Deterministic Dynamic Matching in Worst-Case Update Time
We present deterministic algorithms for maintaining a ( 3 / 2 + ϵ ) and ( 2 + ϵ ) -approximate maximum matching in a fully dynamic graph with worst-case update times O ^ ( n ) and O ~ ( 1 ) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ra...
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| Vydáno v: | Algorithmica Ročník 85; číslo 12; s. 3741 - 3765 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.12.2023
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We present deterministic algorithms for maintaining a
(
3
/
2
+
ϵ
)
and
(
2
+
ϵ
)
-approximate maximum matching in a fully dynamic graph with worst-case update times
O
^
(
n
)
and
O
~
(
1
)
respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio
(
2
-
δ
)
(for any
δ
>
0
) and
(
2
+
ϵ
)
were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times
O
(
n
3
/
4
)
and
O
ϵ
(
n
)
respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are
O
ϵ
(
n
)
and
O
~
(
1
)
which were shown in Bernstein and Stein (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, 2016) and Bhattacharya and Kiss (in: 48th international colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021) respectively. The algorithm achieving
(
3
/
2
+
ϵ
)
approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein (in: International colloquium on automata, languages, and programming, Springer, Berlin, 2015). Say that
H
is a
(
α
,
δ
)
-approximate matching sparsifier if at all times
H
satisfies that
μ
(
H
)
·
α
+
δ
·
n
≥
μ
(
G
)
(define
(
α
,
δ
)
-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a
(
3
/
2
+
ϵ
,
δ
)
-approximate matching sparsifier. We further show how to reduce the maintenance of an
α
-approximate maximum matching to the maintenance of an
(
α
,
δ
)
-approximate maximum matching building based on an observation of Assadi et al. (in: Proceedings of the twenty-seventh annual (ACM-SIAM) symposium on discrete algorithms, (SODA) 2016, Arlington, VA, USA, January 10–12, 2016). Our reduction requires an update time blow-up of
O
^
(
1
)
or
O
~
(
1
)
and is deterministic or randomized against an adaptive adversary respectively. To achieve
(
2
+
ϵ
)
-approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss (in: 48th International colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021). In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak (in: Proceedings of the twenty-seventh annual ACM symposium on theory of computing, 2017) and Bernstein et al. (Fully-dynamic graph sparsifiers against an adaptive adversary, 2020) which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-023-01151-x |