1,015 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
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
Hierarchies of Inefficient Kernelizability
The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam
(STOC 2008) allows us to exclude the existence of polynomial kernels for a
range of problems under reasonable complexity-theoretical assumptions. However,
there are also some issues that are not addressed by this framework, including
the existence of Turing kernels such as the "kernelization" of Leaf Out
Branching(k) into a disjunction over n instances of size poly(k). Observing
that Turing kernels are preserved by polynomial parametric transformations, we
define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of
ordinary parameterized complexity, by the PPT-closure of problems that seem
likely to be fundamentally hard for efficient Turing kernelization. We find
that several previously considered problems are complete for our fundamental
hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected
Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log
n
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
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