31 research outputs found
Improved Bounds for the Graham-Pollak Problem for Hypergraphs
For a fixed , let denote the minimum number of complete
-partite -graphs needed to partition the complete -graph on
vertices. The Graham-Pollak theorem asserts that . An easy
construction shows that ,
and we write for the least number such that .
It was known that for each even , but this was not known
for any odd value of . In this short note, we prove that . Our
method also shows that , answering another open problem
Clique versus Independent Set
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication
complexity asks for the minimum number of cuts separating cliques from stable
sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial
CS-separator, i.e. of size , and addresses the problem of
finding a polynomial CS-separator. This question is still open even for perfect
graphs. We show that a polynomial CS-separator almost surely exists for random
graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a
clique and a stable set) then there exists a constant for which we find a
CS-separator on the class of H-free graphs. This generalizes a
result of Yannakakis on comparability graphs. We also provide a
CS-separator on the class of graphs without induced path of length k and its
complement. Observe that on one side, is of order
resulting from Vapnik-Chervonenkis dimension, and on the other side, is
exponential.
One of the main reason why Yannakakis' CL-IS problem is fascinating is that
it admits equivalent formulations. Our main result in this respect is to show
that a polynomial CS-separator is equivalent to the polynomial
Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition
into k complete bipartite graphs, then its chromatic number is polynomially
bounded in terms of k. We also show that the classical approach to the stubborn
problem (arising in CSP) which consists in covering the set of all solutions by
instances of 2-SAT is again equivalent to the existence of a
polynomial CS-separator
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Constraint Generation Algorithm for the Minimum Connectivity Inference Problem
Given a hypergraph , the Minimum Connectivity Inference problem asks for a
graph on the same vertex set as with the minimum number of edges such that
the subgraph induced by every hyperedge of is connected. This problem has
received a lot of attention these recent years, both from a theoretical and
practical perspective, leading to several implemented approximation, greedy and
heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear
Programming (MILP) formulations have been experimented, all representing
connectivity constraints by the means of graph flows. In this work, we
investigate the efficiency of a constraint generation algorithm, where we
iteratively add cut constraints to a simple ILP until a feasible (and optimal)
solution is found. It turns out that our method is faster than the previous
best flow-based MILP algorithm on random generated instances, which suggests
that a constraint generation approach might be also useful for other
optimization problems dealing with connectivity constraints. At last, we
present the results of an enumeration algorithm for the problem.Comment: 16 pages, 4 tables, 1 figur