256 research outputs found
Solution space structure of random constraint satisfaction problems with growing domains
In this paper we study the solution space structure of model RB, a standard
prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using
rigorous the first and the second moment method, we show that in the solvable
phase close to the satisfiability transition, solutions are clustered into
exponential number of well-separated clusters, with each cluster contains
sub-exponential number of solutions. As a consequence, the system has a
clustering (dynamical) transition but no condensation transition. This picture
of phase diagram is different from other classic random CSPs with fixed domain
size, such as random K-Satisfiability (K-SAT) and graph coloring problems,
where condensation transition exists and is distinct from satisfiability
transition. Our result verifies the non-rigorous results obtained using cavity
method from spin glass theory, and sheds light on the structures of solution
spaces of problems with a large number of states.Comment: 8 pages, 1 figure
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
Noise stability of functions with low influences: invariance and optimality
In this paper we study functions with low influences on product probability
spaces. The analysis of boolean functions with low influences has become a
central problem in discrete Fourier analysis. It is motivated by fundamental
questions arising from the construction of probabilistically checkable proofs
in theoretical computer science and from problems in the theory of social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all product
spaces. Ours is one of the very few known non-linear invariance principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be eliminated
if the polynomials are slightly ``smoothed''; this extension is essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer
science, which was the original motivation for this work, and the ``It Ain't
Over Till It's Over'' conjecture from social choice theory
Beating the random assignment on constraint satisfaction problems of bounded degree
We show that for any odd and any instance of the Max-kXOR constraint
satisfaction problem, there is an efficient algorithm that finds an assignment
satisfying at least a fraction of
constraints, where is a bound on the number of constraints that each
variable occurs in. This improves both qualitatively and quantitatively on the
recent work of Farhi, Goldstone, and Gutmann (2014), which gave a
\emph{quantum} algorithm to find an assignment satisfying a fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for
"triangle-free" instances; i.e., an efficient algorithm that finds an
assignment satisfying at least a fraction of
constraints, where is the fraction that would be satisfied by a uniformly
random assignment.Comment: 14 pages, 1 figur
Les CSP extrĂȘmaux
http://www710.univ-lyon1.fr/~csolnonNous proposons une nouvelle classe de CSP binaires appelĂ©s CSP extrĂȘmaux. Les CSP de cette classe sont inconsistants mais deviendraient consistants si n'importe quel couple de valeurs interdit devenait autorisĂ©. Etant inconsistants, ils ne sont pas traitables avec des mĂ©thodes de rĂ©paration locale. Comme ils autorisent un nombre trĂšs Ă©levĂ© de solutions partielles presque complĂštes, ils peuvent ĂȘtre trĂšs difficiles Ă rĂ©soudre Ă l'aide de mĂ©thodes de recherche arborescente intĂ©grant le filtrage des domaines. Il faudra donc trouver de nouvelles mĂ©thodes pour les rĂ©soudre. Nous prĂ©sentons un algorithme simple de gĂ©nĂ©ration de CSP extrĂȘmaux. Nous constatons expĂ©rimentalement que les CSP extrĂȘmaux Ă©quilibrĂ©s sont beaucoup plus longs Ă rĂ©soudre que les CSP alĂ©atoires dits difficiles de mĂȘme taille. Nous prĂ©sentons aussi un schĂ©ma d'algorithme susceptible d'ĂȘtre performant sur des problĂšmes difficiles, dĂšs lors qu'on sera en mesure de gĂ©nĂ©rer suffisamment rapidement des CSP extrĂȘmaux
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