101 research outputs found
Homometric sets in trees
Let denote a simple graph with the vertex set and the edge
set . The profile of a vertex set denotes the multiset of
pairwise distances between the vertices of . Two disjoint subsets of
are \emph{homometric}, if their profiles are the same. If is a tree on
vertices we prove that its vertex sets contains a pair of disjoint homometric
subsets of size at least . Previously it was known that such a
pair of size at least roughly exists. We get a better result in case
of haircomb trees, in which we are able to find a pair of disjoint homometric
sets of size at least for a constant
Optimal Prefix Codes for Infinite Alphabets with Nonlinear Costs
Let be a measure of strictly positive probabilities on the set
of nonnegative integers. Although the countable number of inputs prevents usage
of the Huffman algorithm, there are nontrivial for which known methods find
a source code that is optimal in the sense of minimizing expected codeword
length. For some applications, however, a source code should instead minimize
one of a family of nonlinear objective functions, -exponential means,
those of the form , where is the length of
the th codeword and is a positive constant. Applications of such
minimizations include a novel problem of maximizing the chance of message
receipt in single-shot communications () and a previously known problem of
minimizing the chance of buffer overflow in a queueing system (). This
paper introduces methods for finding codes optimal for such exponential means.
One method applies to geometric distributions, while another applies to
distributions with lighter tails. The latter algorithm is applied to Poisson
distributions and both are extended to alphabetic codes, as well as to
minimizing maximum pointwise redundancy. The aforementioned application of
minimizing the chance of buffer overflow is also considered.Comment: 14 pages, 6 figures, accepted to IEEE Trans. Inform. Theor
The number of directed k-convex polyominoes
We present a new method to obtain the generating functions for directed
convex polyominoes according to several different statistics including: width,
height, size of last column/row and number of corners. This method can be used
to study different families of directed convex polyominoes: symmetric
polyominoes, parallelogram polyominoes. In this paper, we apply our method to
determine the generating function for directed k-convex polyominoes. We show it
is a rational function and we study its asymptotic behavior
The Parameterized Complexity of the Minimum Shared Edges Problem
We study the NP-complete Minimum Shared Edges (MSE) problem. Given an
undirected graph, a source and a sink vertex, and two integers p and k, the
question is whether there are p paths in the graph connecting the source with
the sink and sharing at most k edges. Herein, an edge is shared if it appears
in at least two paths. We show that MSE is W[1]-hard when parameterized by the
treewidth of the input graph and the number k of shared edges combined. We show
that MSE is fixed-parameter tractable with respect to p, but does not admit a
polynomial-size kernel (unless NP is contained in coNP/poly). In the proof of
the fixed-parameter tractability of MSE parameterized by p, we employ the
treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG
2013].Comment: 35 pages, 16 figure
Proteus: A Hierarchical Portfolio of Solvers and Transformations
In recent years, portfolio approaches to solving SAT problems and CSPs have
become increasingly common. There are also a number of different encodings for
representing CSPs as SAT instances. In this paper, we leverage advances in both
SAT and CSP solving to present a novel hierarchical portfolio-based approach to
CSP solving, which we call Proteus, that does not rely purely on CSP solvers.
Instead, it may decide that it is best to encode a CSP problem instance into
SAT, selecting an appropriate encoding and a corresponding SAT solver. Our
experimental evaluation used an instance of Proteus that involved four CSP
solvers, three SAT encodings, and six SAT solvers, evaluated on the most
challenging problem instances from the CSP solver competitions, involving
global and intensional constraints. We show that significant performance
improvements can be achieved by Proteus obtained by exploiting alternative
view-points and solvers for combinatorial problem-solving.Comment: 11th International Conference on Integration of AI and OR Techniques
in Constraint Programming for Combinatorial Optimization Problems. The final
publication is available at link.springer.co
On the beta-number of forests with isomorphic components
The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1, . . . , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c+ 1, . . . , c+|E (G)| −1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by βs (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m |V (G)| − 1 if G has the additional property that G is a graceful nontrivial tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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