We study the phase diagram and the algorithmic hardness of the random `locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean formulas or graph coloring. The special property of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard constraint satisfaction problems.
Planting a solution into the random RB model, which is a prototype of random constraint satisfaction problem (CSP) with growing domains, can generate very hard satisfiable CSP benchmarks. We study the solution space structure of the planted RB model. With constraint density growing, we find that this model goes through four phase transitions. In the replica symmetric phase, what we call the independent phase transition occurs, after which the planted cluster (cluster containing the planted solution) is separated from the giant cluster. Then the solutions except that in the planted cluster go t…
Relation between problem hardness and solution space structure is an important research aspect. Model d-k-CSP generates very hard instances when $r=1$ and $r$ is near 1, where $r$ represents normalized constraint density. We find that when $r$ is below and close to 1, the solution space contains many widely distributed well-separated small cluster-regions (a cluster-region is a union of some clusters), which should the reason that the generated instances are hard to solve.
Benchmarking Graph Neural Networks in Solving Hard Constraint Satisfaction Problems
Department of Computing Sciences, Bocconi University（博科尼大学计算机科学系）；Bocconi Institute for Data Science and Analytics (BIDSA)（博科尼数据科学与分析研究所）；Department of Physics and Astronomy and INFN, University of Florence（佛罗伦萨大学物理与天文学系及意大利国家核物理研究所）；Dipartimento di Fisica, Sapienza Università di Roma（罗马萨皮恩扎大学物理系）；CNR - Nanotec, unità di Roma, P.le Aldo Moro 5, 00185 Rome, Italy（意大利罗马国家科研委员会纳米技术单元，Aldo Moro大道5号，00185罗马，意大利）；Department of Theoretical Physics, Physics Faculty, University of Havana（古巴哈瓦那大学理论物理系）；INFN, sezione di Roma1, P.le Aldo Moro 5, 00185 Rome, Italy（意大利罗马1研究所，Aldo Moro大道5号，00185罗马，意大利）
Stanford University（斯坦福大学）；Università di Roma La Sapienza（罗马第一大学）；Ecole Normale（巴黎高等师范学院）
Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after each step. Its outcome provides an heuristic to make choices at next step. This approach has been referred to as `decimation,' with reference to analogous procedures in statistical physics. The behavior of decimation procedures is poorly understood. Here we consider a simple randomized decimation algorithm based on belief propagation (BP), and analy…
University of Shanghai for Science and Technology（上海理工大学）；South China Normal University（华南师范大学）
Message passing algorithms, whose iterative nature captures well complicated interactions among interconnected variables in complex systems and extracts information from the fixed point of iterated messages, provide a powerful toolkit in tackling hard computational tasks in optimization, inference, and learning problems. In the context of constraint satisfaction problems (CSPs), when a control parameter (such as constraint density) is tuned, multiple threshold phenomena emerge, signaling fundamental structural transitions in their solution space. Finding solutions around these transition point…
