arXiv subjects

Abdallah Saffidine

Publications and source records attributed to Abdallah Saffidine.

At least 19 recordsLinked to original sources

arXiv2026

Refuting the Direct Sum Conjecture for Total Functions in Deterministic Communication Complexity

In communication complexity the input of a function $f:X\times Y\rightarrow Z$ is distributed between two players Alice and Bob. If Alice knows only $x\in X$ and Bob only $y\in Y$, how much information must Alice and Bob share to be able to elicit the value of $f(x,y)$? Do we need $\ell$ more resources to solve $\ell$ instances of a problem? This question is the direct sum question and has been studied in many computational models. In this paper we focus on the case of 2-party deterministic communication complexity and give a counterexample to the direct sum conjecture in its strongest form. To do so we exhibit a family of functions for which the complexity of solving $\ell$ instances is less than $(1 -ε)\ell$ times the complexity of solving one instance for some small enough $ε>0$. We use a customised method in the analysis of our family of total functions, showing that one can force the alternation of rounds between players. This idea allows us to exploit the integrality of the complexity measure to create an increasing gap between the complexity of solving the instances independently with that of solving them together.

cs.CC↗

arXiv2025

Repairing General Game Descriptions (extended version)

The Game Description Language (GDL) is a widely used formalism for specifying the rules of general games. Writing correct GDL descriptions can be challenging, especially for non-experts. Automated theorem proving has been proposed to assist game design by verifying if a GDL description satisfies desirable logical properties. However, when a description is proved to be faulty, the repair task itself can only be done manually. Motivated by the work on repairing unsolvable planning domain descriptions, we define a more general problem of finding minimal repairs for GDL descriptions that violate formal requirements, and we provide complexity results for various computational problems related to minimal repair. Moreover, we present an Answer Set Programming-based encoding for solving the minimal repair problem and demonstrate its application for automatically repairing ill-defined game descriptions.

cs.LO↗

arXiv2025

Polynomial Prenexing of QBFs with Non-Monotone Boolean Operators

It is well-known that every quantified boolean formula (QBF) can be transformed into a prenex QBF whose only boolean operators are negation, conjunction, and disjunction. It is also well-known that the transformation is polynomial if the boolean operators of the original QBF are restricted to negation, conjunction, and disjunction. In contrast, up to now no polynomial transformation has been found when the original QBF contains other boolean operators such as biconditionals or exclusive disjunction. We define such a transformation and show that it is polynomial and preserves quantifier depth.

cs.CC↗

arXiv2025

An Improved Upper Bound on the Threshold Bias of the Oriented-cycle Game

We study the $b$-biased Oriented-cycle game where two players, OMaker and OBreaker, take turns directing the edges of $K_n$ (the complete graph on $n$ vertices). In each round, OMaker directs one previously undirected edge followed by OBreaker directing between one and $b$ previously undirected edges. The game ends once all edges have been directed, and OMaker wins if and only if the resulting tournament contains a directed cycle. Bollob\'as and Szab\'o asked the following question: what is the largest value of the bias $b$ for which OMaker has a winning strategy? Ben-Eliezer, Krivelevich and Sudakov proved that OMaker has a winning strategy for $b \leq n/2 - 2$. In the other direction, Clemens and Liebenau proved that OBreaker has a winning strategy for $b \geq 5n/6+2$. Inspired by their approach, we propose a significantly stronger strategy for OBreaker which we prove to be winning for $b \geq 0.7841n + O(1)$.

math.CO↗

arXiv2024

Constructions, bounds, and algorithms for peaceable queens

The peaceable queens problem asks to determine the maximum number $a(n)$ such that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an $n \times n$ chessboard so that no queen can capture any queen of the opposite color. In this paper, we consider the peaceable queens problem and its variant on the toroidal board. For the regular board, we show that $a(n) \leq 0.1716n^2$, for all sufficiently large $n$. This improves on the bound $a(n) \leq 0.25n^2$ of van Bommel and MacEachern. For the toroidal board, we provide new upper and lower bounds. Somewhat surprisingly, our bounds show that there is a sharp contrast in behaviour between the odd torus and the even torus. Our lower bounds are given by explicit constructions. For the upper bounds, we formulate the problem as a non-linear optimization problem with at most $100$ variables, regardless of the size of the board. We solve our non-linear program exactly using modern optimization software. We also provide a local search algorithm and a software implementation which converges very rapidly to solutions which appear optimal. Our algorithm is sufficiently robust that it works on both the regular and toroidal boards. For example, for the regular board, the algorithm quickly finds the so-called Ainley construction. Thus, our work provides some further evidence that the Ainley construction is indeed optimal.

math.CO↗

arXiv2024

Fair Railway Network Design

When designing a public transportation network in a country, one may want to minimise the sum of travel duration of all inhabitants. This corresponds to a purely utilitarian view and does not involve any fairness consideration, as the resulting network will typically benefit the capital city and/or large central cities while leaving some peripheral cities behind. On the other hand, a more egalitarian view will allow some people to travel between peripheral cities without having to go through a central city. We define a model, propose algorithms for computing solution networks, and report on experiments based on real data.

