Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond

Authors Georgios Birmpas, Evangelos Markakis, Guido Schäfer

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Georgios Birmpas
  • Department of Computer Science, University of Oxford, UK
  • Department of Informatics, Athens University of Economics and Business, Greece
Evangelos Markakis
  • Department of Informatics, Athens University of Economics and Business, Greece
Guido Schäfer
  • Networks and Optimization group, Centrum Wiskunde & Informatica (CWI), The Netherlands
  • Dept. of Econometrics and Operations Research, Vrije Universiteit Amsterdam, The Netherlands


Part of this work was done while the first author was an intern of the Networks and Optimization group at Centrum Wiskunde & Informatica. The first author was also partially supported by the ERC Advanced Grant 321171 (ALGAME).

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory and mechanism design
  • Approximation Algorithms
  • Combinatorial Cost Sharing
  • Mechanism Design
  • Truthfulness


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