292 research outputs found
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Circular Coloring of Random Graphs: Statistical Physics Investigation
Circular coloring is a constraints satisfaction problem where colors are
assigned to nodes in a graph in such a way that every pair of connected nodes
has two consecutive colors (the first color being consecutive to the last). We
study circular coloring of random graphs using the cavity method. We identify
two very interesting properties of this problem. For sufficiently many color
and sufficiently low temperature there is a spontaneous breaking of the
circular symmetry between colors and a phase transition forwards a
ferromagnet-like phase. Our second main result concerns 5-circular coloring of
random 3-regular graphs. While this case is found colorable, we conclude that
the description via one-step replica symmetry breaking is not sufficient. We
observe that simulated annealing is very efficient to find proper colorings for
this case. The 5-circular coloring of 3-regular random graphs thus provides a
first known example of a problem where the ground state energy is known to be
exactly zero yet the space of solutions probably requires a full-step replica
symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
Random Graph Coloring - a Statistical Physics Approach
The problem of vertex coloring in random graphs is studied using methods of
statistical physics and probability. Our analytical results are compared to
those obtained by exact enumeration and Monte-Carlo simulations. We critically
discuss the merits and shortcomings of the various methods, and interpret the
results obtained. We present an exact analytical expression for the 2-coloring
problem as well as general replica symmetric approximated solutions for the
thermodynamics of the graph coloring problem with p colors and K-body edges.Comment: 17 pages, 9 figure
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