4,036 research outputs found
Proof of the satisfiability conjecture for large k
We establish the satisfiability threshold for random -SAT for all , with an absolute constant. That is, there exists a limiting density
such that a random -SAT formula of clause density is
with high probability satisfiable for , and unsatisfiable for
. We show that the threshold is given explicitly
by the one-step replica symmetry breaking prediction from statistical physics.
The proof develops a new analytic method for moment calculations on random
graphs, mapping a high-dimensional optimization problem to a more tractable
problem of analyzing tree recursions. We believe that our method may apply to a
range of random CSPs in the 1-RSB universality class
Pairs of SAT Assignment in Random Boolean Formulae
We investigate geometrical properties of the random K-satisfiability problem
using the notion of x-satisfiability: a formula is x-satisfiable if there exist
two SAT assignments differing in Nx variables. We show the existence of a sharp
threshold for this property as a function of the clause density. For large
enough K, we prove that there exists a region of clause density, below the
satisfiability threshold, where the landscape of Hamming distances between SAT
assignments experiences a gap: pairs of SAT-assignments exist at small x, and
around x=1/2, but they donot exist at intermediate values of x. This result is
consistent with the clustering scenario which is at the heart of the recent
heuristic analysis of satisfiability using statistical physics analysis (the
cavity method), and its algorithmic counterpart (the survey propagation
algorithm). The method uses elementary probabilistic arguments (first and
second moment methods), and might be useful in other problems of computational
and physical interest where similar phenomena appear
The Power of Unentanglement
The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error?
In this paper, we make progress on all of the above questions.
* We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs.
* We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2.
* We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
We establish the existence of free energy limits for several combinatorial
models on Erd\"{o}s-R\'{e}nyi graph and
random -regular graph . For a variety of models, including
independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy
both at a positive and zero temperature, appropriately rescaled, converges to a
limit as the size of the underlying graph diverges to infinity. In the zero
temperature case, this is interpreted as the existence of the scaling limit for
the corresponding combinatorial optimization problem. For example, as a special
case we prove that the size of a largest independent set in these graphs,
normalized by the number of nodes converges to a limit w.h.p. This resolves an
open problem which was proposed by Aldous (Some open problems) as one of his
six favorite open problems. It was also mentioned as an open problem in several
other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999
(Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob\'{a}s and Riordan
[Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin.
Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on
Discrete Structures (2004) 1-72 Springer].Comment: Published in at http://dx.doi.org/10.1214/12-AOP816 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Relativized Propositional Calculus
Proof systems for the Relativized Propositional Calculus are defined and
compared.Comment: 8 page
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
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