104 research outputs found
Sum of squares lower bounds for refuting any CSP
Let be a nontrivial -ary predicate. Consider a
random instance of the constraint satisfaction problem on
variables with constraints, each being applied to randomly
chosen literals. Provided the constraint density satisfies , such
an instance is unsatisfiable with high probability. The \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate supports a -\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree
(which runs in time ) \emph{cannot} refute a random instance of
. In particular, the polynomial-time SOS algorithm requires
constraints to refute random instances of
CSP when supports a -wise uniform distribution on its satisfying
assignments. Together with recent work of Lee et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least constraints.
Our results (which also extend with no change to CSPs over larger alphabets)
subsume all previously known lower bounds for semialgebraic refutation of
random CSPs. For every constraint predicate~, they give a three-way hardness
tradeoff between the density of constraints, the SOS degree (hence running
time), and the strength of the refutation. By recent algorithmic results of
Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem:Extended Abstract
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propo-sitional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and âŠ(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for âŠ(n1.5) many clauses [13]. We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly autom-atizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n vari-ables and âŠ(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin)
Strong Refutation Heuristics for Random k-SAT
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.A simple first moment argument shows that in a randomly chosen -SAT formula with clauses over boolean variables, the fraction of satisfiable clauses is as almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random -SAT formula, can we find a certificate that the fraction of satisfiable clauses is in polynomial time? We present heuristics based on spectral techniques that in the case and , and in the case and , find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random -uniform hypergraphs from above (resp. from below) for .Peer Reviewe
Short Propositional Refutations for Dense Random 3CNF Formulas
Random 3CNF formulas constitute an important distribution for measuring the
average-case behavior of propositional proof systems. Lower bounds for random
3CNF refutations in many propositional proof systems are known. Most notably
are the exponential-size resolution refutation lower bounds for random 3CNF
formulas with clauses [Chvatal and Szemeredi
(1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known
non-trivial upper bound on the size of random 3CNF refutations in a
non-abstract propositional proof system is for resolution with
clauses, shown by Beame et al. (2002). In this paper we
show that already standard propositional proof systems, within the hierarchy of
Frege proofs, admit short refutations for random 3CNF formulas, for
sufficiently large clause-to-variable ratio. Specifically, we demonstrate
polynomial-size propositional refutations whose lines are formulas
(i.e., -Frege proofs) for random 3CNF formulas with variables and clauses.
The idea is based on demonstrating efficient propositional correctness proofs
of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek
(2006). Since the soundness of these witnesses is verified using spectral
techniques, we develop an appropriate way to reason about eigenvectors in
propositional systems. To carry out the full argument we work inside weak
formal systems of arithmetic and use a general translation scheme to
propositional proofs.Comment: 62 pages; improved introduction and abstract, and a changed title.
Fixed some typo
Generating and Searching Families of FFT Algorithms
A fundamental question of longstanding theoretical interest is to prove the
lowest exact count of real additions and multiplications required to compute a
power-of-two discrete Fourier transform (DFT). For 35 years the split-radix
algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic
operations on real numbers for a size-n DFT, and was widely believed to be the
best possible. Recent work by Van Buskirk et al. demonstrated improvements to
the split-radix operation count by using multiplier coefficients or "twiddle
factors" that are not n-th roots of unity for a size-n DFT. This paper presents
a Boolean Satisfiability-based proof of the lowest operation count for certain
classes of DFT algorithms. First, we present a novel way to choose new yet
valid twiddle factors for the nodes in flowgraphs generated by common
power-of-two fast Fourier transform algorithms, FFTs. With this new technique,
we can generate a large family of FFTs realizable by a fixed flowgraph. This
solution space of FFTs is cast as a Boolean Satisfiability problem, and a
modern Satisfiability Modulo Theory solver is applied to search for FFTs
requiring the fewest arithmetic operations. Surprisingly, we find that there
are FFTs requiring fewer operations than the split-radix even when all twiddle
factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on
Satisfiability, Boolean Modeling and Computatio
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