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Connected Subgraph Defense Games

Akrida, Eleni C.; Deligkas, Argyrios; Melissourgos, Themistoklis; Spirakis, Paul G.

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Authors

Argyrios Deligkas

Themistoklis Melissourgos

Paul G. Spirakis



Abstract

We study a security game over a network played between a defender and k attackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for λ constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is NP-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any λ; this is the worst equilibrium defense ratio over all graphs.

Citation

Akrida, E. C., Deligkas, A., Melissourgos, T., & Spirakis, P. G. (2021). Connected Subgraph Defense Games. Algorithmica, 83(11), 3403-3431. https://doi.org/10.1007/s00453-021-00858-z

Journal Article Type Article
Acceptance Date Jul 16, 2021
Online Publication Date Aug 1, 2021
Publication Date 2021-11
Deposit Date Nov 23, 2021
Publicly Available Date Nov 23, 2021
Journal Algorithmica
Print ISSN 0178-4617
Electronic ISSN 1432-0541
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 83
Issue 11
Pages 3403-3431
DOI https://doi.org/10.1007/s00453-021-00858-z

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Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/

Copyright Statement
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.





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