Network Modelling Group An Overview of the Minimum-Congestion Hypergraph Embedding in a Cycle (MCHEC) Problem By Yong Wang Jun 4, 2002 2018/11/9 Network Modelling Group

Network Modelling Group Road Map Introduction to MCHEC Some other variants of MCHEC The previous research results References 2018/11/9 Network Modelling Group

Network Modelling Group Introduction What is MCHEC ? The MCHEC problem is to embed the hyperedges of the hypergraph as paths on a cycle with the same number of vertices, such that congestion (the maximum number of paths that use any single edge in the cycle) is minimized. Electronic design automation Parallel computing Computer communication on a ring network 2018/11/9 Network Modelling Group

MCHEC Problem Variants Weighted/Unweighted & Hypergraph/Graph Weighted Hypergraph Embedding(WHEC) (NP-Complete) Weighted Graph Embedding(WGEC) (NP) Unweighted Hypergraph Embedding(HEC) (NP) Unweighted Graph Embedding(GEC) (P) 2018/11/9 Network Modelling Group

Previous works – MCHEC (1) NP Complete [1] A linear approximation algorithm [1] to compute an embedding whose congestion is at most three times optimal (i.e. 3-approximation) 2018/11/9 Network Modelling Group

Previous works – MCHEC (2) 2-approximation algorithm 1 [3] to transform the MCHEC problem to a LP 2-approximation algorithm 2 [3] to solve the problem in linear time by transforming hyperedges to the normal edges 2018/11/9 Network Modelling Group

Previous works – MCHEC (3) A very simple 2-approximation algorithm [5] runs in linear time routes the hyperedges in the clockwise direction starting from the lowest numbered vertex to the highest numbered vertex. 2018/11/9 Network Modelling Group

Network Modelling Group Previous works - GEC A simple version of MCHEC For normal graph but not hypergraph, i.e. all hyperedges contain exactly only two vertices Be solvable in polynomial time using a linear time algorithm, i.e. the GEC problem is in the class of P but not NP [6] 2018/11/9 Network Modelling Group

Network Modelling Group Previous works - WGEC NP Complete [2] Reducing PARTITION to DECISION-WGEC 2018/11/9 Network Modelling Group

Network Modelling Group Previous works - WHEC NP Complete [2] WGEC is a subproblem of WHEC && WGEC is NP complete 2 appro. algorithm 1: Formulated as ILP [2] Get idea from [3] LP-relaxation && Rounding heuristic Disadvantage: Time Complexity is high, so … 2 appro. Algorithm 2: linear-time [2] greedily remove the longest adjacent-path of each hyperedge 2018/11/9 Network Modelling Group

Network Modelling Group References (1) [1] J. L. Ganley and J. P. Cohoon, “Minimum-congestion hypergraph embedding in a cycle”, IEEE Trans. Comput. 46 (5) (1997) 600-602 [2] SingLing Lee and Hann-Jang Ho, “Algorithms and Complexity for Weighted Hypergraph Embedding in a Cycle", [3] T. Gonzalez. “Improved approximation algorithms for embedding hyperedges in a cycle", Information Processing Letters 67 (4) (1998) 267-271 [4] J. L. Ganley and J. P. Cohoon, “A provably good moat routing algorithm”, Proc. Sixth Grate lakes Symp. VLSI (1996) 86-91 [5] T. Carpenter, S. Cosares, J. L. Ganley and I. Saniee, “A simple approximation algorithm for two problems in circuit design”, IEEE Trans. Comput. 47 (11) (1998) 1310-1312 2018/11/9 Network Modelling Group

Network Modelling Group References (2) [6] A. Frank, T. Nishizeki, N. Saito, H. Suzuki, E. Tardos, “Algorithms for routing around a rectangle”, Discrete Appl. Math. 40 (3) (1992) 363-378 2018/11/9 Network Modelling Group