17 research outputs found
Steiner Point Removal with Distortion
In the Steiner point removal (SPR) problem, we are given a weighted graph
and a set of terminals of size . The objective is to
find a minor of with only the terminals as its vertex set, such that
the distance between the terminals will be preserved up to a small
multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a
ball-growing algorithm with exponential distributions to show that the
distortion is at most . Cheung [Che17] improved the analysis of
the same algorithm, bounding the distortion by . We improve the
analysis of this ball-growing algorithm even further, bounding the distortion
by
Mimicking Networks and Succinct Representations of Terminal Cuts
Given a large edge-weighted network with terminal vertices, we wish
to compress it and store, using little memory, the value of the minimum cut (or
equivalently, maximum flow) between every bipartition of terminals. One
appealing methodology to implement a compression of is to construct a
\emph{mimicking network}: a small network with the same terminals, in
which the minimum cut value between every bipartition of terminals is the same
as in . This notion was introduced by Hagerup, Katajainen, Nishimura, and
Ragde [JCSS '98], who proved that such of size at most always
exists. Obviously, by having access to the smaller network , certain
computations involving cuts can be carried out much more efficiently.
We provide several new bounds, which together narrow the previously known gap
from doubly-exponential to only singly-exponential, both for planar and for
general graphs. Our first and main result is that every -terminal planar
network admits a mimicking network of size , which is
moreover a minor of . On the other hand, some planar networks require
. For general networks, we show that certain bipartite
graphs only admit mimicking networks of size , and
moreover, every data structure that stores the minimum cut value between all
bipartitions of the terminals must use machine words
On Communication Complexity of Fixed Point Computation
Brouwer's fixed point theorem states that any continuous function from a
compact convex space to itself has a fixed point. Roughgarden and Weinstein
(FOCS 2016) initiated the study of fixed point computation in the two-player
communication model, where each player gets a function from to
, and their goal is to find an approximate fixed point of the
composition of the two functions. They left it as an open question to show a
lower bound of for the (randomized) communication complexity of
this problem, in the range of parameters which make it a total search problem.
We answer this question affirmatively.
Additionally, we introduce two natural fixed point problems in the two-player
communication model.
Each player is given a function from to ,
and their goal is to find an approximate fixed point of the concatenation of
the functions.
Each player is given a function from to , and
their goal is to find an approximate fixed point of the interpolation of the
functions.
We show a randomized communication complexity lower bound of
for these problems (for some constant approximation factor).
Finally, we initiate the study of finding a panchromatic simplex in a
Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the
two-player communication model: A triangulation of the -simplex is
publicly known and one player is given a set and a coloring
function from to , and the other player is given a set
and a coloring function from to ,
such that , and their goal is to find a panchromatic
simplex. We show a randomized communication complexity lower bound of
for the aforementioned problem as well (when is large)
Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomialtime algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log 2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we prove a lower bound of Ω ( √ log k / log log k) for flow sparsifiers and a lower bound of Ω ( 4 √ log k / log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ( √ log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than ˜ Ω ( √ log k) would imply a negative answer to this question.