79,297 research outputs found
Long Alternating Paths Exist
p_? of ? points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ? j.
We show that there is an absolute constant ? > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + ?)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + ?)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n)
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width
Parity games are a much researched class of games in NP intersect CoNP that
are not known to be in P. Consequently, researchers have considered specialised
algorithms for the case where certain graph parameters are small. In this
paper, we study parity games on graphs with bounded treewidth, and graphs with
bounded DAG width. We show that parity games with bounded DAG width can be
solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the
size, treewidth, and number of priorities in the parity game. This is an
improvement over the previous best algorithm, given by Berwanger et al., which
runs in n^O(k^2) time. We also show that, if a tree decomposition is provided,
then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d +
1)^(3k + 5)) time. This improves over previous best algorithm, given by
Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be
adapted to show that the problem of solving parity games with bounded treewidth
lies in the complexity class NC^2, which is the class of problems that can be
efficiently parallelized. This is in stark contrast to the general parity game
problem, which is known to be P-hard, and thus unlikely to be contained in NC
Long properly colored cycles in edge colored complete graphs
Let denote a complete graph on vertices whose edges are
colored in an arbitrary way. Let denote the
maximum number of edges of the same color incident with a vertex of
. A properly colored cycle (path) in is a cycle (path)
in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s
(1976) proposed the following conjecture: if , then contains a properly
colored Hamiltonian cycle. Li, Wang and Zhou proved that if
, then
contains a properly colored cycle of length at least . In this paper, we improve the bound to .Comment: 8 page
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