_n)-approximate with respect to the social cost. Further, we show that our mechanism is budget-balanced and H_n-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible. Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost sharing problems, including Vertex Cover and Set Cover.
Cite as

Georgios Birmpas, Evangelos Markakis, and Guido Schäfer. Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{birmpas_et_al:LIPIcs.ESA.2019.20,
  author =	{Birmpas, Georgios and Markakis, Evangelos and Sch\"{a}fer, Guido},
  title =	{{Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.20},
  URN =		{urn:nbn:de:0030-drops-111419},
  doi =		{10.4230/LIPIcs.ESA.2019.20},
  annote =	{Keywords: Approximation Algorithms, Combinatorial Cost Sharing, Mechanism Design, Truthfulness}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.50
The Ground-Set-Cost Budgeted Maximum Coverage Problem

Authors: Irving van Heuven van Staereling, Bart de Keijzer, and Guido Schäfer

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V, E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T subseteq E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords. It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1-1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows: - We obtain a (1 - 1/sqrt(e))/2-approximation algorithm for graphs. - We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set. - We give a (1-epsilon)/(2d^2)-approximation algorithm for every epsilon > 0, where d is the maximum degree of a vertex in the hypergraph.
Cite as

Irving van Heuven van Staereling, Bart de Keijzer, and Guido Schäfer. The Ground-Set-Cost Budgeted Maximum Coverage Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{vanheuvenvanstaereling_et_al:LIPIcs.MFCS.2016.50,
  author =	{van Heuven van Staereling, Irving and de Keijzer, Bart and Sch\"{a}fer, Guido},
  title =	{{The Ground-Set-Cost Budgeted Maximum Coverage Problem}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.50},
  URN =		{urn:nbn:de:0030-drops-65020},
  doi =		{10.4230/LIPIcs.MFCS.2016.50},
  annote =	{Keywords: maximum coverage problem, approximation algorithms, hypergraphs, submodular optimization, sponsored search}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.433
Computing Optimal Tolls with Arc Restrictions and Heterogeneous Players

Authors: Tomas Jelinek, Marcus Klaas, and Guido Schäfer

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
The problem of computing optimal network tolls that induce a Nash equilibrium of minimum total cost has been studied intensively in the literature, but mostly under the assumption that these tolls are unrestricted. Here we consider this problem under the more realistic assumption that the tolls have to respect some given upper bound restrictions on the arcs. The problem of taxing subnetworks optimally constitutes an important special case of this problem. We study the restricted network toll problem for both non-atomic and atomic (unweighted and weighted) players; our studies are the first that also incorporate heterogeneous players, i.e., players with different sensitivities to tolls. For non-atomic and heterogeneous players, we prove that the problem is NP-hard even for single-commodity networks and affine latency functions. We therefore focus on parallel-arc networks and give an algorithm for optimally taxing subnetworks with affine latency functions. For weighted atomic players, the problem is NP-hard already for parallel-arc networks and linear latency functions, even if players are homogeneous. In contrast, for unweighted atomic and homogeneous players, we develop an algorithm to compute optimal restricted tolls for parallel-arc networks and arbitrary (standard) latency functions. Similarly, for unweighted atomic and heterogeneous players, we derive an algorithm for optimally taxing subnetworks for parallel-arc networks and arbitrary (standard) latency functions. The key to most of our results is to derive (combinatorial) characterizations of flows that are inducible by restricted tolls. These characterizations might be of independent interest.
Cite as

Tomas Jelinek, Marcus Klaas, and Guido Schäfer. Computing Optimal Tolls with Arc Restrictions and Heterogeneous Players. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 433-444, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{jelinek_et_al:LIPIcs.STACS.2014.433,
  author =	{Jelinek, Tomas and Klaas, Marcus and Sch\"{a}fer, Guido},
  title =	{{Computing Optimal Tolls with Arc Restrictions and Heterogeneous Players}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{433--444},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.433},
  URN =		{urn:nbn:de:0030-drops-44771},
  doi =		{10.4230/LIPIcs.STACS.2014.433},
  annote =	{Keywords: Network routing games, coordination mechanisms, network tolls, taxing subnetworks, heterogeneous players}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2011.122
On the Smoothed Price of Anarchy of the Traffic Assignment Problem

