Learning Pseudo-Backdoors for Mixed Integer Programs

We propose a machine learning approach for quickly solving Mixed Integer Programs (MIP) by learning to prioritize a set of decision variables, which we call pseudo-backdoors, for branching that results in faster solution times. Learning-based approaches have seen success in the area of solving combinatorial optimization problems by being able to flexibly leverage common structures in a given distribution of problems. Our approach takes inspiration from the concept of strong backdoors, which corresponds to a small set of variables such that only branching on these variables yields an optimal integral solution and a proof of optimality. Our notion of pseudo-backdoors corresponds to a small set of variables such that only branching on them leads to faster solve time (which can be solver dependent). A key advantage of pseudo-backdoors over strong backdoors is that they are much amenable to data-driven identification or prediction. Our proposed method learns to estimate the solver performance of a proposed pseudo-backdoor, using a labeled dataset collected on a set of training MIP instances. This model can then be used to identify high-quality pseudo-backdoors on new MIP instances from the same distribution. We evaluate our method on the generalized independent set problems and find that our approach can efficiently identify high-quality pseudo-backdoors. In addition, we compare our learned approach against Gurobi, a state-of-the-art MIP solver, demonstrating that our method can be used to improve solver performance

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

DPLL+ROBDD Derivation Applied to Inversion of Some Cryptographic Functions

Abstract. The paper presents logical derivation algorithms that can be applied to inversion of polynomially computable discrete functions. The proposed approach is based on the fact that it is possible to organize DPLL derivation on a small subset of variables appeared in a CNF which encodes the algorithm computing the function. The experimental results showed that arrays of conflict clauses generated by this mode of derivation, as a rule, have efficient ROBDD representations. This fact is the departing point of development of a hybrid DPLL+ROBDD derivation strategy: derivation techniques for ROBDD representations of conflict databases are the same as those ones in common DPLL (variable assignments and unit propagation). In addition, compact ROBDD representations of the conflict databases can be shared effectively in a distributed computing environment

Translation of Algorithmic Descriptions of Discrete Functions to SAT with Applications to Cryptanalysis Problems

In the present paper, we propose a technology for translating algorithmic descriptions of discrete functions to SAT. The proposed technology is aimed at applications in algebraic cryptanalysis. We describe how cryptanalysis problems are reduced to SAT in such a way that it should be perceived as natural by the cryptographic community. In~the theoretical part of the paper we justify the main principles of general reduction to SAT for discrete functions from a class containing the majority of functions employed in cryptography. Then, we describe the Transalg software tool developed based on these principles with SAT-based cryptanalysis specifics in mind. We demonstrate the results of applications of Transalg to construction of a number of attacks on various cryptographic functions. Some of the corresponding attacks are state of the art. We compare the functional capabilities of the proposed tool with that of other domain-specific software tools which can be used to reduce cryptanalysis problems to SAT, and also with the CBMC system widely employed in symbolic verification. The paper also presents vast experimental data, obtained using the SAT solvers that took first places at the SAT competitions in the recent several years