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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Bibliographic Details
Published in:Algorithmica Vol. 81; no. 10; pp. 4029 - 4042
Main Authors: Karpov, Nikolai, Pilipczuk, Marcin, Zych-Pawlewicz, Anna
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2019
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graphs
Lower bounds
Mathematics of Computing
Special Issue: Parameterized and Exact Computation
Terminals
Theory of Computation
Upper bounds
Sparsifier
Mimicking network
Planar graph
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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 ).
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-018-0504-8