Abstract
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 mathcal {O}(k 2^{2k}) 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 mathcal {O}(k^2) and mathcal {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).
Highlights
One of the most popular paradigms when designing effective algorithms is preprocessing
In this work we focus on this kind of preprocessing, known as graph compression, for flows and cuts
Theorem 1.1 For every integer k ≥ 3, there exists a planar graph G with a set Q of k terminals and edge cost function under which every mimicking network for G has at least 2k−2 edges
Summary
One of the most popular paradigms when designing effective algorithms is preprocessing. Hagerup et al [6] observed the following simple preprocessing step: if two vertices u and v are always on the same side of the minimum cut between S and Sfor every choice of the partition Q = S S, they can be merged without changing the size of any minimum S-separating cut This procedure always leads to a mimicking network with at most 22k vertices. Theorem 1.1 For every integer k ≥ 3, there exists a planar graph G with a set Q of k terminals and edge cost function under which every mimicking network for G has at least 2k−2 edges This nearly matches the upper bound of O(k22k) of Krauthgamer and Rika [10] and is in sharp contrast with the polynomial bounds when the terminals lie on a constant number of faces [5,10].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
