104,399 research outputs found
On the Lovász theta function for independent sets in sparse graphs
We consider the maximum independent set problem on graphs with maximum degree~. We show that the integrality gap of the Lov\'asz -function based SDP is . This improves on the previous best result of , and almost matches the integrality gap of recently shown for stronger SDPs, namely those obtained using poly- levels of the semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of -free graphs for large values of . We also show how to obtain an algorithmic version of the above-mentioned -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of matches the best unique-games-based hardness result up to lower-order poly- factors
Average-energy games
Two-player quantitative zero-sum games provide a natural framework to
synthesize controllers with performance guarantees for reactive systems within
an uncontrollable environment. Classical settings include mean-payoff games,
where the objective is to optimize the long-run average gain per action, and
energy games, where the system has to avoid running out of energy.
We study average-energy games, where the goal is to optimize the long-run
average of the accumulated energy. We show that this objective arises naturally
in several applications, and that it yields interesting connections with
previous concepts in the literature. We prove that deciding the winner in such
games is in NP inter coNP and at least as hard as solving mean-payoff games,
and we establish that memoryless strategies suffice to win. We also consider
the case where the system has to minimize the average-energy while maintaining
the accumulated energy within predefined bounds at all times: this corresponds
to operating with a finite-capacity storage for energy. We give results for
one-player and two-player games, and establish complexity bounds and memory
requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
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
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