arXiv subjects

Pierre Coucheney

Publications and source records attributed to Pierre Coucheney.

7 recordsLinked to original sources

arXiv2024

Linear Extensions of Rotor-Routing in Directed Graphs: Reachability Problems

We develop a unified framework for rotor-routing that extends the classical model to a broad class of multigraphs equipped with Generalized Rotor Mechanisms (GRM). This perspective places rotor-routing on the same footing as abelian sandpiles by interpreting both as conservative instances of Vector Addition Systems (VAS). Within this framework, routing becomes a linear transformation governed by arc mechanisms, while legality is enforced through non-negativity constraints. We introduce four routing models -- free routing, standard rotor-routing, cyclic GRM routing, and fully general GRM routing -- and study their reachability problems in both the linear and legal settings. Our results generalize previous characterizations for standard rotor-routing and extend them to the GRM setting. In particular, we show that legal reachability in GRM multigraphs is NP-complete, whereas the cyclic GRM routing model, which includes the classical rotor-router, admits a polynomial-time algorithm.

cs.DM↗

arXiv2023

Generalized ARRIVAL Problem for Rotor Walks in Path Multigraphs

Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical properties similar to random walks. In this study, we investigate a generalized version of the reachability problem known as ARRIVAL in Path Multigraphs, which involves predicting the number of particles that will reach designated target vertices. We show that this problem is in NP and co-NP in the general case. However, we exhibit algebraic invariants for Path Multigraphs that allow us to solve the problem efficiently, even for an exponential configuration of particles. These invariants are based on harmonic functions and are connected to the decomposition of integers in rational bases.

cs.DM↗

arXiv2022

Polynomial Time Algorithm for ARRIVAL on Tree-like Multigraphs

A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, is reached. The ARRIVAL problem, as defined by Dohrau and al., consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP$\cap$co-NP without being known to be in P. Several variants have been studied where we add one or two players to the model, defining deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games) with rotor-routing rules instead of random transitions. The corresponding decision problem address the existence of strategies for players that ensure some condition on the reached sink. These problems are known to be $NP$-complete for one player and $PSPACE$-complete for two players. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that the different variants of the reachability problem with zero, one, or two players can be solved in linear time, while the number of steps of rotor walks can still be exponential. To achieve this, we define a notion of return flow, which counts the number of times the pebble will bounce back in subtrees of the graph.

cs.GT↗

arXiv2019

Solving Simple Stochastic Games with few Random Nodes faster using Bland's Rule

The best algorithm so far for solving Simple Stochastic Games is Ludwig's randomized algorithm which works in expected $2^{O(\sqrt{n})}$ time. We first give a simpler iterative variant of this algorithm, using Bland's rule from the simplex algorithm, which uses exponentially less random bits than Ludwig's version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn whose worst case complexity is $O(k!)$, where $k$ is the number of random nodes. Our algorithm has an expected running time of $2^{O(k)}$, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs.

cs.DS↗

arXiv2017

Optimal Distributed Channel Assignment in D2D Networks Using Learning in Noisy Potential Games

We present a novel solution for Channel Assignment Problem (CAP) in Device-to-Device (D2D) wireless networks that takes into account the throughput estimation noise. CAP is known to be NP-hard in the literature and there is no practical optimal learning algorithm that takes into account the estimation noise. In this paper, we first formulate the CAP as a stochastic optimization problem to maximize the expected sum data rate. To capture the estimation noise, CAP is modeled as a noisy potential game, a novel notion we introduce in this paper. Then, we propose a distributed Binary Log-linear Learning Algorithm (BLLA) that converges to the optimal channel assignments. Convergence of BLLA is proved for bounded and unbounded noise. Proofs for fixed and decreasing temperature parameter of BLLA are provided. A sufficient number of estimation samples is given that guarantees the convergence to the optimal state. We assess the performance of BLLA by extensive simulations, which show that the sum data rate increases with the number of channels and users. Contrary to the better response algorithm, the proposed algorithm achieves the optimal channel assignments distributively even in presence of estimation noise.

cs.NI↗

arXiv2014

Penalty-regulated dynamics and robust learning procedures in games

Starting from a heuristic learning scheme for N-person games, we derive a new class of continuous-time learning dynamics consisting of a replicator-like drift adjusted by a penalty term that renders the boundary of the game's strategy space repelling. These penalty-regulated dynamics are equivalent to players keeping an exponentially discounted aggregate of their on-going payoffs and then using a smooth best response to pick an action based on these performance scores. Owing to this inherent duality, the proposed dynamics satisfy a variant of the folk theorem of evolutionary game theory and they converge to (arbitrarily precise) approximations of Nash equilibria in potential games. Motivated by applications to traffic engineering, we exploit this duality further to design a discrete-time, payoff-based learning algorithm which retains these convergence properties and only requires players to observe their in-game payoffs: moreover, the algorithm remains robust in the presence of stochastic perturbations and observation errors, and it does not require any synchronization between players.

math.OC↗

arXiv2009

A Distributed Algorithm for Fair and Efficient User-Network Association in Multi-Technology Wireless Networks

Recent mobile equipment (as well as the norm IEEE 802.21) now offers the possibility for users to switch from one technology to another (vertical handover). This allows flexibility in resource assignments and, consequently, increases the potential throughput allocated to each user. In this paper, we design a fully distributed algorithm based on trial and error mechanisms that exploits the benefits of vertical handover by finding fair and efficient assignment schemes. On the one hand, mobiles gradually update the fraction of data packets they send to each network based on the rewards they receive from the stations. On the other hand, network stations send rewards to each mobile that represent the impact each mobile has on the cell throughput. This reward function is closely related to the concept of marginal cost in the pricing literature. Both the station and the mobile algorithms are simple enough to be implemented in current standard equipment. Based on tools from evolutionary games, potential games and replicator dynamics, we analytically show the convergence of the algorithm to solutions that are efficient and fair in terms of throughput. Moreover, we show that after convergence, each user is connected to a single network cell which avoids costly repeated vertical handovers. Several simple heuristics based on this algorithm are proposed to achieve fast convergence. Indeed, for implementation purposes, the number of iterations should remain in the order of a few tens. We also compare, for different loads, the quality of their solutions.

cs.GT↗