Authors: Luciana Buriol, Marcus Ritt, Felix Rodrigues, and Guido Schäfer

Published in: OASIcs, Volume 20, 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2011)

Abstract
We study the effect of perturbations on the Price of Anarchy for the Traffic Assignment Problem. Adopting the smoothed analysis approach, we randomly perturb the latency functions of the given network and estimate the expected Price of Anarchy on the perturbed instances. We provide both theoretical and experimental results that show that the Smoothed Price of Anarchy is of the same order of magnitude as the original one.
Cite as

Luciana Buriol, Marcus Ritt, Felix Rodrigues, and Guido Schäfer. On the Smoothed Price of Anarchy of the Traffic Assignment Problem. In 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 20, pp. 122-133, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{buriol_et_al:OASIcs.ATMOS.2011.122,
  author =	{Buriol, Luciana and Ritt, Marcus and Rodrigues, Felix and Sch\"{a}fer, Guido},
  title =	{{On the Smoothed Price of Anarchy of the Traffic Assignment Problem}},
  booktitle =	{11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{122--133},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-33-0},
  ISSN =	{2190-6807},
  year =	{2011},
  volume =	{20},
  editor =	{Caprara, Alberto and Kontogiannis, Spyros},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2011.122},
  URN =		{urn:nbn:de:0030-drops-32727},
  doi =		{10.4230/OASIcs.ATMOS.2011.122},
  annote =	{Keywords: Traffic Assignment Problem, Smoothed Analysis, Price of Anarchy}
}
Document
DOI: 10.4230/DagSemProc.05031.29
Topology Matters: Smoothed Competitiveness of Metrical Task Systems

Authors: Guido Schäfer and Naveen Sivadasan

Published in: Dagstuhl Seminar Proceedings, Volume 5031, Algorithms for Optimization with Incomplete Information (2005)

Abstract
We consider online problems that can be modeled as metrical task systems: An online algorithm resides in a graph of n nodes and may move in this graph at a cost equal to the distance. The algorithm has to service a sequence of tasks that arrive over time; each task specifies for each node a request cost that is incurred if the algorithm services the task in this particular node. The objective is to minimize the total request plus travel cost. Borodin, Linial and Saks gave a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n-1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we add some random noise to the request costs and analyze the competitive ratio of WFA on the perturbed sequence. We prove upper and matching lower bounds. Our analysis reveals that the smoothed competitive ratio of WFA is much better than its (worst case) competitive ratio and that it depends on several topological parameters of the graph underlying the metric, such as maximum degree, diameter, etc. For example, already for moderate perturbations, the smoothed competitive ratio of WFA is O(log(n)) on a clique and O(\sqrt{n}) on a line. We also provide the first average case analysis of WFA. For a large class of probability distributions, we prove that WFA has O(log(D)) expected competitive ratio, where D is the maximum degree of the underlying graph.
Cite as

Guido Schäfer and Naveen Sivadasan. Topology Matters: Smoothed Competitiveness of Metrical Task Systems. In Algorithms for Optimization with Incomplete Information. Dagstuhl Seminar Proceedings, Volume 5031, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{schafer_et_al:DagSemProc.05031.29,
  author =	{Sch\"{a}fer, Guido and Sivadasan, Naveen},
  title =	{{Topology Matters: Smoothed Competitiveness of Metrical Task Systems}},
  booktitle =	{Algorithms for Optimization with Incomplete Information},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{5031},
  editor =	{Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05031.29},
  URN =		{urn:nbn:de:0030-drops-684},
  doi =		{10.4230/DagSemProc.05031.29},
  annote =	{Keywords: online algorithm, metrical task systems, work function algorithm, smoothed competitive analysis}
}
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