100,472 research outputs found
Sublinear Estimation of Weighted Matchings in Dynamic Data Streams
This paper presents an algorithm for estimating the weight of a maximum
weighted matching by augmenting any estimation routine for the size of an
unweighted matching. The algorithm is implementable in any streaming model
including dynamic graph streams. We also give the first constant estimation for
the maximum matching size in a dynamic graph stream for planar graphs (or any
graph with bounded arboricity) using space which also
extends to weighted matching. Using previous results by Kapralov, Khanna, and
Sudan (2014) we obtain a approximation for general graphs
using space in random order streams, respectively. In
addition, we give a space lower bound of for any
randomized algorithm estimating the size of a maximum matching up to a
factor for adversarial streams
Distributed Approximate Maximum Matching in the CONGEST Model
We study distributed algorithms for the maximum matching problem in the CONGEST model, where each message must be bounded in size. We give new deterministic upper bounds, and a new lower bound on the problem.
We begin by giving a distributed algorithm that computes an exact maximum (unweighted) matching in bipartite graphs, in O(n log n) rounds. Next, we give a distributed algorithm that approximates the fractional weighted maximum matching problem in general graphs. In a graph with maximum degree at most Delta, the algorithm computes a (1-epsilon)-approximation for the problem in time O(log(Delta W)/epsilon^2), where W is a bound on the ratio between the largest and the smallest edge weight. Next, we show a slightly improved and generalized version of the deterministic rounding algorithm of Fischer [DISC \u2717]. Given a fractional weighted maximum matching solution of value f for a given graph G, we show that in time O((log^2(Delta)+log^*n)/epsilon), the fractional solution can be turned into an integer solution of value at least (1-epsilon)f for bipartite graphs and (1-epsilon) * (g-1)/g * f for general graphs, where g is the length of the shortest odd cycle of G. Together with the above fractional maximum matching algorithm, this implies a deterministic algorithm that computes a (1-epsilon)* (g-1)/g-approximation for the weighted maximum matching problem in time O(log(Delta W)/epsilon^2 + (log^2(Delta)+log^* n)/epsilon).
On the lower-bound front, we show that even for unweighted fractional maximum matching in bipartite graphs, computing an (1 - O(1/sqrt{n}))-approximate solution requires at least Omega~(D+sqrt{n}) rounds in CONGEST. This lower bound requires the introduction of a new 2-party communication problem, for which we prove a tight lower bound
Deterministic Algorithms for Maximum Matching on General Graphs in the Semi-Streaming Model
We present an improved deterministic algorithm for Maximum Cardinality Matching on general graphs in the Semi-Streaming Model. In the Semi-Streaming Model, a graph is presented as a sequence of edges, and an algorithm must access the edges in the given sequence. It can only use O(n polylog n) space to perform computations, where n is the number of vertices of the graph. If the algorithm goes over the stream k times, it is called a k-pass algorithm. In this model, McGregor [McGregor, 2005] gave the currently best known randomized (1+epsilon)-approximation algorithm for maximum cardinality matching on general graphs, that uses (1/epsilon)^{O(1/epsilon)} passes. Ahn and Guha [Ahn and Guha, 2013] later gave the currently best known deterministic (1+epsilon)-approximation algorithms for maximum cardinality matching: one on bipartite graphs that uses O(log log(1/epsilon)/epsilon^2) passes, and the other on general graphs that uses O(log n *poly(1/epsilon)) passes (note that, for general graphs, the number of passes is dependent on the size of the input). We present the first deterministic algorithm that achieves a (1+epsilon)-approximation on general graphs in only a constant number ((1/epsilon)^{O(1/epsilon)}) of passes
New deterministic approximation algorithms for fully dynamic matching
We present two deterministic dynamic algorithms for the maximum matching
problem. (1) An algorithm that maintains a -approximate maximum
matching in general graphs with update
time. (2) An algorithm that maintains an approximation of the {\em
value} of the maximum matching with update time in bipartite
graphs, for every sufficiently large constant positive integer . Here,
is a constant determined by the value of . Result (1)
is the first deterministic algorithm that can maintain an -approximate maximum matching with polylogarithmic update time, improving
the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee
almost matches the guarantee of the best {\em randomized} polylogarithmic
update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a
better-than-two approximation with {\em arbitrarily small polynomial} update
time on bipartite graphs. Previously the best update time for this problem was
[Bernstein et al. ICALP 2015], where is the current number of
edges in the graph.Comment: To appear in STOC 201
- …