Su-Doku, a popular combinatorial puzzle, provides an excellent testbench for heuristic explorations. Several interesting questions arise from its deceptively simple set of rules. How many distinct Su-Doku grids are there? How to find a solution to a Su-Doku puzzle? Is there a unique solution to a given Su-Doku puzzle? What is a good estimation of a puzzle's difficulty? What is the minimum puzzle size (the number of "givens")?
This paper explores how these questions are related to the well-known alldifferent constraint which emerges in a wide variety of Constraint Satisfaction Problems (CSP) and compares various algorithmic approaches based on different formulations of Su-Doku.
Royal Holloway College（皇家霍洛威学院）；University of London（伦敦大学）
Combinatorial optimization is widely applied in a number of areas nowadays. Unfortunately, many combinatorial optimization problems are NP-hard which usually means that they are unsolvable in practice. However, it is often unnecessary to have an exact solution. In this case one may use heuristic approach to obtain a near-optimal solution in some reasonable time. We focus on two combinatorial optimization problems, namely the Generalized Traveling Salesman Problem and the Multidimensional Assignment Problem. The first problem is an important generalization of the Traveling Salesman Problem; the…
A key challenge in solving a combinatorial optimization problem is how to guide the agent (i.e., solver) to efficiently explore the enormous search space. Conventional approaches often rely on enumeration (e.g., exhaustive, random, or tabu search) or have to restrict the exploration to rather limited regions (e.g., a single path as in iterative algorithms). In this paper, we show it is possible to use machine learning to speedup the exploration. In particular, a value network is trained to evaluate solution candidates, which provides a useful structure (i.e., an approximate value surface) over…
The combinatorial pure exploration of causal bandits is the following online learning task: given a causal graph with unknown causal inference distributions, in each round we choose a subset of variables to intervene or do no intervention, and observe the random outcomes of all random variables, with the goal that using as few rounds as possible, we can output an intervention that gives the best (or almost best) expected outcome on the reward variable $Y$ with probability at least $1-δ$, where $δ$ is a given confidence level. We provide the first gap-dependent and fully adaptive pure explorati…
From logistics to the natural sciences, combinatorial optimisation on graphs underpins numerous real-world applications. Reinforcement learning (RL) has shown particular promise in this setting as it can adapt to specific problem structures and does not require pre-solved instances for these, often NP-hard, problems. However, state-of-the-art (SOTA) approaches typically suffer from severe scalability issues, primarily due to their reliance on expensive graph neural networks (GNNs) at each decision step. We introduce ECORD; a novel RL algorithm that alleviates this expense by restricting the GN…
On Training-Test (Mis)alignment in Unsupervised Combinatorial Optimization: Observation, Empirical Exploration, and Analysis
School of Electrical Engineering, KAIST（韩国庆熙大学电气工程学院）；AI4CO Open-Source Community（AI4CO开源社区）；Kim Jaechul Graduate School of Artificial Intelligence, KAIST（韩国庆熙大学人工智能研究生院）
Partial monitoring games are repeated games where the learner receives feedback that might be different from adversary's move or even the reward gained by the learner. Recently, a general model of combinatorial partial monitoring (CPM) games was proposed \cite{lincombinatorial2014}, where the learner's action space can be exponentially large and adversary samples its moves from a bounded, continuous space, according to a fixed distribution. The paper gave a confidence bound based algorithm (GCB) that achieves $O(T^{2/3}\log T)$ distribution independent and $O(\log T)$ distribution dependent re…
Text-based games -- in which an agent interacts with the world through textual natural language -- present us with the problem of combinatorially-sized action-spaces. Most current reinforcement learning algorithms are not capable of effectively handling such a large number of possible actions per turn. Poor sample efficiency, consequently, results in agents that are unable to pass bottleneck states, where they are unable to proceed because they do not see the right action sequence to pass the bottleneck enough times to be sufficiently reinforced. Building on prior work using knowledge graphs i…
University of Kassel（卡塞尔大学）；University of Passau（帕绍大学）
In domains with high knowledge distribution a natural objective is to create principle foundations for collaborative interactive learning environments. We present a first mathematical characterization of a collaborative learning group, a consortium, based on closure systems of attribute sets and the well-known attribute exploration algorithm from formal concept analysis. To this end, we introduce (weak) local experts for subdomains of a given knowledge domain. These entities are able to refute and potentially accept a given (implicational) query for some closure system that is a restriction of…
Georgia Institute of Technology（佐治亚理工学院）；Microsoft Research（微软研究院）
Text-based games are long puzzles or quests, characterized by a sequence of sparse and potentially deceptive rewards. They provide an ideal platform to develop agents that perceive and act upon the world using a combinatorially sized natural language state-action space. Standard Reinforcement Learning agents are poorly equipped to effectively explore such spaces and often struggle to overcome bottlenecks---states that agents are unable to pass through simply because they do not see the right action sequence enough times to be sufficiently reinforced. We introduce Q*BERT, an agent that learns t…
