418 research outputs found
Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
We revisit a well-known family of polynomial ideals encoding the problem of
graph--colorability. Our paper describes how the inherent combinatorial
structure of the ideals implies several interesting algebraic properties.
Specifically, we provide lower bounds on the difficulty of computing Gr\"obner
bases and Nullstellensatz certificates for the coloring ideals of general
graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis
for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Polynomial iterative algorithms for coloring and analyzing random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters. Furthermore, we
extended our considerations to the case of single instances showing consistent
results. This lead us to propose a new algorithm able to color in polynomial
time random graphs in the hard but colorable region, i.e when .Comment: 23 pages, 10 eps figure
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