150 research outputs found
On Approximability of Bounded Degree Instances of Selected Optimization Problems
In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics
Packing k-partite k-uniform hypergraphs
Let and be -graphs (-uniform hypergraphs); then a perfect
-packing in is a collection of vertex-disjoint copies of in
which together cover every vertex of . For any fixed let
be the minimum such that any -graph on vertices with
minimum codegree contains a perfect -packing. The
problem of determining has been widely studied for graphs (i.e.
-graphs), but little is known for . Here we determine the
asymptotic value of for all complete -partite -graphs ,
as well as a wide class of other -partite -graphs. In particular, these
results provide an asymptotic solution to a question of R\"odl and Ruci\'nski
on the value of when is a loose cycle. We also determine
asymptotically the codegree threshold needed to guarantee an -packing
covering all but a constant number of vertices of for any complete
-partite -graph .Comment: v2: Updated with minor corrections. Accepted for publication in
Journal of Combinatorial Theory, Series
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
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