183 research outputs found
On the -matching polytope and the fractional -chromatic index
Our motivation is the question of how similar the -colouring problem is to
the classic edge-colouring problem, particularly with regard to graph
parameters. In 2010, Zhang, Yu, and Liu gave a new description of the
-matching polytope and derived a formula for the fractional -chromatic
index, stating that the fractional -chromatic index equals the maximum of
the fractional maximum -degree and the fractional -density.
Unfortunately, this formula is incorrect. We present counterexamples for both
the description of the -matching polytope and the formula for the fractional
-chromatic index. Finally, we prove a short lemma concerning the
generalization of Goldberg's conjecture
Integer round-up property for the chromatic number of some h-perfect graphs
A graph is h-perfect if its stable set polytope can be completely described
by non-negativity, clique and odd-hole constraints. It is t-perfect if it
furthermore has no clique of size 4. For every graph and every
, the weighted chromatic number of is the
minimum cardinality of a multi-set of stable sets of such
that every belongs to at least members of .
We prove that every h-perfect line-graph and every t-perfect claw-free graph
has the integer round-up property for the chromatic number: for every
non-negative integer weight on the vertices of , the weighted chromatic
number of can be obtained by rounding up its fractional relaxation. In
other words, the stable set polytope of has the integer decomposition
property.
Our results imply the existence of a polynomial-time algorithm which computes
the weighted chromatic number of t-perfect claw-free graphs and h-perfect
line-graphs. Finally, they yield a new case of a conjecture of Goldberg and
Seymour on edge-colorings.Comment: 20 pages, 13 figure
Matching polytons
Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions
of matching and fractional vertex covers in graphons. These are counterparts to
the corresponding notions in finite graphs.
Combinatorial optimization studies the structure of the matching polytope and
the fractional vertex cover polytope of a graph. Here, in analogy, we initiate
the study of the structure of the set of all matchings and of all fractional
vertex covers in a graphon. We call these sets the matching polyton and the
fractional vertex cover polyton.
We also study properties of matching polytons and fractional vertex cover
polytons along convergent sequences of graphons.
As an auxiliary tool of independent interest, we prove that a graphon is
-partite if and only if it contains no graph of chromatic number . This
in turn gives a characterization of bipartite graphons as those having a
symmetric spectrum.Comment: 32 pages, 2 figures; more background on graphons and analysis, a new
section on spectra of bipartite graphons, numerous corrections based on
suggestion by an anonymous refere
Independent sets, cliques, and colorings in graphons
We study graphon counterparts of the chromatic and the clique number, the
fractional chromatic number, the b-chromatic number, and the fractional clique
number. We establish some basic properties of the independence set polytope in
the graphon setting, and duality properties between the fractional chromatic
number and the fractional clique number. We present a notion of perfect
graphons and characterize them in terms of induced densities of odd cycles and
its complements.Comment: 18 pages, accepted to European Journal of Combinatorics (special
issue Eurocomb 2017
Ear-decompositions and the complexity of the matching polytope
The complexity of the matching polytope of graphs may be measured with the
maximum length of a starting sequence of odd ears in an
ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its
facets are defined by 2-connected factor-critical graphs, which have an odd
ear-decomposition (according to a theorem of Lov\'asz). In particular,
if and only if the matching polytope of the graph is
completely described by non-negativity, star and odd-circuit inequalities. This
is essentially equivalent to the h-perfection of the line-graph of , as
observed by Cao and Nemhauser.
The complexity of computing is apparently not known. We show that
deciding whether can be executed efficiently by looking at any
ear-decomposition starting with an odd circuit and performing basic modulo-2
computations. Such a greedy-approach is surprising in view of the complexity of
the problem in more special cases by Bruhn and Schaudt, and it is simpler than
using the Parity Minor Algorithm.
Our results imply a simple polynomial-time algorithm testing h-perfection in
line-graphs (deciding h-perfection is open in general). We also generalize our
approach to binary matroids and show that computing is a
Fixed-Parameter-Tractable problem (FPT)
Asymptotics of the chromatic number for quasi-line graphs
As proved by Kahn, the chromatic number and fractional chromatic number of a
line graph agree asymptotically. That is, for any line graph we have
. We extend this result to quasi-line graphs,
an important subclass of claw-free graphs. Furthermore we prove that we can
construct a colouring that achieves this bound in polynomial time, giving us an
asymptotic approximation algorithm for the chromatic number of quasi-line
graphs.Comment: 20 pages, 2 figure
Fractional matching preclusion number of graphs
Let be a graph with an even number of vertices. The matching preclusion
number of , denoted by , is the minimum number of edges whose
deletion leaves the resulting graph without a perfect matching. We introduced a
- linear programming which can be used to find matching preclusion number
of graphs. In this paper, by relaxing of the - linear programming we
obtain a linear programming and call its optimal objective value as fractional
matching preclusion number of graph , denoted by . We show
can be computed in polynomial time for any graph . By using
perfect matching polytope, we transform it as a new linear programming whose
optimal value equals the reciprocal of . For bipartite graph , we
obtain an explicit formula for and show that is the maximum integer such that has a -factor. Moreover,
for any two bipartite graphs and , we show , where is the Cartesian product
of and .Comment: 18 pages, 1 figur
Performance Guarantees of Distributed Algorithms for QoS in Wireless Ad Hoc Networks
Consider a wireless network where each communication link has a minimum
bandwidth quality-of-service requirement. Certain pairs of wireless links
interfere with each other due to being in the same vicinity, and this
interference is modeled by a conflict graph. Given the conflict graph and link
bandwidth requirements, the objective is to determine, using only localized
information, whether the demands of all the links can be satisfied. At one
extreme, each node knows the demands of only its neighbors; at the other
extreme, there exists an optimal, centralized scheduler that has global
information. The present work interpolates between these two extremes by
quantifying the tradeoff between the degree of decentralization and the
performance of the distributed algorithm. This open problem is resolved for the
primary interference model, and the following general result is obtained: if
each node knows the demands of all links in a ball of radius centered at
the node, then there is a distributed algorithm whose performance is away from
that of an optimal, centralized algorithm by a factor of at most
. The tradeoff between performance and complexity of the
distributed algorithm is also analyzed. It is shown that for line networks
under the protocol interference model, the row constraints are a factor of at
most away from optimal. Both bounds are best possible
List Colouring Squares of Planar Graphs
In 1977, Wegner conjectured that the chromatic number of the square of every
planar graph with maximum degree is at most
. We show that it is at most (where the is as ), and indeed that
this is true for the list chromatic number and for more general classes of
graphs.Comment: 34 page
Analysis of Sparse Cutting-planes for Sparse MILPs with Applications to Stochastic MILPs
In this paper, we present an analysis of the strength of sparse
cutting-planes for mixed integer linear programs (MILP) with sparse
formulations. We examine three kinds of problems: packing problems, covering
problems, and more general MILPs with the only assumption that the objective
function is non-negative. Given a MILP instance of one of these three types,
assume that we decide on the support of cutting-planes to be used and the
strongest inequalities on these supports are added to the linear programming
relaxation. Call the optimal objective function value of the linear programming
relaxation together with these cuts as . We present bounds on the
ratio of and the optimal objective function value of the MILP that
depends only on the sparsity structure of the constraint matrix and the support
of sparse cuts selected, that is, these bounds are completely data independent.
These results also shed light on the strength of scenario-specific cuts for two
stage stochastic MILPs
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