Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 113126
author:  David Gamarnik 

title:  Linear Phase Transition in Random Linear Constraint Satisfaction Problems 
keywords:  Random KSAT, Satisfiability Threshold, Linear Programming, Sparse Random Graphs 
abstract: 
Our model is a generalized linear programming relaxation of
a much studied random KSAT problem. Specifically, a set of
linear constraints
C
on
K
variables is fixed. From a pool of
n
variables,
K
variables are chosen uniformly at random and a
constraint is chosen from
C
also uniformly at random. This procedure is repeated
m
times independently. We are interested in whether the
resulting linear programming problem is feasible. We prove
that the feasibility property experiences a linear phase
transition, when
n→∞
and
m=cn
for a constant
c
. Namely, there exists a critical value
c
such that, when
*
c < c
, the problem is feasible or is asymptotically almost
feasible, as
*
n→∞
, but, when
c>c
, the "distance" to feasibility is at least a
positive constant independent of
*
n
. Our result is obtained using the combination of a
powerful local weak convergence method developed in Aldous
[1992, 2000], Aldous and Steele [2003], Steele [2002] and
martingale techniques. By exploiting a linear programming
duality, our theorem implies the following result in the
context of sparse random graphs
G(n, cn)
on
n
nodes with
cn
edges, where edges are equipped with randomly
generated weights. Let
M(n,c)
denote maximum weight matching in
G(n, cn)
. We prove that when
c
is a constant and
n→∞
, the limit
lim
exists, with high probability. We further extend this
result to maximum weight
n→∞
M(n,c)/n,
b
matchings also in
G(n,cn)
.

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reference:  David Gamarnik (2003), Linear Phase Transition in Random Linear Constraint Satisfaction Problems, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 113126 
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