Discrete hopfield neural network in restricted maximum k-satisfiability logic programming

Maximum k-Satisfiability (MAX-kSAT) consists of the most consistent interpretation that generate the maximum number of satisfied clauses. MAX-kSAT is an important logic representation in logic programming since not all combinatorial problem is satisfiable in nature. This paper presents Hopfield Neural Network based on MAX-kSAT logical rule. Learning of Hopfield Neural Network will be integrated with Wan Abdullah method and Sathasivam relaxation method to obtain the correct final state of the neurons. The computer simulation shows that MAX-kSAT can be embedded optimally in Hopfield Neural Network

Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.Comment: 17 pages, Conference version to appear in ESA 201

NSNet: A General Neural Probabilistic Framework for Satisfiability Problems

We present the Neural Satisfiability Network (NSNet), a general neural framework that models satisfiability problems as probabilistic inference and meanwhile exhibits proper explainability. Inspired by the Belief Propagation (BP), NSNet uses a novel graph neural network (GNN) to parameterize BP in the latent space, where its hidden representations maintain the same probabilistic interpretation as BP. NSNet can be flexibly configured to solve both SAT and #SAT problems by applying different learning objectives. For SAT, instead of directly predicting a satisfying assignment, NSNet performs marginal inference among all satisfying solutions, which we empirically find is more feasible for neural networks to learn. With the estimated marginals, a satisfying assignment can be efficiently generated by rounding and executing a stochastic local search. For #SAT, NSNet performs approximate model counting by learning the Bethe approximation of the partition function. Our evaluations show that NSNet achieves competitive results in terms of inference accuracy and time efficiency on multiple SAT and #SAT datasets

Monte Carlo Forest Search: UNSAT Solver Synthesis via Reinforcement learning

We introduce Monte Carlo Forest Search (MCFS), an offline algorithm for automatically synthesizing strong tree-search solvers for proving \emph{unsatisfiability} on given distributions, leveraging ideas from the Monte Carlo Tree Search (MCTS) algorithm that led to breakthroughs in AlphaGo. The crucial difference between proving unsatisfiability and existing applications of MCTS, is that policies produce trees rather than paths. Rather than finding a good path (solution) within a tree, the search problem becomes searching for a small proof tree within a forest of candidate proof trees. We introduce two key ideas to adapt to this setting. First, we estimate tree size with paths, via the unbiased approximation from Knuth (1975). Second, we query a strong solver at a user-defined depth rather than learning a policy across the whole tree, in order to focus our policy search on early decisions, which offer the greatest potential for reducing tree size. We then present MCFS-SAT, an implementation of MCFS for learning branching policies for solving the Boolean satisfiability (SAT) problem that required many modifications from AlphaGo. We matched or improved performance over a strong baseline on two well-known SAT distributions (\texttt{sgen}, \texttt{random}). Notably, we improved running time by 9\% on \texttt{sgen} over the \texttt{kcnfs} solver and even further over the strongest UNSAT solver from the 2021 SAT competition

Multi-Objective Probabilistically Constrained Programming with Variable Risk: New Models and Applications

We consider a class of multi-objective probabilistically constrained problems MOPCP with a joint chance constraint, a multi-row random technology matrix, and a risk parameter (i.e., the reliability level) defined as a decision variable. We propose a Boolean modeling framework and derive a series of new equivalent mixed-integer programming formulations. We demonstrate the computational efficiency of the formulations that contain a small number of binary variables. We provide modeling insights pertaining to the most suitable reformulation, to the trade-off between the conflicting cost/revenue and reliability objectives, and to the scalarization parameter determining the relative importance of the objectives. Finally, we propose several MOPCP variants of multi-portfolio financial optimization models that implement a downside risk measure and can be used in a centralized or decentralized investment context. We study the impact of the model parameters on the portfolios, show, via a cross-validation study, the robustness of the proposed models, and perform a comparative analysis of the optimal investment decisions