366 research outputs found
Depth Lower Bounds in Stabbing Planes for Combinatorial Principles
We prove logarithmic depth lower bounds in Stabbing Planes for the classes of
combinatorial principles known as the Pigeonhole principle and the Tseitin
contradictions. The depth lower bounds are new, obtained by giving almost
linear length lower bounds which do not depend on the bit-size of the
inequalities and in the case of the Pigeonhole principle are tight.
The technique known so far to prove depth lower bounds for Stabbing Planes is
a generalization of that used for the Cutting Planes proof system. In this work
we introduce two new approaches to prove length/depth lower bounds in Stabbing
Planes: one relying on Sperner's Theorem which works for the Pigeonhole
principle and Tseitin contradictions over the complete graph; a second proving
the lower bound for Tseitin contradictions over a grid graph, which uses a
result on essential coverings of the boolean cube by linear polynomials, which
in turn relies on Alon's combinatorial Nullenstellensatz
Stabbing Planes
We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
On the Power and Limitations of Branch and Cut
The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes.
In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas
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Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
Depth lower bounds in Stabbing Planes for combinatorial principles
Stabbing Planes (also known as Branch and Cut) is a proof system introduced
very recently which, informally speaking, extends the DPLL method by branching
on integer linear inequalities instead of single variables. The techniques
known so far to prove size and depth lower bounds for Stabbing Planes are
generalizations of those used for the Cutting Planes proof system. For size
lower bounds these are established by monotone circuit arguments, while for
depth these are found via communication complexity and protection. As such
these bounds apply for lifted versions of combinatorial statements. Rank lower
bounds for Cutting Planes are also obtained by geometric arguments called
protection lemmas.
In this work we introduce two new geometric approaches to prove size/depth
lower bounds in Stabbing Planes working for any formula: (1) the antichain
method, relying on Sperner's Theorem and (2) the covering method which uses
results on essential coverings of the boolean cube by linear polynomials, which
in turn relies on Alon's combinatorial Nullenstellensatz.
We demonstrate their use on classes of combinatorial principles such as the
Pigeonhole principle, the Tseitin contradictions and the Linear Ordering
Principle. By the first method we prove almost linear size lower bounds and
optimal logarithmic depth lower bounds for the Pigeonhole principle and
analogous lower bounds for the Tseitin contradictions over the complete graph
and for the Linear Ordering Principle. By the covering method we obtain a
superlinear size lower bound and a logarithmic depth lower bound for Stabbing
Planes proof of Tseitin contradictions over a grid graph
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Complexity of optimizing over the integers
In the first part of this paper, we present a unified framework for analyzing
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avoiding context dependent definitions which is one of the sources of
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{\em mixed-integer convex optimization}, which contains as a special case
continuous convex optimization on the one hand and pure integer optimization on
the other. We strive for the maximum possible generality in our exposition.
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