2,605 research outputs found
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
The Satisfiability Threshold for k-XORSAT
We consider "unconstrained" random -XORSAT, which is a uniformly random
system of linear non-homogeneous equations in over
variables, each equation containing variables, and also consider a
"constrained" model where every variable appears in at least two equations.
Dubois and Mandler proved that is a sharp threshold for satisfiability
of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform
hypergraph to extend this result to find the threshold for unconstrained
3-XORSAT.
We show that remains a sharp threshold for satisfiability of
constrained -XORSAT for every , and we use standard results on the
2-core of a random -uniform hypergraph to extend this result to find the
threshold for unconstrained -XORSAT. For constrained -XORSAT we narrow
the phase transition window, showing that implies almost-sure
satisfiability, while implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in
literature survey, and improvements in presentation; removes Appendix
treating k=
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
Threshold values of Random K-SAT from the cavity method
Using the cavity equations of
\cite{mezard:parisi:zecchina:02,mezard:zecchina:02}, we derive the various
threshold values for the number of clauses per variable of the random
-satisfiability problem, generalizing the previous results to . We
also give an analytic solution of the equations, and some closed expressions
for these thresholds, in an expansion around large . The stability of the
solution is also computed. For any , the satisfiability threshold is found
to be in the stable region of the solution, which adds further credit to the
conjecture that this computation gives the exact satisfiability threshold.Comment: 38 pages; extended explanations and derivations; this version is
going to appear in Random Structures & Algorithm
The backtracking survey propagation algorithm for solving random K-SAT problems
Discrete combinatorial optimization has a central role in many scientific
disciplines, however, for hard problems we lack linear time algorithms that
would allow us to solve very large instances. Moreover, it is still unclear
what are the key features that make a discrete combinatorial optimization
problem hard to solve. Here we study random K-satisfiability problems with
, which are known to be very hard close to the SAT-UNSAT threshold,
where problems stop having solutions. We show that the backtracking survey
propagation algorithm, in a time practically linear in the problem size, is
able to find solutions very close to the threshold, in a region unreachable by
any other algorithm. All solutions found have no frozen variables, thus
supporting the conjecture that only unfrozen solutions can be found in linear
time, and that a problem becomes impossible to solve in linear time when all
solutions contain frozen variables.Comment: 11 pages, 10 figures. v2: data largely improved and manuscript
rewritte
Reweighted belief propagation and quiet planting for random K-SAT
We study the random K-satisfiability problem using a partition function where
each solution is reweighted according to the number of variables that satisfy
every clause. We apply belief propagation and the related cavity method to the
reweighted partition function. This allows us to obtain several new results on
the properties of random K-satisfiability problem. In particular the
reweighting allows to introduce a planted ensemble that generates instances
that are, in some region of parameters, equivalent to random instances. We are
hence able to generate at the same time a typical random SAT instance and one
of its solutions. We study the relation between clustering and belief
propagation fixed points and we give a direct evidence for the existence of
purely entropic (rather than energetic) barriers between clusters in some
region of parameters in the random K-satisfiability problem. We exhibit, in
some large planted instances, solutions with a non-trivial whitening core; such
solutions were known to exist but were so far never found on very large
instances. Finally, we discuss algorithmic hardness of such planted instances
and we determine a region of parameters in which planting leads to satisfiable
benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression
correcte
Hiding Satisfying Assignments: Two are Better than One
The evaluation of incomplete satisfiability solvers depends critically on the
availability of hard satisfiable instances. A plausible source of such
instances consists of random k-SAT formulas whose clauses are chosen uniformly
from among all clauses satisfying some randomly chosen truth assignment A.
Unfortunately, instances generated in this manner tend to be relatively easy
and can be solved efficiently by practical heuristics. Roughly speaking, as the
formula's density increases, for a number of different algorithms, A acts as a
stronger and stronger attractor. Motivated by recent results on the geometry of
the space of satisfying truth assignments of random k-SAT and NAE-k-SAT
formulas, we introduce a simple twist on this basic model, which appears to
dramatically increase its hardness. Namely, in addition to forbidding the
clauses violated by the hidden assignment A, we also forbid the clauses
violated by its complement, so that both A and complement of A are satisfying.
It appears that under this "symmetrization'' the effects of the two attractors
largely cancel out, making it much harder for algorithms to find any truth
assignment. We give theoretical and experimental evidence supporting this
assertion.Comment: Preliminary version appeared in AAAI 200
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