27 research outputs found
The biased odd cycle game
In this paper we consider biased Maker-Breaker games played on the edge set
of a given graph . We prove that for every and large enough ,
there exists a constant for which if and
, then Maker can build an odd cycle in the game for
. We also consider the analogous game where Maker and
Breaker claim vertices instead of edges. This is a special case of the
following well known and notoriously difficult problem due to Duffus, {\L}uczak
and R\"{o}dl: is it true that for any positive constants and , there
exists an integer such that for every graph , if , then
Maker can build a graph which is not -colorable, in the
Maker-Breaker game played on the vertices of ?Comment: 10 page
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , we determine the
number of random edges required to add to an arbitrary graph of linear minimum
degree in order to ensure the resulting graph contains a perfect -tiling
with high probability. Our proof utilises Szemer\'edi's Regularity lemma as
well as a special case of a result of Koml\'os concerning almost perfect
-tilings in dense graphs.Comment: 19 pages, to appear in CP
Smoothed Complexity Theory
Smoothed analysis is a new way of analyzing algorithms introduced by Spielman
and Teng (J. ACM, 2004). Classical methods like worst-case or average-case
analysis have accompanying complexity classes, like P and AvgP, respectively.
While worst-case or average-case analysis give us a means to talk about the
running time of a particular algorithm, complexity classes allows us to talk
about the inherent difficulty of problems.
Smoothed analysis is a hybrid of worst-case and average-case analysis and
compensates some of their drawbacks. Despite its success for the analysis of
single algorithms and problems, there is no embedding of smoothed analysis into
computational complexity theory, which is necessary to classify problems
according to their intrinsic difficulty.
We propose a framework for smoothed complexity theory, define the relevant
classes, and prove some first hardness results (of bounded halting and tiling)
and tractability results (binary optimization problems, graph coloring,
satisfiability). Furthermore, we discuss extensions and shortcomings of our
model and relate it to semi-random models.Comment: to be presented at MFCS 201
Smoothed Analysis of the Koml\'os Conjecture: Rademacher Noise
The {\em discrepancy} of a matrix is given by
. An outstanding conjecture, attributed to Koml\'os,
stipulates that , whenever is a Koml\'os matrix,
that is, whenever every column of lies within the unit sphere. Our main
result asserts that holds
asymptotically almost surely, whenever is
Koml\'os, is a Rademacher random matrix, , and . We conjecture that suffices for the same assertion to hold. The factor
normalising is essentially best possible.Comment: For version 2, the bound on the discrepancy is improve