A 7/3-approximation for feedback vertex sets in tournaments

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]

A 7/3-Approximation for Feedback Vertex Sets in Tournaments

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]

On minimum integer representations of weighted games

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for $t\ge 4$ types of voters without a minimum integer representation preserving types, i.e., where we additionally require that the weights are equal within equivalence classes of voters.Comment: 29 page

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the "yea" voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. Freixas and Molinero have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.Comment: 7 pages, 6 tables; enumerations correcte

Limit Your Consumption! Finding Bounds in Average-energy Games

Energy games are infinite two-player games played in weighted arenas with quantitative objectives that restrict the consumption of a resource modeled by the weights, e.g., a battery that is charged and drained. Typically, upper and/or lower bounds on the battery capacity are part of the problem description. Here, we consider the problem of determining upper bounds on the average accumulated energy or on the capacity while satisfying a given lower bound, i.e., we do not determine whether a given bound is sufficient to meet the specification, but if there exists a sufficient bound to meet it. In the classical setting with positive and negative weights, we show that the problem of determining the existence of a sufficient bound on the long-run average accumulated energy can be solved in doubly-exponential time. Then, we consider recharge games: here, all weights are negative, but there are recharge edges that recharge the energy to some fixed capacity. We show that bounding the long-run average energy in such games is complete for exponential time. Then, we consider the existential version of the problem, which turns out to be solvable in polynomial time: here, we ask whether there is a recharge capacity that allows the system player to win the game. We conclude by studying tradeoffs between the memory needed to implement strategies and the bounds they realize. We give an example showing that memory can be traded for bounds and vice versa. Also, we show that increasing the capacity allows to lower the average accumulated energy.Comment: In Proceedings QAPL'16, arXiv:1610.0769