An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs
Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networ...
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| Vydáno v: | Algorithmica Ročník 81; číslo 10; s. 4029 - 4042 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.10.2019
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Given an edge-weighted graph
G
with a set
Q
of
k
terminals, a
mimicking network
is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph
G
being either an arbitrary graph or coming from a specific graph class. We show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with
k
terminals that require
2
k
-
2
edges in any mimicking network. This nearly matches an upper bound of
O
(
k
2
2
k
)
of Krauthgamer and Rika (in: Khanna (ed) Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans,
2013
) and is in sharp contrast with the upper bounds of
O
(
k
2
)
and
O
(
k
4
)
under the assumption that all terminals lie on a single face (Goranci et al., in: Pruhs and Sohler (eds) 25th Annual European symposium on algorithms (ESA 2017),
2017
,
arXiv:1702.01136
; Krauthgamer and Rika in Refined vertex sparsifiers of planar graphs,
2017
,
arXiv:1702.05951
). As a side result we show a tight example for double-exponential upper bounds given by Hagerup et al. (J Comput Syst Sci 57(3):366–375,
1998
), Khan and Raghavendra (Inf Process Lett 114(7):365–371,
2014
), and Chambers and Eppstein (J Gr Algorithms Appl 17(3):201–220,
2013
). |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-018-0504-8 |