Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut
In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure...
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| Veröffentlicht in: | Proceedings of the ACM on management of data Jg. 2; H. 2 |
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01.05.2024
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| Abstract | In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that
-approximates cut values in graphs with
vertices. For arbitrary directed graphs, such a data structure requires
bits even for constant
. To circumvent this, recent works study
-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most
times that in the other direction. We consider two models: the
model, where the goal is to approximate each cut with constant probability, and the
model, where all cuts must be preserved simultaneously. We improve the previous
lower bound to
in the for-each model, and we improve the previous
lower bound to
in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous
query complexity lower bound to
for this problem, where
is the number of edges,
is the size of the minimum cut, and we seek a
-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors. |
|---|---|
| AbstractList | In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that
-approximates cut values in graphs with
vertices. For arbitrary directed graphs, such a data structure requires
bits even for constant
. To circumvent this, recent works study
-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most
times that in the other direction. We consider two models: the
model, where the goal is to approximate each cut with constant probability, and the
model, where all cuts must be preserved simultaneously. We improve the previous
lower bound to
in the for-each model, and we improve the previous
lower bound to
in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous
query complexity lower bound to
for this problem, where
is the number of edges,
is the size of the minimum cut, and we seek a
-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors. In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that ( 1 ± ε ) -approximates cut values in graphs with n vertices. For arbitrary directed graphs, such a data structure requires Ω n 2 bits even for constant ε . To circumvent this, recent works study β -balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times that in the other direction. We consider two models: the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ω ( n β / ε ) lower bound to Ω ~ ( n β / ε ) in the for-each model, and we improve the previous Ω ( n β / ε ) lower bound to Ω n β / ε 2 in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous Ω m k query complexity lower bound to Ω ( m i n { m , m ε 2 k } ) for this problem, where m is the number of edges, k is the size of the minimum cut, and we seek a ( 1 + ε ) -approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors.In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that ( 1 ± ε ) -approximates cut values in graphs with n vertices. For arbitrary directed graphs, such a data structure requires Ω n 2 bits even for constant ε . To circumvent this, recent works study β -balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times that in the other direction. We consider two models: the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ω ( n β / ε ) lower bound to Ω ~ ( n β / ε ) in the for-each model, and we improve the previous Ω ( n β / ε ) lower bound to Ω n β / ε 2 in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous Ω m k query complexity lower bound to Ω ( m i n { m , m ε 2 k } ) for this problem, where m is the number of edges, k is the size of the minimum cut, and we seek a ( 1 + ε ) -approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors. |
| Author | Li, Max Woodruff, David P Tai, Zi-Yi Lin, Honghao Cheng, Yu Zhang, Jason |
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| Title | Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut |
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