289 research outputs found
Applications of Discrepancy Theory in Multiobjective Approximation
We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2-approximable on directed graphs and randomized 2/3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1/2-approximable
On the Beck-Fiala Conjecture for Random Set Systems
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for
random sparse set systems. Concretely, these are set systems ,
where each element lies in randomly selected sets of ,
where is an integer parameter. We provide new bounds in two regimes of
parameters. We show that when the hereditary discrepancy of
is with high probability ; and when the hereditary discrepancy of is with high probability
. The first bound combines the Lov{\'a}sz Local Lemma with a new argument
based on partial matchings; the second follows from an analysis of the lattice
spanned by sparse vectors
Bounds for approximate discrete tomography solutions
In earlier papers we have developed an algebraic theory of discrete
tomography. In those papers the structure of the functions
and having given line sums in certain directions have
been analyzed. Here was a block in with sides parallel to
the axes. In the present paper we assume that there is noise in the
measurements and (only) that is an arbitrary or convex finite set in
. We derive generalizations of earlier results. Furthermore we
apply a method of Beck and Fiala to obtain results of he following type: if the
line sums in directions of a function are known, then
there exists a function such that its line sums differ by at
most from the corresponding line sums of .Comment: 16 page
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
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