7,083 research outputs found
Empty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in
diagonally opposite corners of empty axis-aligned rectangles. The maximum
number of edges of such a graph on points is shown to be 1/4 n^2 +n -2.
This number also has other interpretations:
* It is the maximum number of edges of a graph of dimension
\bbetween{3}{4}, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}.
* It is the number of 1-faces in a special Scarf complex.
The last of these interpretations allows to deduce the maximum number of
empty axis-aligned rectangles spanned by 4-element subsets of a set of
points. Moreover, it follows that the extremal point sets for the two problems
coincide.
We investigate the maximum number of of edges of a graph of dimension
, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be .
Box graphs are defined as the 3-dimensional analog of rectangle graphs. The
maximum number of edges of such a graph on points is shown to be
Simple and Optimal Randomized Fault-Tolerant Rumor Spreading
We revisit the classic problem of spreading a piece of information in a group
of fully connected processors. By suitably adding a small dose of
randomness to the protocol of Gasienic and Pelc (1996), we derive for the first
time protocols that (i) use a linear number of messages, (ii) are correct even
when an arbitrary number of adversarially chosen processors does not
participate in the process, and (iii) with high probability have the
asymptotically optimal runtime of when at least an arbitrarily
small constant fraction of the processors are working. In addition, our
protocols do not require that the system is synchronized nor that all
processors are simultaneously woken up at time zero, they are fully based on
push-operations, and they do not need an a priori estimate on the number of
failed nodes.
Our protocols thus overcome the typical disadvantages of the two known
approaches, algorithms based on random gossip (typically needing a large number
of messages due to their unorganized nature) and algorithms based on fair
workload splitting (which are either not {time-efficient} or require intricate
preprocessing steps plus synchronization).Comment: This is the author-generated version of a paper which is to appear in
Distributed Computing, Springer, DOI: 10.1007/s00446-014-0238-z It is
available online from
http://link.springer.com/article/10.1007/s00446-014-0238-z This version
contains some new results (Section 6
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
Vertex Cover Gets Faster and Harder on Low Degree Graphs
The problem of finding an optimal vertex cover in a graph is a classic
NP-complete problem, and is a special case of the hitting set question. On the
other hand, the hitting set problem, when asked in the context of induced
geometric objects, often turns out to be exactly the vertex cover problem on
restricted classes of graphs. In this work we explore a particular instance of
such a phenomenon. We consider the problem of hitting all axis-parallel slabs
induced by a point set P, and show that it is equivalent to the problem of
finding a vertex cover on a graph whose edge set is the union of two
Hamiltonian Paths. We show the latter problem to be NP-complete, and we also
give an algorithm to find a vertex cover of size at most k, on graphs of
maximum degree four, whose running time is 1.2637^k n^O(1)
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