13,618 research outputs found
All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected -vertex graph with non-negative edge-weights, we
consider the following type of query: given two vertices and in ,
what is the weight of a minimum -cut in ? We solve this problem in
preprocessing time for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem
We prove that with high probability over the choice of a random graph
from the Erd\H{o}s-R\'enyi distribution , the -time degree
Sum-of-Squares semidefinite programming relaxation for the clique problem
will give a value of at least for some constant
. This yields a nearly tight bound on the value of this
program for any degree . Moreover we introduce a new framework
that we call \emph{pseudo-calibration} to construct Sum of Squares lower
bounds. This framework is inspired by taking a computational analog of Bayesian
probability theory. It yields a general recipe for constructing good
pseudo-distributions (i.e., dual certificates for the Sum-of-Squares
semidefinite program), and sheds further light on the ways in which this
hierarchy differs from others.Comment: 55 page
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