1,658 research outputs found
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
Waiting Time Distribution for the Emergence of Superpatterns
Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking
values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of
X_1,...,X_t that contains, as an embedded subsequence, each of the
non-order-isomorphic subpatterns of length k. We focus on the non-trivial case
of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t
is a superpattern}Comment: 17 page
Tableau sequences, open diagrams, and Baxter families
Walks on Young's lattice of integer partitions encode many objects of
algebraic and combinatorial interest. Chen et al. established connections
between such walks and arc diagrams. We show that walks that start at
, end at a row shape, and only visit partitions of bounded height
are in bijection with a new type of arc diagram -- open diagrams. Remarkably
two subclasses of open diagrams are equinumerous with well known objects:
standard Young tableaux of bounded height, and Baxter permutations. We give an
explicit combinatorial bijection in the former case.Comment: 20 pages; Text overlap with arXiv:1411.6606. This is the full version
of that extended abstract. Conjectures from that work are proved in this wor
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