arXiv subjects

Mikael Touati

Publications and source records attributed to Mikael Touati.

3 recordsLinked to original sources

arXiv2020

Solving Random Parity Games in Polynomial Time

We consider the problem of solving random parity games. We prove that parity games exibit a phase transition threshold above $d_P$, so that when the degree of the graph that defines the game has a degree $d > d_P$ then there exists a polynomial time algorithm that solves the game with high probability when the number of nodes goes to infinity. We further propose the SWCP (Self-Winning Cycles Propagation) algorithm and show that, when the degree is large enough, SWCP solves the game with high probability. Furthermore, the complexity of SWCP is polynomial $O\Big(|{\cal V}|^2 + |{\cal V}||{\cal E}|\Big)$. The design of SWCP is based on the threshold for the appearance of particular types of cycles in the players' respective subgraphs. We further show that non-sparse games can be solved in time $O(|{\cal V}|)$ with high probability, and emit a conjecture concerning the hardness of the $d=2$ case.

cs.LO↗

arXiv2019

Computationally Efficient Estimation of the Spectral Gap of a Markov Chain

We consider the problem of estimating from sample paths the absolute spectral gap $γ_*$ of a reversible, irreducible and aperiodic Markov chain $(X_t)_{t \in \mathbb{N}}$ over a finite state space $Ω$. We propose the ${\tt UCPI}$ (Upper Confidence Power Iteration) algorithm for this problem, a low-complexity algorithm which estimates the spectral gap in time ${\cal O}(n)$ and memory space ${\cal O}((\ln n)^2)$ given $n$ samples. This is in stark contrast with most known methods which require at least memory space ${\cal O}(|Ω|)$, so that they cannot be applied to large state spaces. Furthermore, ${\tt UCPI}$ is amenable to parallel implementation.

stat.ML↗

arXiv2016

Controlled Matching Game for Resource Allocation and User Association in WLANs

In multi-rate IEEE 802.11 WLANs, the traditional user association based on the strongest received signal and the well known anomaly of the MAC protocol can lead to overloaded Access Points (APs), and poor or heterogeneous performance. Our goal is to propose an alternative game-theoretic approach for association. We model the joint resource allocation and user association as a matching game with complementarities and peer effects consisting of selfish players solely interested in their individual throughputs. Using recent game-theoretic results we first show that various resource sharing protocols actually fall in the scope of the set of stability-inducing resource allocation schemes. The game makes an extensive use of the Nash bargaining and some of its related properties that allow to control the incentives of the players. We show that the proposed mechanism can greatly improve the efficiency of 802.11 with heterogeneous nodes and reduce the negative impact of peer effects such as its MAC anomaly. The mechanism can be implemented as a virtual connectivity management layer to achieve efficient APs-user associations without modification of the MAC layer.

cs.GT↗