Pseudo-Boolean Optimization by Implicit Hitting Sets

Recent developments in applying and extending Boolean satisfiability (SAT) based techniques have resulted in new types of approaches to pseudo-Boolean optimization (PBO), complementary to the more classical integer programming techniques. In this paper, we develop the first approach to pseudo-Boolean optimization based on instantiating the so-called implicit hitting set (IHS) approach, motivated by the success of IHS implementations for maximum satisfiability (MaxSAT). In particular, we harness recent advances in native reasoning techniques for pseudo-Boolean constraints, which enable efficiently identifying inconsistent assignments over subsets of objective function variables (i.e. unsatisfiable cores in the context of PBO), as a basis for developing a native IHS approach to PBO, and study the impact of various search techniques applicable in the context of IHS for PBO. Through an extensive empirical evaluation, we show that the IHS approach to PBO can outperform other currently available PBO solvers, and also provides a complementary approach to PBO when compared to classical integer programming techniques

Pseudo-Boolean Optimization by Implicit Hitting Sets

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Improvements to the Implicit Hitting Set Approach to Pseudo-Boolean Optimization

Peer reviewe

MaxSAT-Based Bi-Objective Boolean Optimization

Peer reviewe

Incremental Maximum Satisfiability

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Investigating Constraint Programming and Hybrid Methods for Real World Industrial Test Laboratory Scheduling

In this paper we deal with a complex real world scheduling problem closely related to the well-known Resource-Constrained Project Scheduling Problem (RCPSP). The problem concerns industrial test laboratories in which a large number of tests has to be performed by qualified personnel using specialised equipment, while respecting deadlines and other constraints. We present different constraint programming models and search strategies for this problem. Furthermore, we propose a Very Large Neighborhood Search approach based on our CP methods. Our models are evaluated using CP solvers and a MIP solver both on real-world test laboratory data and on a set of generated instances of different sizes based on the real-world data. Further, we compare the exact approaches with VLNS and a Simulated Annealing heuristic. We could find feasible solutions for all instances and several optimal solutions and we show that using VLNS we can improve upon the results of the other approaches

Sequential and parallel solution-biased search for subgraph algorithms

Funding: This work was supported by the Engineering and Physical Sciences Research Council (grant numbers EP/P026842/1, EP/M508056/1, and EP/N007565).The current state of the art in subgraph isomorphism solving involves using degree as a value-ordering heuristic to direct backtracking search. Such a search makes a heavy commitment to the first branching choice, which is often incorrect. To mitigate this, we introduce and evaluate a new approach, which we call “solution-biased search”. By combining a slightly-random value-ordering heuristic, rapid restarts, and nogood recording, we design an algorithm which instead uses degree to direct the proportion of search effort spent in different subproblems. This increases performance by two orders of magnitude on satisfiable instances, whilst not affecting performance on unsatisfiable instances. This algorithm can also be parallelised in a very simple but effective way: across both satisfiable and unsatisfiable instances, we get a further speedup of over thirty from thirty-six cores, and over one hundred from ten distributed-memory hosts. Finally, we show that solution-biased search is also suitable for optimisation problems, by using it to improve two maximum common induced subgraph algorithms.Postprin

A Maximum Satisfiability Based Approach to Bi-Objective Boolean Optimization

Many real-world problem settings give rise to NP-hard combinatorial optimization problems. This results in a need for non-trivial algorithmic approaches for finding optimal solutions to such problems. Many such approaches—ranging from probabilistic and meta-heuristic algorithms to declarative programming—have been presented for optimization problems with a single objective. Less work has been done on approaches for optimization problems with multiple objectives. We present BiOptSat, an exact declarative approach for finding so-called Pareto-optimal solutions to bi-objective optimization problems. A bi-objective optimization problem arises for example when learning interpretable classifiers and the size, as well as the classification error of the classifier should be taken into account as objectives. Using propositional logic as a declarative programming language, we seek to extend the progress and success in maximum satisfiability (MaxSAT) solving to two objectives. BiOptSat can be viewed as an instantiation of the lexicographic method and makes use of a single SAT solver that is preserved throughout the entire search procedure. It allows for solving three tasks for bi-objective optimization: finding a single Pareto-optimal solution, finding one representative solution for each Pareto point, and enumerating all Pareto-optimal solutions. We provide an open-source implementation of five variants of BiOptSat, building on different algorithms proposed for MaxSAT. Additionally, we empirically evaluate these five variants, comparing their runtime performance to that of three key competing algorithmic approaches. The empirical comparison in the contexts of learning interpretable decision rules and bi-objective set covering shows practical benefits of our approach. Furthermore, for the best-performing variant of BiOptSat, we study the effects of proposed refinements to determine their effectiveness

Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems

Pseudo-Booleanilainen optimisaatio käyttäen implisiittisiä osumisjoukkoja

There are many computationally difficult problems where the task is to find a solution with the lowest cost possible that fulfills a given set of constraints. Such problems are often NP-hard and are encountered in a variety of real-world problem domains, including planning and scheduling. NP-hard problems are often solved using a declarative approach by encoding the problem into a declarative constraint language and solving the encoding using a generic algorithm for that language. In this thesis we focus on pseudo-Boolean optimization (PBO), a special class of integer programs (IP) that only contain variables that admit the values 0 and 1. We propose a novel approach to PBO that is based on the implicit hitting set (IHS) paradigm, which uses two separate components. An IP solver is used to find an optimal solution under an incomplete set of constraints. A pseudo-Boolean satisfiability solver is used to either validate the feasibility of the solution or to extract more constraints to the integer program. The IHS-based PBO algorithm iteratively invokes the two algorithms until an optimal solution to a given PBO instance is found. In this thesis we lay out the IHS-based PBO solving approach in detail. We implement the algorithm as the PBO-IHS solver by making use of recent advances in reasoning techniques for pseudo-Boolean constraints. Through extensive empirical evaluation we show that our PBO-IHS solver outperforms other available specialized PBO solvers and has complementary performance compared to classical integer programming techniques