An Atypical Survey of Typical-Case Heuristic Algorithms

Heuristic approaches often do so well that they seem to pretty much always give the right answer. How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences? This article first discusses how a series of results by Berman, Buhrman, Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the early 1970s through the early 1990s, explicitly or implicitly limited how well heuristic algorithms can do on NP-hard problems. In particular, many desirable levels of heuristic success cannot be obtained unless severe, highly unlikely complexity class collapses occur. Second, we survey work initiated by Goldreich and Wigderson, who showed how under plausible assumptions deterministic heuristics for randomized computation can achieve a very high frequency of correctness. Finally, we consider formal ways in which theory can help explain the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012 issue of SIGACT New

Extending ACL2 with SMT Solvers

We present our extension of ACL2 with Satisfiability Modulo Theories (SMT) solvers using ACL2's trusted clause processor mechanism. We are particularly interested in the verification of physical systems including Analog and Mixed-Signal (AMS) designs. ACL2 offers strong induction abilities for reasoning about sequences and SMT complements deduction methods like ACL2 with fast nonlinear arithmetic solving procedures. While SAT solvers have been integrated into ACL2 in previous work, SMT methods raise new issues because of their support for a broader range of domains including real numbers and uninterpreted functions. This paper presents Smtlink, our clause processor for integrating SMT solvers into ACL2. We describe key design and implementation issues and describe our experience with its use.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

Automated generation of computationally hard feature models using evolutionary algorithms

This is the post-print version of the final paper published in Expert Systems with Applications. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2014 Elsevier B.V.A feature model is a compact representation of the products of a software product line. The automated extraction of information from feature models is a thriving topic involving numerous analysis operations, techniques and tools. Performance evaluations in this domain mainly rely on the use of random feature models. However, these only provide a rough idea of the behaviour of the tools with average problems and are not sufficient to reveal their real strengths and weaknesses. In this article, we propose to model the problem of finding computationally hard feature models as an optimization problem and we solve it using a novel evolutionary algorithm for optimized feature models (ETHOM). Given a tool and an analysis operation, ETHOM generates input models of a predefined size maximizing aspects such as the execution time or the memory consumption of the tool when performing the operation over the model. This allows users and developers to know the performance of tools in pessimistic cases providing a better idea of their real power and revealing performance bugs. Experiments using ETHOM on a number of analyses and tools have successfully identified models producing much longer executions times and higher memory consumption than those obtained with random models of identical or even larger size.European Commission (FEDER), the Spanish Government and the Andalusian Government

On vanishing of Kronecker coefficients

We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P=NP. We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples $(\lambda, \mu, \pi)$ such that the Kronecker coefficient $k^\lambda_{\mu, \pi} = 0$ but the Kronecker coefficient $k^{l \lambda}_{l \mu, l \pi} > 0$ for some integer $l>1$. Such "holes" are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any $0<\epsilon\leq1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda_{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|=|\mu| \le m^3$. The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur