We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Boolean Satisfiability (SAT) has seen many successful applications in various fields such as Electronic Design Automation and Artificial Intelligence. However, in some cases, it may be required/preferable to use variations of the general SAT problem. In this paper, we consider one important variation, the Partial MAX-SAT problem. Unlike SAT, Partial MAX-SAT has certain constraints (clauses) that are marked as relaxable and the rest are hard, i.e. non-relaxable. The objective is to find a variable assignment that satisfies all non-relaxable clauses together with the maximum number of relaxable ones. We have implemented two solvers for the Partial MAX-SAT problem using a contemporary SAT solver, zChaff. The first approach is a novel diagnosis based algorithm; it iteratively analyzes the UNSAT core of the current SAT instance and eliminates the core through a modification of the problem instance by adding relaxation variables. The second approach is encoding based; it constructs an efficient auxiliary counter that constrains the number of relaxed clauses and supports binary search or linear scan for the optimal solution. Both solvers are complete as they guarantee the optimality of the solution. We discuss the relative strengths and thus applicability of the two solvers for different solution scenarios. Further, we show how both techniques benefit from the persistent learning techniques of incremental SAT. Experiments using practical instances of this problem show significant improvements over the best known solvers.

On solving the partial MAX-SAT problem

Maximum Satisfiability (MaxSAT) is an optimization version of SAT, and many real world applications can be naturally encoded as such. Solving MaxSAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in developing efficient algorithms and several families of algorithms have been proposed. This paper overviews recent approaches to handle MaxSAT and presents a survey of MaxSAT algorithms based on iteratively calling a SAT solver which are particularly effective to solve problems arising in industrial settings. First, classic algorithms based on iteratively calling a SAT solver and updating a bound are overviewed. Such algorithms are referred to as iterative MaxSAT algorithms. Then, more sophisticated algorithms that additionally take advantage of unsatisfiable cores are described, which are referred to as core-guided MaxSAT algorithms. Core-guided MaxSAT algorithms use the information provided by unsatisfiable cores to relax clauses on demand and to create simpler constraints. Finally, a comprehensive empirical study on non-random benchmarks is conducted, including not only the surveyed algorithms, but also other state-of-the-art MaxSAT solvers. The results indicate that (i) core-guided MaxSAT algorithms in general abort in less instances than classic solvers based on iteratively calling a SAT solver and that (ii) core-guided MaxSAT algorithms are fairly competitive compared to other approaches.

/pdf/iterative-and-core-guided-maxsat-solving-a-survey-and-hl9eruxu7b.pdf

Iterative and core-guided MaxSAT solving: A survey and assessment

This paper presents open-wbo, a new MaxSAT solver. open-wbo has two main features. First, it is an open-source solver that can be easily modified and extended. Most MaxSAT solvers are not available in open-source, making it hard to extend and improve current MaxSAT algorithms. Second, open-wbo may use any MiniSAT-like solver as the underlying SAT solver. As many other MaxSAT solvers, open-wbo relies on successive calls to a SAT solver. Even though new techniques are proposed for SAT solvers every year, for many MaxSAT solvers it is hard to change the underlying SAT solver. With open-wbo, advances in SAT technology will result in a free improvement in the performance of the solver. In addition, the paper uses open-wbo to evaluate the impact of using different SAT solvers in the performance of MaxSAT algorithms.

/pdf/open-wbo-a-modular-maxsat-solver-13hcnvpher.pdf

Open-WBO: A Modular MaxSAT Solver ,

Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

https://www.informatik.uni-leipzig.de/~lohrey/12-SURVEY.pdf

Algorithmics on SLP-compressed strings: A survey

https://www.iiia.csic.es/~levy/papers/ArtInt2.pdf

SAT-based MaxSAT algorithms

Recently, Fu and Malik described an unweighted Partial MaxSAT solver based on successive calls to a SAT solver. At the k th iteration the SAT solver tries to certify that there exist an assignment that satisfies all but k clauses. Later Marques-Silva and Planes implemented and extended these ideas. In this paper we present and implement two Partial MaxSAT solvers and the weighted variant of one of them. Both are based on Fu and Malik ideas. We prove the correctness of our algorithm and compare our solver with other (Weighted) MaxSAT and (Weighted) Partial MaxSAT solvers.

/pdf/solving-weighted-partial-maxsat-through-satisfiability-2mkpj6ep49.pdf

Solving (Weighted) Partial MaxSAT through Satisfiability Testing

Resolution for Max-SAT

The research community on complex networks has developed techniques of analysis and algorithms that can be used by the SAT community to improve our knowledge about the structure of industrial SAT instances. It is often argued that modern SAT solvers are able to exploit this hidden structure, without a precise definition of this notion.

In this paper, we show that most industrial SAT instances have a high modularity that is not present in random instances. We also show that successful techniques, like learning, (indirectly) take into account this community structure. Our experimental study reveal that most learnt clauses are local on one of those modules or communities.

/pdf/the-community-structure-of-sat-formulas-4a7nu32a7q.pdf

The community structure of SAT formulas

We present and implement a Weighted Partial MaxSAT solver based on successive calls to a SAT solver. We prove the correctness of our algorithm and compare our solver with other Weighted Partial MaxSAT solvers.

/pdf/a-new-algorithm-for-weighted-partial-maxsat-4iogevwzm1.pdf

Jordi Levy

Papers

SAT-based MaxSAT algorithms

Solving (Weighted) Partial MaxSAT through Satisfiability Testing

Resolution for Max-SAT

The community structure of SAT formulas

A new algorithm for weighted partial MaxSAT