182 research outputs found
Finite metric spaces--combinatorics, geometry and algorithms
Finite metric spaces arise in many different contexts. Enormous bodies of
data, scientific, commercial and others can often be viewed as large metric
spaces. It turns out that the metric of graphs reveals a lot of interesting
information. Metric spaces also come up in many recent advances in the theory
of algorithms. Finally, finite submetrics of classical geometric objects such
as normed spaces or manifolds reflect many important properties of the
underlying structure. In this paper we review some of the recent advances in
this area
Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap
We present a new explicit construction for expander graphs with nearly
optimal spectral gap. The construction is based on a series of 2-lift
operations.
Let be a graph on vertices. A 2-lift of is a graph on
vertices, with a covering map . It is not hard to see that all
eigenvalues of are also eigenvalues of . In addition, has
``new'' eigenvalues. We conjecture that every -regular graph has a 2-lift
such that all new eigenvalues are in the range (If
true, this is tight, e.g. by the Alon-Boppana bound). Here we show that every
graph of maximal degree has a 2-lift such that all ``new'' eigenvalues are
in the range for some constant .
This leads to a polynomial time algorithm for constructing arbitrarily large
-regular graphs, with second eigenvalue .
The proof uses the following lemma: Let be a real symmetric matrix such
that the norm of each row in is at most . Let . Then
the spectral radius of is at most , for some
universal constant .
An interesting consequence of this lemma is a converse to the Expander Mixing
Lemma.Comment: 29 pages, 1 figur
A counterexample to a conjecture of Bj\"{o}rner and Lov\'asz on the -coloring complex
Associated with every graph of chromatic number is another graph
. The vertex set of consists of all -colorings of , and two
-colorings are adjacent when they differ on exactly one vertex. According
to a conjecture of Bj\"{o}rner and Lov\'asz, this graph must be
disconnected. In this note we give a counterexample to this conjecture.Comment: To appear in JCT
Discrepancy of High-Dimensional Permutations
Let be an order- Latin square. For ,
let be the number of triples such that
. We conjecture that asymptotically almost every Latin square
satisfies for every
and . Let when . The
above conjecture implies that holds asymptotically
almost surely (this bound is obviously tight). We show that there exist Latin
squares with , and that for almost every order- Latin square. On the other hand, we
recall that if is the multiplication
table of an order- group. Some of these results extend to higher dimensions.
Many open problems remain
Are stable instances easy?
We introduce the notion of a stable instance for a discrete optimization
problem, and argue that in many practical situations only sufficiently stable
instances are of interest. The question then arises whether stable instances of
NP--hard problems are easier to solve. In particular, whether there exist
algorithms that solve correctly and in polynomial time all sufficiently stable
instances of some NP--hard problem. The paper focuses on the Max--Cut problem,
for which we show that this is indeed the case.Comment: 14 page
Tight products and Expansion
In this paper we study a new product of graphs called {\em tight product}. A
graph is said to be a tight product of two (undirected multi) graphs
and , if and both projection maps and are covering maps. It is not a priori clear when
two given graphs have a tight product (in fact, it is -hard to decide). We
investigate the conditions under which this is possible. This perspective
yields a new characterization of class-1 -regular graphs. We also
obtain a new model of random -regular graphs whose second eigenvalue is
almost surely at most . This construction resembles random graph
lifts, but requires fewer random bits
On the phase transition in random simplicial complexes
It is well-known that the model of random graphs undergoes a
dramatic change around . It is here that the random graph is,
almost surely, no longer a forest, and here it first acquires a giant (i.e.,
order ) connected component. Several years ago, Linial and Meshulam
have introduced the model, a probability space of -vertex
-dimensional simplicial complexes, where coincides with .
Within this model we prove a natural -dimensional analog of these graph
theoretic phenomena. Specifically, we determine the exact threshold for the
nonvanishing of the real -th homology of complexes from . We also
compute the real Betti numbers of for . Finally, we establish
the emergence of giant shadow at this threshold. (For a giant shadow and
a giant component are equivalent). Unlike the case for graphs, for the
emergence of the giant shadow is a first order phase transition
Asymptotically Almost Every -regular Graph has an Internal Partition
An internal partition of a graph is a partitioning of the vertex set into two
parts such that for every vertex, at least half of its neighbors are on its
side. We prove that for every positive integer , asymptotically almost every
-regular graph has an internal partition.Comment: 7 page
The expected genus of a random chord diagram
To any generic curve in an oriented surface there corresponds an oriented
chord diagram, and any oriented chord diagram may be realized by a curve in
some oriented surface. The genus of an oriented chord diagram is the minimal
genus of an oriented surface in which it may be realized. Let g_n denote the
expected genus of a randomly chosen oriented chord diagram of order n. We show
that g_n satisfies: g_n = n/2 - Theta(ln n)
Monotone Subsequences in High-Dimensional Permutations
This paper is part of the ongoing effort to study high-dimensional
permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For
every , every order- -dimensional permutation contains a monotone
subsequence of length , and this is tight. On
the other hand, and unlike the classical case, the longest monotone subsequence
in a random -dimensional permutation of order is asymptotically almost
surely .Comment: 12 pages, 1 figur
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