We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete [28, 29], we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP-hard, and deciding whether there exists a solution with fewer than n cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP ⊆ BU ⊆ TFETR and that LinearBU = PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.

In this paper, we study contention resolution protocols from a game-theoretic perspective. We focus on acknowledgment-based protocols, where a user gets feedback from the channel only when she attempts transmission. In this case she will learn whether her transmission was successful or not. Users that do not transmit will not receive any feedback. We are interested in equilibrium protocols, where no player has an incentive to deviate.The limited feedback makes the design of equilibrium protocols a hard task as best response policies usually have to be modeled as Partially Observable Markov Decision Processes, which are hard to analyze. Nevertheless, we show how to circumvent this for the case of two players and present an equilibrium protocol. For many players, we give impossibility results for a large class of acknowledgment-based protocols, namely age-based and backoff protocols with finite expected finishing time. Finally, we provide an age-based equilibrium protocol, which has infinite expected finishing time, but every player finishes in linear time with high probability.

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The Moran process, as studied by Lieberman, Hauert and Nowak [10], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen uniformly at random (u.a.r.) is a mutant, with fitness r > 0, while all other individuals are residents, with fitness 1. In every step, an individual is chosen with probability proportional to its fitness, and its state (mutant or resident) is passed on to a neighbor which is chosen u.a.r. In this paper, we introduce and study for the first time a generalization of the model of [10] by assuming that different types of individuals perceive the population through different graphs defined on the same vertex set, namely GR = (V, ER) for residents and GM = (V, EM ) for mutants. In this model, we study the fixation probability, namely the probability that eventually only mutants remain in the population, for various pairs of graphs. In particular, in the first part of the paper, we examine how known results from the original single-graph model of [10] can be transferred to our 2-graph model. In that direction, by using a Markov chain abstraction, we provide a generalization of the Isothermal Theorem of [10], that gives sufficient conditions for a pair of graphs to have fixation probability equal to the fixation probability of a pair of cliques; this corresponds to the absorption probability of a birth-death process with forward bias r. In the second part of the paper, we give a 2-player strategic game view of the process where player payoffs correspond to fixation and/or extinction probabilities. In this setting, we attempt to identify best responses for each player. We give evidence that the clique is the most beneficial graph for both players, by proving bounds on the fixation probability when one of the two graphs is complete and the other graph belongs to various natural graph classes. In the final part of the paper, we examine the possibility of efficient approximation of the fixation probability. Interestingly, we show that This work can be found at the proceedings of LATIN 2018 [11]. The work of this author was partially supported by the ERC Project ALGAME. arXiv:1710.07365v2 [q-bio.PE] 23 May 2018 2 T. Melissourgos, S. Nikoletseas, C. Raptopoulos and P. Spirakis there is a pair of graphs for which the fixation probability is exponentially small. This implies that the fixation probability in the general case of an arbitrary pair of graphs cannot be approximated via a method similar to [2]. Nevertheless, we prove that, in the special case when the mutant graph is complete, an efficient approximation of the fixation probability is possible through an FPRAS which we describe.

The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using ε-GCircuit as an intermediate problem.We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as ε-GCircuit pushed to the limit as ε → 1, and we show that the problem is PPAD-complete. We then prove that ε-GCircuit is PPAD-hard for all ε < 0.1 by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games.