224 research outputs found
Deterministic fully dynamic approximate vertex cover and fractional matching in O(1) update time
We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting
with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011],
Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the
assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption
on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question.
We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously,
the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of (2 + e) and an amortized update time of O(logn=e2 ).
Our result can be generalized to give a fully dynamic O( f3 )-approximate algorithm with O( f2 ) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices
Incremental (1-?)-Approximate Dynamic Matching in O(poly(1/?)) Update Time
In the dynamic approximate maximum bipartite matching problem we are given bipartite graph G undergoing updates and our goal is to maintain a matching of G which is large compared the maximum matching size ?(G). We define a dynamic matching algorithm to be ? (respectively (?, ?))-approximate if it maintains matching M such that at all times |M | ? ?(G) ? ? (respectively |M| ? ?(G) ? ? - ?).
We present the first deterministic (1-?)-approximate dynamic matching algorithm with O(poly(?^{-1})) amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS\u2714, Bhattacharya-Kiss-Saranurak SODA\u2723] or exponential in 1/? [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA\u2719] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive (1, ??n)-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for (1-?)-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of G in a fully dynamic manner.
Our algorithm relies on the weighted variant of the celebrated Edge-Degree-Constrained-Subgraph (EDCS) datastructure introduced by [Bernstein-Stein ICALP\u2715]. As far as we are aware we introduce the first application of the weighted-EDCS for arbitrarily dense graphs. We also present a significantly simplified proof for the approximation ratio of weighed-EDCS as a matching sparsifier compared to [Bernstein-Stein], as well as simple descriptions of a fractional matching and fractional vertex cover defined on top of the EDCS. Considering the wide range of applications EDCS has found in settings such as streaming, sub-linear, stochastic and more we hope our techniques will be of independent research interest outside of the dynamic setting
Incremental -approximate dynamic matching in update time
In the dynamic approximate maximum bipartite matching problem we are given
bipartite graph undergoing updates and our goal is to maintain a matching
of which is large compared the maximum matching size . We define a
dynamic matching algorithm to be (respectively )-approximate if it maintains matching such that at all times (respectively ).
We present the first deterministic -approximate dynamic
matching algorithm with amortized update time for
graphs undergoing edge insertions. Previous solutions either required
super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or
exponential in
[Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our
implementation is arguably simpler than the mentioned algorithms and its
description is self contained. Moreover, we show that if we allow for additive
-approximation our algorithm seamlessly extends to also
handle vertex deletions, on top of edge insertions. This makes our algorithm
one of the few small update time algorithms for -approximate
dynamic matching allowing for updates both increasing and decreasing the
maximum matching size of in a fully dynamic manner
Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching
We present the first deterministic data structures for maintaining
approximate minimum vertex cover and maximum matching in a fully dynamic graph
, with and , in time per update.
In particular, for minimum vertex cover we provide deterministic data
structures for maintaining a (2+\eps) approximation in O(\log n/\eps^2)
amortized time per update.
For maximum matching, we show how to maintain a (3+\eps) approximation in
O(\min(\sqrt{n}/\epsilon, m^{1/3}/\eps^2)) {\em amortized} time per update,
and a (4+\eps) approximation in O(m^{1/3}/\eps^2) {\em worst-case} time per
update. Our data structure for fully dynamic minimum vertex cover is
essentially near-optimal and settles an open problem by Onak and Rubinfeld from
STOC' 2010.Comment: An extended abstract of this paper will appear in SODA' 201
Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs
We study dynamic -approximate rounding of fractional matchings
-- a key ingredient in numerous breakthroughs in the dynamic graph algorithms
literature. Our first contribution is a surprisingly simple deterministic
rounding algorithm in bipartite graphs with amortized update time
, matching an (unconditional)
recourse lower bound of up to logarithmic factors.
Moreover, this algorithm's update time improves provided the minimum (non-zero)
weight in the fractional matching is lower bounded throughout. Combining this
algorithm with novel dynamic \emph{partial rounding} algorithms to increase
this minimum weight, we obtain several algorithms that improve this dependence
on . For example, we give a high-probability randomized algorithm with
-update time against adaptive
adversaries. (We use Soft-Oh notation, , to suppress polylogarithmic
factors in the argument, i.e., .)
Using our rounding algorithms, we also round known -decremental
fractional bipartite matching algorithms with no asymptotic overhead, thus
improving on state-of-the-art algorithms for the decremental bipartite matching
problem. Further, we provide extensions of our results to general graphs and to
maintaining almost-maximal matchings.Comment: Full version of STOC 2024 pape
An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph G=(V,E), which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v\u27s incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.\u27s work [SODA\u2715 and ICALP\u2715] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with O(log n / epsilon) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem
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