11 research outputs found
Improved Bounds for the Graham-Pollak Problem for Hypergraphs
For a fixed , let denote the minimum number of complete
-partite -graphs needed to partition the complete -graph on
vertices. The Graham-Pollak theorem asserts that . An easy
construction shows that ,
and we write for the least number such that .
It was known that for each even , but this was not known
for any odd value of . In this short note, we prove that . Our
method also shows that , answering another open problem
Exact values and improved bounds on -neighborly families of boxes
A finite family of -dimensional convex polytopes is called
-neighborly if for any two distinct
members . In 1997, Alon initiated the study of the general
function , which is defined to be the maximum size of -neighborly
families of standard boxes in . Based on a weighted count of
vectors in , we improve a recent upper bound on by Alon,
Grytczuk, Kisielewicz, and Przes\l awski for any positive integers and
with . In particular, when is sufficiently large and , our upper bound on improves the bound
shown by Huang and Sudakov exponentially.
Furthermore, we determine that , , ,
, , and . The stability result of Kleitman's
isodiametric inequality plays an important role in the proofs.Comment: 17 pages. The main results were further improve
Neighborly boxes and bipartite coverings; constructions and conjectures
Two axis-aligned boxes in are \emph{-neighborly} if their
intersection has dimension at least and at most . The maximum number
of pairwise -neighborly boxes in is denoted by . It
is known that , for fixed , but
exact formulas are known only in three cases: , , and . In
particular, the formula is equivalent to the famous theorem of
Graham and Pollak on bipartite partitions of cliques.
In this paper we are dealing with the case . We give a new construction
of -neighborly \emph{codes} giving better lower bounds on . The
construction is recursive in nature and uses a kind of ``algebra'' on
\emph{lists} of ternary strings, which encode neighborly boxes in a familiar
way. Moreover, we conjecture that our construction is optimal and gives an
explicit formula for . This supposition is supported by some numerical
experiments and some partial results on related open problems which are
recalled
A polynomial space proof of the Graham–Pollak theorem
This note describes a polynomial space proof of the Graham–Pollak theorem.© Elsevie
-free Partition and Cover Numbers and Application
-free graphs-- also known as cographs, complement-reducible graphs, or hereditary Dacey graphs--have been well studied in graph theory.
Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite -free graphs.
For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant.
Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems.
Interestingly, our covering problem is closely related to the well-studied problem of product/Prague dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension.
We prove that computing these numbers is \npol-complete, even for bipartite graphs.
We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size- partite sets such that these numbers are at least , for .
Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality.
For applications in information theory and communication \& cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample.
Alice and Bob\u27s objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness.
A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective.
Lower bounds to the minimum size of the genie\u27s assistance translate into communication and cryptographic lower bounds.
We show that (the of) the -free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie\u27s assistance.
Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high -free partition numbers correspond to joint distributions requiring more assistance from the genie.
As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output.
We show that (the of) the -free cover number of the bipartite graph encoding a Boolean function is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of in this model).
Consequently, the functions corresponding to the bipartite graphs with high -free cover numbers have high communication complexity.
Furthermore, there are functions with communication complexity close to the \naive protocol where the nondeterministic input reveals a party\u27s input.
Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle.
To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal.
On the other hand, access to the equality oracle is (nearly) useless for computing set intersection