The impact of heterogeneity and geometry on the proof complexity of random satisfiability

Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random -SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random -SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time

Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i) ∼ i −β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k) ∼ k −δ , where δ = 1 + 1/β. This property has been observed in most real-world SAT instances. For β = 0, our model extends classical random SAT instances. We prove the existence of a SAT– UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β < 1/2 when the clause/variable ratio is m/n = 1−2β (1−β) 2 . We also prove that scale-free random k-SAT instances are unsatisfiable with a high probability when the number of clauses exceeds ω(n (1−β)k ). The proof of this result suggests that, when β > 1 − 1/k, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes.This research was supported by the project PROOFS, Grant PID2019-109137GB-C21 funded by MCIN/AEI/10.13039/501100011033

Understanding and Improving SAT Solvers via Proof Complexity and Reinforcement Learning

Despite the fact that the Boolean satisfiability (SAT) problem is NP-complete and believed to be intractable, SAT solvers are routinely used by practitioners to solve hard problems in wide variety of fields such as software engineering, formal methods, security, and AI. This gap between theory and practice has motivated an entire line of research whose primary goals are twofold: first, to develop a better theoretical framework aimed at accurately capturing solver behavior and thus prove tighter complexity bounds; and second, to further experimentally verify the soundness of the theory thus developed via rigorous empirical analysis and design theory-inspired techniques to improve solver performance. This interplay between theory and practice is at the heart of the work presented here. More precisely, this thesis contains a collection of results which attempt to resolve the above-described discrepancy between theory and practice. The first two sets of results are centered around the restart problem. Restarts are classes of methods which aim at erasing part of the progress a solver may have made at run time, in order to help solvers escape from the ``bad parts'' of the search space. We provide a detailed theoretical analysis of the power of restarts used in modern Conflict-Driven Clause Learning (CDCL) SAT solvers, where we prove a series of equivalence and separation results for various configurations of solvers with and without restarts. From the intuition developed via this theoretical analysis, we design and implement a machine learning based reset policy, where resets are variants of restarts that erase activity scores in addition to the parts of the solver state erased by restarts. We perform extensive experimental work to show that our reset policy outperforms both baseline and state-of-the-art solvers over a class of cryptographic instances derived from bitcoin mining problems. In a different direction, we propose the concept of hierarchical community structure (HCS) for Boolean formulas. We first theoretically show that formulas with ``good'' HCS parameter values have short CDCL proofs. Then we construct an Empirical Hardness Model using the HCS parameters. These HCS parameters exhibit a robust correlation with solver run time, leading to the development of a classifier capable of accurately distinguishing between easily solvable industrial instances and challenging random/crafted scenarios. We also present scaling studies of formulas with HCS structures to further support of theoretical analysis. In the latter part of the thesis, the focus shifts to satisfaction-driven clause-learning (SDCL) solvers, known to be being exponentially more powerful than CDCL solvers. Despite the theoretical strength of SDCL, it remains a challenge to automate and determinize such solvers. To address this, we again leverage machine learning techniques to strategically decide when to invoke an SDCL subroutine, with the goal of minimizing the associated overhead. The resulting SDCL solver, enhanced with MaxSAT techniques and conflict analysis, outperforms existing solvers on combinatorial benchmarks, particularly demonstrating superior efficacy on Mutilated Chess Board (MCB) problems

Proceedings of SAT Competition 2020 : Solver and Benchmark Descriptions

Non peer reviewe

Geometric Inhomogeneous Random Graphs for Algorithm Engineering

The design and analysis of graph algorithms is heavily based on the worst case. In practice, however, many algorithms perform much better than the worst case would suggest. Furthermore, various problems can be tackled more efficiently if one assumes the input to be, in a sense, realistic. The field of network science, which studies the structure and emergence of real-world networks, identifies locality and heterogeneity as two frequently occurring properties. A popular model that captures these properties are geometric inhomogeneous random graphs (GIRGs), which is a generalization of hyperbolic random graphs (HRGs). Aside from their importance to network science, GIRGs can be an immensely valuable tool in algorithm engineering. Since they convincingly mimic real-world networks, guarantees about quality and performance of an algorithm on instances of the model can be transferred to real-world applications. They have model parameters to control the amount of heterogeneity and locality, which allows to evaluate those properties in isolation while keeping the rest fixed. Moreover, they can be efficiently generated which allows for experimental analysis. While realistic instances are often rare, generated instances are readily available. Furthermore, the underlying geometry of GIRGs helps to visualize the network, e.g.,~for debugging or to improve understanding of its structure. The aim of this work is to demonstrate the capabilities of geometric inhomogeneous random graphs in algorithm engineering and establish them as routine tools to replace previous models like the Erd\H{o}s-R{\\u27e}nyi model, where each edge exists with equal probability. We utilize geometric inhomogeneous random graphs to design, evaluate, and optimize efficient algorithms for realistic inputs. In detail, we provide the currently fastest sequential generator for GIRGs and HRGs and describe algorithms for maximum flow, directed spanning arborescence, cluster editing, and hitting set. For all four problems, our implementations beat the state-of-the-art on realistic inputs. On top of providing crucial benchmark instances, GIRGs allow us to obtain valuable insights. Most notably, our efficient generator allows us to experimentally show sublinear running time of our flow algorithm, investigate the solution structure of cluster editing, complement our benchmark set of arborescence instances with a density for which there are no real-world networks available, and generate networks with adjustable locality and heterogeneity to reveal the effects of these properties on our algorithms

Application of spin glass ideas in social sciences, economics and finance

Classical economics has developed an arsenal of methods, based on the idea of representative agents, to come up with precise numbers for next year's GDP, inflation and exchange rates, among (many) other things. Few, however, will disagree with the fact that the economy is a complex system, with a large number of strongly heterogeneous, interacting units of different types (firms, banks, households, public institutions) and different sizes. Now, the main issue in economics is precisely the emergent organization, cooperation and coordination of such a motley crowd of micro-units. Treating them as a unique ``representative'' firm or household clearly risks throwing the baby with the bathwater. As we have learnt from statistical physics, understanding and characterizing such emergent properties can be difficult. Because of feedback loops of different signs, heterogeneities and non-linearities, the macro-properties are often hard to anticipate. In particular, these situations generically lead to a very large number of possible equilibria, or even the lack thereof. Spin-glasses and other disordered systems give a concrete example of such difficulties. In order to tackle these complex situations, new theoretical and numerical tools have been invented in the last 50 years, including of course the replica method and replica symmetry breaking, and the cavity method, both static and dynamic. In this chapter we review the application of such ideas and methods in economics and social sciences. Of particular interest are the proliferation (and fragility) of equilibria, the analogue of satisfiability phase transitions in games and random economies, and condensation (or concentration) effects in opinion, wealth, etcComment: Contribution to the edited volume "Spin Glass Theory & Far Beyond - Replica Symmetry Breaking after 40 Years", World Scientific, 202