154 research outputs found
On the critical exponents of random k-SAT
There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from satisfiability to unsatisfiability; in the case of 2-SAT this happens when
m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The
sharpness of this transition is characterized by a critical exponent, sometimes
called \nu=\nu_k (the smaller the value of \nu the sharper the transition).
Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05,
\nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k
--> 1 as k --> infinity. We give here a simple proof that each of these
exponents is at least 2 (provided the exponent is well-defined). This result
holds for each of the three standard ensembles of random k-SAT formulas: m
clauses selected uniformly at random without replacement, m clauses selected
uniformly at random with replacement, and each clause selected with probability
p independent of the other clauses. We also obtain similar results for
q-colorability and the appearance of a q-core in a random graph.Comment: 11 pages. v2 has revised introduction and updated reference
The random K-satisfiability problem: from an analytic solution to an efficient algorithm
We study the problem of satisfiability of randomly chosen clauses, each with
K Boolean variables. Using the cavity method at zero temperature, we find the
phase diagram for the K=3 case. We show the existence of an intermediate phase
in the satisfiable region, where the proliferation of metastable states is at
the origin of the slowdown of search algorithms. The fundamental order
parameter introduced in the cavity method, which consists of surveys of local
magnetic fields in the various possible states of the system, can be computed
for one given sample. These surveys can be used to invent new types of
algorithms for solving hard combinatorial optimizations problems. One such
algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo
Distance Preserving Graph Simplification
Large graphs are difficult to represent, visualize, and understand. In this
paper, we introduce "gate graph" - a new approach to perform graph
simplification. A gate graph provides a simplified topological view of the
original graph. Specifically, we construct a gate graph from a large graph so
that for any "non-local" vertex pair (distance higher than some threshold) in
the original graph, their shortest-path distance can be recovered by
consecutive "local" walks through the gate vertices in the gate graph. We
perform a theoretical investigation on the gate-vertex set discovery problem.
We characterize its computational complexity and reveal the upper bound of
minimum gate-vertex set using VC-dimension theory. We propose an efficient
mining algorithm to discover a gate-vertex set with guaranteed logarithmic
bound. We further present a fast technique for pruning redundant edges in a
gate graph. The detailed experimental results using both real and synthetic
graphs demonstrate the effectiveness and efficiency of our approach.Comment: A short version of this paper will be published for ICDM'11, December
201
Foundations of Reasoning with Uncertainty via Real-valued Logics
Real-valued logics underlie an increasing number of neuro-symbolic
approaches, though typically their logical inference capabilities are
characterized only qualitatively. We provide foundations for establishing the
correctness and power of such systems. For the first time, we give a sound and
complete axiomatization for a broad class containing all the common real-valued
logics. This axiomatization allows us to derive exactly what information can be
inferred about the combinations of real values of a collection of formulas
given information about the combinations of real values of several other
collections of formulas. We then extend the axiomatization to deal with
weighted subformulas. Finally, we give a decision procedure based on linear
programming for deciding, under certain natural assumptions, whether a set of
our sentences logically implies another of our sentences.Comment: 9 pages (incl. references), 9 pages supplementary. In submission to
NeurIPS 202
A pearl on SAT solving in Prolog
A succinct SAT solver is presented that exploits the control provided by delay declarations to implement watched literals and unit propagation. Despite its brevity the solver is surprisingly powerful and its elegant use of Prolog constructs is presented as a programming pearl
Locally Stable Marriage with Strict Preferences
We study stable matching problems with locality of information and control.
In our model, each agent is a node in a fixed network and strives to be matched
to another agent. An agent has a complete preference list over all other agents
it can be matched with. Agents can match arbitrarily, and they learn about
possible partners dynamically based on their current neighborhood. We consider
convergence of dynamics to locally stable matchings -- states that are stable
with respect to their imposed information structure in the network. In the
two-sided case of stable marriage in which existence is guaranteed, we show
that the existence of a path to stability becomes NP-hard to decide. This holds
even when the network exists only among one partition of agents. In contrast,
if one partition has no network and agents remember a previous match every
round, a path to stability is guaranteed and random dynamics converge with
probability 1. We characterize this positive result in various ways. For
instance, it holds for random memory and for cache memory with the most recent
partner, but not for cache memory with the best partner. Also, it is crucial
which partition of the agents has memory. Finally, we present results for
centralized computation of locally stable matchings, i.e., computing maximum
locally stable matchings in the two-sided case and deciding existence in the
roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat
Several Issues on the Boolean Satisfiability (SAT) Problem
Boolean Satisfiability (SAT) is often used as the model for a significant and increasing number of applications in Electronics Design Automation (EDA) and many other fields of computer science and engineering. Although the SAT problem belongs to the class NP-complete - problems that do not have a polynomial run time algorithm but answers for which can be checked for correctness, by an algorithm with run time is polynomial in the size of the input - typical SAT instances are easy to solve. Both theoretical and empirical studies have been conducted on various SAT models to investigate when SAT instances become hard to solve. For more than a decade, crossover point has been the only parameter considered for hardness. Existing results state that for random SAT, the problems becomes relatively harder when the clause to variable ratio is 4.3. This thesis work is motivated by the observation that not all benchmarks at the crossover point are hard. We conjecture that the structure of the solution space is also related to the hardness. We provide an empirical framework for the validation of this conjecture. Firstly, we show by experiments that (1) the crossover point is not the only metric to characterize hardness and (2) existing benchmarks are inefficient in providing solution space information We present a novel approach to generate the SAT instances with known solution space structure. Another related issue on how to obtain the solution space information is also discussed, where we propose two probabilistic techniques for quick estimation of the solution space with high accuracy
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