30 research outputs found
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
Reducing SAT to Max2SAT
In the literature we find reductions from 3SAT to
Max2SAT. These reductions are based on the usage
of a gadget, i.e., a combinatorial structure that allows translating constraints of one problem to constraints of another. Unfortunately, the generation of
these gadgets lacks an intuitive or efficient method.
In this paper, we provide an efficient and constructive method for Reducing SAT to Max2SAT and
show empirical results of how MaxSAT solvers are
more efficient than SAT solvers solving the translation of hard formulas for Resolution.Supported by projects PROOFS (PID2019-109137GB-C21) and EU-H2020-RIP LOGISTAR (No. 769142)
Unifying Reasoning and Core-Guided Search for Maximum Satisfiability
A central algorithmic paradigm in maximum satisfiability solving geared towards real-world optimization problems is the core-guided approach. Furthermore, recent progress on preprocessing techniques is bringing in additional reasoning techniques to MaxSAT solving. Towards realizing their combined potential, understanding formal underpinnings of interleavings of preprocessing-style reasoning and core-guided algorithms is important. It turns out that earlier proposed notions for establishing correctness of core-guided algorithms and preprocessing, respectively, are not enough for capturing correctness of interleavings of the techniques. We provide an in-depth analysis of these and related MaxSAT instance transformations, and propose correction set reducibility as a notion that captures inprocessing MaxSAT solving within a state-transition style abstract MaxSAT solving framework. Furthermore, we establish a general theorem of correctness for applications of SAT-based preprocessing techniques in MaxSAT. The results pave way for generic techniques for arguing about the formal correctness of MaxSAT algorithms.Peer reviewe
On solving the MAX-SAT using sum of squares
We consider semidefinite programming (SDP) approaches for solving the maximum
satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely
known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows
the potential of SDP also for other satisfiability problems, by being
competitive with some of the best solvers in the yearly MAX-SAT competition.
Our solver combines sum of squares (SOS) based SDP bounds and an efficient
parser within a branch & bound scheme.
On the theoretical side, we propose a family of semidefinite feasibility
problems, and show that a member of this family provides the rank two
guarantee. We also provide a parametric family of semidefinite relaxations for
the MAX-SAT, and derive several properties of monomial bases used in the SOS
approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an
elegant way. Moreover, we relate our SOS-SDP relaxations for the partial
MAX-SAT to the known SAT relaxations.Comment: 26 pages, 5 figures, 8 tables, 2 appendix page