An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs - Karpov, Nikolai; Pilipczuk, Marcin; Zych-Pawlewicz, Anna (Algorithmica, 2019)
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 ).