39 research outputs found
On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed
For , let be an -uniform linear system. The
transversal number of is the minimum
number of points that intersect every line of . The 2-packing
number of is the maximum number of
lines such that the intersection of any three of them is empty. In [Discrete
Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it
true that if is a -uniform linear system then
holds for
all ?. In this paper, some results about of -uniform linear systems
whose 2-packing number is fixed which satisfies the inequality are given
Multipartite hypergraphs achieving equality in Ryser's conjecture
A famous conjecture of Ryser is that in an -partite hypergraph the
covering number is at most times the matching number. If true, this is
known to be sharp for for which there exists a projective plane of order
. We show that the conjecture, if true, is also sharp for the smallest
previously open value, namely . For , we find the minimal
number of edges in an intersecting -partite hypergraph that has
covering number at least . We find that is achieved only by linear
hypergraphs for , but that this is not the case for . We
also improve the general lower bound on , showing that .
We show that a stronger form of Ryser's conjecture that was used to prove the
case fails for all . We also prove a fractional version of the
following stronger form of Ryser's conjecture: in an -partite hypergraph
there exists a set of size at most , contained either in one side of
the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac
On local search and LP and SDP relaxations for k-Set Packing
Set packing is a fundamental problem that generalises some well-known
combinatorial optimization problems and knows a lot of applications. It is
equivalent to hypergraph matching and it is strongly related to the maximum
independent set problem. In this thesis we study the k-set packing problem
where given a universe U and a collection C of subsets over U, each of
cardinality k, one needs to find the maximum collection of mutually disjoint
subsets. Local search techniques have proved to be successful in the search for
approximation algorithms, both for the unweighted and the weighted version of
the problem where every subset in C is associated with a weight and the
objective is to maximise the sum of the weights. We make a survey of these
approaches and give some background and intuition behind them. In particular,
we simplify the algebraic proof of the main lemma for the currently best
weighted approximation algorithm of Berman ([Ber00]) into a proof that reveals
more intuition on what is really happening behind the math. The main result is
a new bound of k/3 + 1 + epsilon on the integrality gap for a polynomially
sized LP relaxation for k-set packing by Chan and Lau ([CL10]) and the natural
SDP relaxation [NOTE: see page iii]. We provide detailed proofs of lemmas
needed to prove this new bound and treat some background on related topics like
semidefinite programming and the Lovasz Theta function. Finally we have an
extended discussion in which we suggest some possibilities for future research.
We discuss how the current results from the weighted approximation algorithms
and the LP and SDP relaxations might be improved, the strong relation between
set packing and the independent set problem and the difference between the
weighted and the unweighted version of the problem.Comment: There is a mistake in the following line of Theorem 17: "As an
induced subgraph of H with more edges than vertices constitutes an improving
set". Therefore, the proofs of Theorem 17, and hence Theorems 19, 23 and 24,
are false. It is still open whether these theorems are tru