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 |
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| Abstract | 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. |
|---|---|
| AbstractList | We present deterministic algorithms for maintaining a
$$(3/2 + \epsilon )$$
(
3
/
2
+
ϵ
)
and
$$(2 + \epsilon )$$
(
2
+
ϵ
)
-approximate maximum matching in a fully dynamic graph with worst-case update times
$${\hat{O}}(\sqrt{n})$$
O
^
(
n
)
and
$${\tilde{O}}(1)$$
O
~
(
1
)
respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio
$$(2 - \delta )$$
(
2
-
δ
)
(for any
$$\delta > 0$$
δ
>
0
) and
$$(2 + \epsilon )$$
(
2
+
ϵ
)
were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times
$$O(n^{3/4})$$
O
(
n
3
/
4
)
and
$$O_\epsilon (\sqrt{n})$$
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
$$O_\epsilon (\sqrt{n})$$
O
ϵ
(
n
)
and
$${\tilde{O}}(1)$$
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
$$(3/2 + \epsilon )$$
(
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
$$(\alpha , \delta )$$
(
α
,
δ
)
-approximate matching sparsifier if at all times
H
satisfies that
$$\mu (H) \cdot \alpha + \delta \cdot n \ge \mu (G)$$
μ
(
H
)
·
α
+
δ
·
n
≥
μ
(
G
)
(define
$$(\alpha , \delta )$$
(
α
,
δ
)
-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a
$$(3/2 + \epsilon , \delta )$$
(
3
/
2
+
ϵ
,
δ
)
-approximate matching sparsifier. We further show how to reduce the maintenance of an
$$\alpha $$
α
-approximate maximum matching to the maintenance of an
$$(\alpha , \delta )$$
(
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
$${\hat{O}}(1)$$
O
^
(
1
)
or
$${\tilde{O}}(1)$$
O
~
(
1
)
and is deterministic or randomized against an adaptive adversary respectively. To achieve
$$(2 + \epsilon )$$
(
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. 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(n3/4) and Oϵ(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 Oϵ(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. 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. |
| Author | Kiss, Peter |
| Author_xml | – sequence: 1 givenname: Peter surname: Kiss fullname: Kiss, Peter email: peter.kiss@warwick.ac.uk organization: Department of Computer Science, University of Warwick |
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| Keywords | Matching Dynamic algorithms Approximate matching EDCS |
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-approximate maximum matching in a fully dynamic graph with worst-case update... We present deterministic algorithms for maintaining a $$(3/2 + \epsilon )$$ ( 3 / 2 + ϵ ) and $$(2 + \epsilon )$$ ( 2 + ϵ ) -approximate maximum matching in a... 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... |
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