4 research outputs found
New bounds on the classical and quantum communication complexity of some graph properties
We study the communication complexity of a number of graph properties where
the edges of the graph are distributed between Alice and Bob (i.e., each
receives some of the edges as input). Our main results are:
* An Omega(n) lower bound on the quantum communication complexity of deciding
whether an n-vertex graph G is connected, nearly matching the trivial classical
upper bound of O(n log n) bits of communication.
* A deterministic upper bound of O(n^{3/2}log n) bits for deciding if a
bipartite graph contains a perfect matching, and a quantum lower bound of
Omega(n) for this problem.
* A Theta(n^2) bound for the randomized communication complexity of deciding
if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for the quantum
communication complexity of this problem.
The first two quantum lower bounds are obtained by exhibiting a reduction
from the n-bit Inner Product problem to these graph problems, which solves an
open question of Babai, Frankl and Simon. The third quantum lower bound comes
from recent results about the quantum communication complexity of composed
functions. We also obtain essentially tight bounds for the quantum
communication complexity of a few other problems, such as deciding if G is
triangle-free, or if G is bipartite, as well as computing the determinant of a
distributed matrix.Comment: 12 pages LaTe
Nearly Optimal Communication and Query Complexity of Bipartite Matching
We settle the complexities of the maximum-cardinality bipartite matching
problem (BMM) up to poly-logarithmic factors in five models of computation: the
two-party communication, AND query, OR query, XOR query, and quantum edge query
models. Our results answer open problems that have been raised repeatedly since
at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck,
Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan
SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21]
and Zhang [ICALP'04]. We also settle the communication complexity of the
generalizations of BMM, such as maximum-cost bipartite -matching and
transshipment; and the query complexity of unique bipartite perfect matching
(answering an open question by Beniamini [2022]). Our algorithms and lower
bounds follow from simple applications of known techniques such as cutting
planes methods and set disjointness.Comment: Accepted in FOCS 202
The Message Complexity of Distributed Graph Optimization
The message complexity of a distributed algorithm is the total number of
messages sent by all nodes over the course of the algorithm. This paper studies
the message complexity of distributed algorithms for fundamental graph
optimization problems. We focus on four classical graph optimization problems:
Maximum Matching (MaxM), Minimum Vertex Cover (MVC), Minimum Dominating Set
(MDS), and Maximum Independent Set (MaxIS). In the sequential setting, these
problems are representative of a wide spectrum of hardness of approximation.
While there has been some progress in understanding the round complexity of
distributed algorithms (for both exact and approximate versions) for these
problems, much less is known about their message complexity and its relation
with the quality of approximation. We almost fully quantify the message
complexity of distributed graph optimization by showing the following
results...[see paper for full abstract