cs.SI↗

arXiv2024

Enhancing Reinforcement Learning Through Guided Search

With the aim of improving performance in Markov Decision Problem in an Off-Policy setting, we suggest taking inspiration from what is done in Offline Reinforcement Learning (RL). In Offline RL, it is a common practice during policy learning to maintain proximity to a reference policy to mitigate uncertainty, reduce potential policy errors, and help improve performance. We find ourselves in a different setting, yet it raises questions about whether a similar concept can be applied to enhance performance ie, whether it is possible to find a guiding policy capable of contributing to performance improvement, and how to incorporate it into our RL agent. Our attention is particularly focused on algorithms based on Monte Carlo Tree Search (MCTS) as a guide.MCTS renowned for its state-of-the-art capabilities across various domains, catches our interest due to its ability to converge to equilibrium in single-player and two-player contexts. By harnessing the power of MCTS as a guide for our RL agent, we observed a significant performance improvement, surpassing the outcomes achieved by utilizing each method in isolation. Our experiments were carried out on the Atari 100k benchmark.

cs.AI↗

arXiv2024

Perfect Information Monte Carlo with Postponing Reasoning

Imperfect information games, such as Bridge and Skat, present challenges due to state-space explosion and hidden information, posing formidable obstacles for search algorithms. Determinization-based algorithms offer a resolution by sampling hidden information and solving the game in a perfect information setting, facilitating rapid and effective action estimation. However, transitioning to perfect information introduces challenges, notably one called strategy fusion.This research introduces `Extended Perfect Information Monte Carlo' (EPIMC), an online algorithm inspired by the state-of-the-art determinization-based approach Perfect Information Monte Carlo (PIMC). EPIMC enhances the capabilities of PIMC by postponing the perfect information resolution, reducing alleviating issues related to strategy fusion. However, the decision to postpone the leaf evaluator introduces novel considerations, such as the interplay between prior levels of reasoning and the newly deferred resolution. In our empirical analysis, we investigate the performance of EPIMC across a range of games, with a particular focus on those characterized by varying degrees of strategy fusion. Our results demonstrate notable performance enhancements, particularly in games where strategy fusion significantly impacts gameplay. Furthermore, our research contributes to the theoretical foundation of determinization-based algorithms addressing challenges associated with strategy fusion.%, thereby enhancing our understanding of these algorithms within the context of imperfect information game scenarios.

cs.AI↗

arXiv2024

A Piecewise Approach for the Analysis of Exact Algorithms

To analyze the worst-case running time of branching algorithms, the majority of work in exponential time algorithms focuses on designing complicated branching rules over developing better analysis methods for simple algorithms. In the mid-$2000$s, Fomin et al. [2005] introduced measure & conquer, an advanced general analysis method, sparking widespread adoption for obtaining tighter worst-case running time upper bounds for many fundamental NP-complete problems. Yet, much potential in this direction remains untapped, as most subsequent work applied it without further advancement. Motivated by this, we present piecewise analysis, a new general method that analyzes the running time of branching algorithms. Our approach is to define a similarity ratio that divides instances into groups and then analyze the running time within each group separately. The similarity ratio is a scale between two parameters of an instance I. Instead of relying on a single measure and a single analysis for the whole instance space, our method allows to take advantage of different intrinsic properties of instances with different similarity ratios. To showcase its potential, we reanalyze two $17$-year-old algorithms from Fomin et al. [2007] that solve $4$-Coloring and #$3$-Coloring respectively. The original analysis in their paper gave running times of $O(1.7272^n)$ and $O(1.6262^n)$ respectively for these algorithms, our analysis improves these running times to $O(1.7207^n)$ and $O(1.6225^n)$.

cs.DS↗

arXiv2024

Deep Reinforcement Learning for 5*5 Multiplayer Go

In recent years, much progress has been made in computer Go and most of the results have been obtained thanks to search algorithms (Monte Carlo Tree Search) and Deep Reinforcement Learning (DRL). In this paper, we propose to use and analyze the latest algorithms that use search and DRL (AlphaZero and Descent algorithms) to automatically learn to play an extended version of the game of Go with more than two players. We show that using search and DRL we were able to improve the level of play, even though there are more than two players.

cs.AI↗

arXiv2024

Mixture of Public and Private Distributions in Imperfect Information Games

In imperfect information games (e.g. Bridge, Skat, Poker), one of the fundamental considerations is to infer the missing information while at the same time avoiding the disclosure of private information. Disregarding the issue of protecting private information can lead to a highly exploitable performance. Yet, excessive attention to it leads to hesitations that are no longer consistent with our private information. In our work, we show that to improve performance, one must choose whether to use a player's private information. We extend our work by proposing a new belief distribution depending on the amount of private and public information desired. We empirically demonstrate an increase in performance and, with the aim of further improving performance, the new distribution should be used according to the position in the game. Our experiments have been done on multiple benchmarks and in multiple determinization-based algorithms (PIMC and IS-MCTS).

