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

Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

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

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Choosing the root of the tree decomposition when solving WCSPs: preliminary results

In this paper we analyze the effect of selecting the root in a tree decomposition when using decomposition-based backtracking algorithms. We focus on optimization tasks for Graphical Models using the BTD algorithm. We show that the choice of the root typically has a dramatic effect in the solving performance. Then we investigate different simple measures to predict near optimal roots. Our study shows that correlations are often low, so the automatic selection of a near optimal root will require more sophisticated techniques.Projects RTI2018-094403-B-C33, funded by: FEDER/Ministerio de Ciencia e Innovación Agencia Estatal de Investigación,SpainPeer ReviewedPostprint (published version

MaxSAT-Based Bi-Objective Boolean Optimization

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Exploiting short supports for improved encoding of arbitrary constraints into SAT

Encoding to SAT and applying a highly efficient modern SAT solver is an increasingly popular method of solving finite-domain constraint problems. In this paper we study encodings of arbitrary constraints where unit propagation on the encoding provides strong reasoning. Specifically, unit propagation on the encoding simulates generalised arc consistency on the original constraint. To create compact and efficient encodings we use the concept of short support. Short support has been successfully applied to create efficient propagation algorithms for arbitrary constraints. A short support of a constraint is similar to a satisfying tuple however a short support is not required to assign every variable in scope. Some variables are left free to take any value. In some cases a short support representation is smaller than the table of satisfying tuples by an exponential factor. We present two encodings based on short supports and evaluate them on a set of benchmark problems, demonstrating a substantial improvement over the state of the art

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