Boosting Haplotype Inference with Local Search

Abstract. A very challenging problem in the genetics domain is to infer haplotypes from genotypes. This process is expected to identify genes affecting health, disease and response to drugs. One of the approaches to haplotype inference aims to minimise the number of different haplotypes used, and is known as haplotype inference by pure parsimony (HIPP). The HIPP problem is computationally difficult, being NP-hard. Recently, a SAT-based method (SHIPs) has been proposed to solve the HIPP problem. This method iteratively considers an increasing number of haplotypes, starting from an initial lower bound. Hence, one important aspect of SHIPs is the lower bounding procedure, which reduces the number of iterations of the basic algorithm, and also indirectly simplifies the resulting SAT model. This paper describes the use of local search to improve existing lower bounding procedures. The new lower bounding procedure is guaranteed to be as tight as the existing procedures. In practice the new procedure is in most cases considerably tighter, allowing significant improvement of performance on challenging problem instances.

SAT Modulo Monotonic Theories

We define the concept of a monotonic theory and show how to build efficient SMT (SAT Modulo Theory) solvers, including effective theory propagation and clause learning, for such theories. We present examples showing that monotonic theories arise from many common problems, e.g., graph properties such as reachability, shortest paths, connected components, minimum spanning tree, and max-flow/min-cut, and then demonstrate our framework by building SMT solvers for each of these theories. We apply these solvers to procedural content generation problems, demonstrating major speed-ups over state-of-the-art approaches based on SAT or Answer Set Programming, and easily solving several instances that were previously impractical to solve

Quantum adiabatic optimization and combinatorial landscapes

In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, $\gamma=M/N$. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (versus only energy) is used, and are able to show the existence of a dynamic threshold $\gamma=\gamma_d$ starting with some value of K -- the number of variables in each clause. Beyond dynamic threshold, the algorithm should take exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz. We have been able to map the ensemble of random graphs onto another ensemble with fluctuations significantly reduced. This enabled us to obtain tight upper bounds on satisfiability transition and to recompute the dynamical transition using the extended set of landscapes.Comment: 41 pages, 10 figures; added a paragraph on paper's organization to the introduction, fixed reference

On the Hardness of SAT with Community Structure

Recent attempts to explain the effectiveness of Boolean satisfiability (SAT) solvers based on conflict-driven clause learning (CDCL) on large industrial benchmarks have focused on the concept of community structure. Specifically, industrial benchmarks have been empirically found to have good community structure, and experiments seem to show a correlation between such structure and the efficiency of CDCL. However, in this paper we establish hardness results suggesting that community structure is not sufficient to explain the success of CDCL in practice. First, we formally characterize a property shared by a wide class of metrics capturing community structure, including "modularity". Next, we show that the SAT instances with good community structure according to any metric with this property are still NP-hard. Finally, we study a class of random instances generated from the "pseudo-industrial" community attachment model of Gir\'aldez-Cru and Levy. We prove that, with high probability, instances from this model that have relatively few communities but are still highly modular require exponentially long resolution proofs and so are hard for CDCL. We also present experimental evidence that our result continues to hold for instances with many more communities. This indicates that actual industrial instances easily solved by CDCL may have some other relevant structure not captured by the community attachment model.Comment: 23 pages. Full version of a SAT 2016 pape