cs.AI↗

arXiv2024

Generalizing Roberts' characterization of unit interval graphs

For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs are unit and balanced $d$-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for $d$-interval graphs. In particular, we prove that for any $d \geq 2$, if $G$ is a $K_{1,2d+1}$-free interval graph, then $G$ is a unit $d$-interval graph. However, somehow surprisingly, under the same assumptions, $G$ is not always a \emph{disjoint} unit $d$-interval graph. This implies that the class of disjoint unit $d$-interval graphs is strictly included in the class of unit $d$-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint $d$-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for $d>2$.

cs.DM↗

arXiv2023

Positional Games and QBF: A Polished Encoding

Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBFs) such that a game instance admits a winning strategy for the first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is generic and lets us encode other positional games, such as Hex. Second, the structural properties of positional games, together with careful treatment of illegal moves, let us generate more compact instances that can be solved faster by state-of-the-art QBF solvers. We establish the latter fact through extensive experiments. Finally, the compactness of our new encoding makes it feasible to translate realistic game problems. We identify a few such problems of historical significance and put them forward to the QBF community as milestones of increasing difficulty.

cs.LO↗

arXiv2023

Vision Transformers for Computer Go

Motivated by the success of transformers in various fields, such as language understanding and image analysis, this investigation explores their application in the context of the game of Go. In particular, our study focuses on the analysis of the Transformer in Vision. Through a detailed analysis of numerous points such as prediction accuracy, win rates, memory, speed, size, or even learning rate, we have been able to highlight the substantial role that transformers can play in the game of Go. This study was carried out by comparing them to the usual Residual Networks.

cs.AI↗

arXiv2023

Towards Tackling MaxSAT by Combining Nested Monte Carlo with Local Search

Recent work proposed the UCTMAXSAT algorithm to address Maximum Satisfiability Problems (MaxSAT) and shown improved performance over pure Stochastic Local Search algorithms (SLS). UCTMAXSAT is based on Monte Carlo Tree Search but it uses SLS instead of purely random playouts. In this work, we introduce two algorithmic variations over UCTMAXSAT. We carry an empirical analysis on MaxSAT benchmarks from recent competitions and establish that both ideas lead to performance improvements. First, a nesting of the tree search inspired by the Nested Monte Carlo Search algorithm is effective on most instance types in the benchmark. Second, we observe that using a static flip limit in SLS, the ideal budget depends heavily on the instance size and we propose to set it dynamically. We show that it is a robust way to achieve comparable performance on a variety of instances without requiring additional tuning.

cs.AI↗

arXiv2023

Implicit State and Goals in QBF Encodings for Positional Games (extended version)

We address two bottlenecks for concise QBF encodings of maker-breaker positional games, like Hex and Tic-Tac-Toe. Our baseline is a QBF encoding with explicit variables for board positions and an explicit representation of winning configurations. The first improvement is inspired by lifted planning and avoids variables for explicit board positions, introducing a universal quantifier representing a symbolic board state. The second improvement represents the winning configurations implicitly, exploiting their structure. The paper evaluates the size of several encodings, depending on board size and game depth. It also reports the performance of QBF solvers on these encodings. We evaluate the techniques on Hex instances and also apply them to Harary's Tic-Tac-Toe. In particular, we study scalability to 19$\times$19 boards, played in human Hex tournaments.

cs.AI↗

arXiv2022

On Bellman's Optimality Principle for zs-POSGs

Many non-trivial sequential decision-making problems are efficiently solved by relying on Bellman's optimality principle, i.e., exploiting the fact that sub-problems are nested recursively within the original problem. Here we show how it can apply to (infinite horizon) 2-player zero-sum partially observable stochastic games (zs-POSGs) by (i) taking a central planner's viewpoint, which can only reason on a sufficient statistic called occupancy state, and (ii) turning such problems into zero-sum occupancy Markov games (zs-OMGs). Then, exploiting the Lipschitz-continuity of the value function in occupancy space, one can derive a version of the HSVI algorithm (Heuristic Search Value Iteration) that provably finds an $ε$-Nash equilibrium in finite time.

cs.AI↗

arXiv2022

HSVI can solve zero-sum Partially Observable Stochastic Games

State-of-the-art methods for solving 2-player zero-sum imperfect information games rely on linear programming or regret minimization, though not on dynamic programming (DP) or heuristic search (HS), while the latter are often at the core of state-of-the-art solvers for other sequential decision-making problems. In partially observable or collaborative settings (e.g., POMDPs and Dec- POMDPs), DP and HS require introducing an appropriate statistic that induces a fully observable problem as well as bounding (convex) approximators of the optimal value function. This approach has succeeded in some subclasses of 2-player zero-sum partially observable stochastic games (zs- POSGs) as well, but how to apply it in the general case still remains an open question. We answer it by (i) rigorously defining an equivalent game to work with, (ii) proving mathematical properties of the optimal value function that allow deriving bounds that come with solution strategies, (iii) proposing for the first time an HSVI-like solver that provably converges to an $ε$-optimal solution in finite time, and (iv) empirically analyzing it. This opens the door to a novel family of promising approaches complementing those relying on linear programming or iterative methods.

cs.AI↗