5,260 research outputs found
Automated Discharging Arguments for Density Problems in Grids
Discharging arguments demonstrate a connection between local structure and
global averages. This makes it an effective tool for proving lower bounds on
the density of special sets in infinite grids. However, the minimum density of
an identifying code in the hexagonal grid remains open, with an upper bound of
and a lower bound of . We present a new, experimental framework for producing discharging
arguments using an algorithm. This algorithm replaces the lengthy case analysis
of human-written discharging arguments with a linear program that produces the
best possible lower bound using the specified set of discharging rules. We use
this framework to present a lower bound of on
the density of an identifying code in the hexagonal grid, and also find several
sharp lower bounds for variations on identifying codes in the hexagonal,
square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables,
and 2 figure
An improved lower bound for (1,<=2)-identifying codes in the king grid
We call a subset of vertices of a graph a -identifying
code if for all subsets of vertices with size at most , the sets
are distinct. The concept of
identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin.
Identifying codes have been studied in various grids. In particular, it has
been shown that there exists a -identifying code in the king grid
with density 3/7 and that there are no such identifying codes with density
smaller than 5/12. Using a suitable frame and a discharging procedure, we
improve the lower bound by showing that any -identifying code of
the king grid has density at least 47/111
Uniform hypergraphs containing no grids
A hypergraph is called an rĂr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that AiâŠAj=BiâŠBj=Ď for 1â¤i<jâ¤r and {pipe}AiâŠBj{pipe}=1 for 1â¤i, jâ¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1âŠC2{pipe}={pipe}C2âŠC3{pipe}={pipe}C3âŠC1{pipe}=1, C1âŠC2â C1âŠC3. A hypergraph is linear, if {pipe}EâŠF{pipe}â¤1 holds for every pair of edges Eâ F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For râĽ. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Š 2013 Elsevier Ltd
Finding codes on infinite grids automatically
We apply automata theory and Karp's minimum mean weight cycle algorithm to
minimum density problems in coding theory. Using this method, we find the new
upper bound for the minimum density of an identifying
code on the infinite hexagonal grid, down from the previous record of .Comment: 18 pages, 5 figure
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