@inproceedings{12101,
  abstract     = {Spatial games form a widely-studied class of games from biology and physics modeling the evolution of social behavior. Formally, such a game is defined by a square (d by d) payoff matrix M and an undirected graph G. Each vertex of G represents an individual, that initially follows some strategy i ∈ {1,2,…,d}. In each round of the game, every individual plays the matrix game with each of its neighbors: An individual following strategy i meeting a neighbor following strategy j receives a payoff equal to the entry (i,j) of M. Then, each individual updates its strategy to its neighbors' strategy with the highest sum of payoffs, and the next round starts. The basic computational problems consist of reachability between configurations and the average frequency of a strategy. For general spatial games and graphs, these problems are in PSPACE. In this paper, we examine restricted setting: the game is a prisoner’s dilemma; and G is a subgraph of grid. We prove that basic computational problems for spatial games with prisoner’s dilemma on a subgraph of a grid are PSPACE-hard.},
  author       = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Jecker, Ismael R and Svoboda, Jakub},
  booktitle    = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  isbn         = {9783959772617},
  issn         = {1868-8969},
  location     = {Madras, India},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Complexity of spatial games}},
  doi          = {10.4230/LIPIcs.FSTTCS.2022.11},
  volume       = {250},
  year         = {2022},
}

@inproceedings{12102,
  abstract     = {Given a Markov chain M = (V, v_0, δ), with state space V and a starting state v_0, and a probability threshold ε, an ε-core is a subset C of states that is left with probability at most ε. More formally, C ⊆ V is an ε-core, iff ℙ[reach (V\C)] ≤ ε. Cores have been applied in a wide variety of verification problems over Markov chains, Markov decision processes, and probabilistic programs, as a means of discarding uninteresting and low-probability parts of a probabilistic system and instead being able to focus on the states that are likely to be encountered in a real-world run. In this work, we focus on the problem of computing a minimal ε-core in a Markov chain. Our contributions include both negative and positive results: (i) We show that the decision problem on the existence of an ε-core of a given size is NP-complete. This solves an open problem posed in [Jan Kretínský and Tobias Meggendorfer, 2020]. We additionally show that the problem remains NP-complete even when limited to acyclic Markov chains with bounded maximal vertex degree; (ii) We provide a polynomial time algorithm for computing a minimal ε-core on Markov chains over control-flow graphs of structured programs. A straightforward combination of our algorithm with standard branch prediction techniques allows one to apply the idea of cores to find a subset of program lines that are left with low probability and then focus any desired static analysis on this core subset.},
  author       = {Ahmadi, Ali and Chatterjee, Krishnendu and Goharshady, Amir Kafshdar and Meggendorfer, Tobias and Safavi Hemami, Roodabeh and Zikelic, Dorde},
  booktitle    = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  isbn         = {9783959772617},
  issn         = {1868-8969},
  location     = {Madras, India},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Algorithms and hardness results for computing cores of Markov chains}},
  doi          = {10.4230/LIPIcs.FSTTCS.2022.29},
  volume       = {250},
  year         = {2022},
}