TY - GEN

T1 - Graph bisection algorithms with good average case behavior

AU - Bui, Thang

AU - Chaudhuri, Soma

AU - Leighton, Tom

AU - Sipser, Mike

N1 - Funding Information:
This research was supported by Air Force contract AFOSR-82-0326 and DARPA contract N000M-S0-C-0622.
Publisher Copyright:
© 1984 IEEE.

PY - 1984

Y1 - 1984

N2 - In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d ≥ 3, all sufficiently large n and all b = o(n1-1/[d+1/2]), the algorithm finds the minimum bisection for almost all d-regular labelled simple graphs with 2n nodes and bisection width b. For example, the algorithm succeeds for almost all 5-regular graphs with 2n nodes and bisection width o(n 2/3 ). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably: (i) the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection, (ii) whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection), (iii) the algorithm works well both theoretically and experimentally, (iv) the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and (v) the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

AB - In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d ≥ 3, all sufficiently large n and all b = o(n1-1/[d+1/2]), the algorithm finds the minimum bisection for almost all d-regular labelled simple graphs with 2n nodes and bisection width b. For example, the algorithm succeeds for almost all 5-regular graphs with 2n nodes and bisection width o(n 2/3 ). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably: (i) the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection, (ii) whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection), (iii) the algorithm works well both theoretically and experimentally, (iv) the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and (v) the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

UR - http://www.scopus.com/inward/record.url?scp=84951077116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951077116&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84951077116

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 181

EP - 192

BT - 25th Annual Symposium on Foundations of Computer Science, FOCS 1984

PB - IEEE Computer Society

T2 - 25th Annual Symposium on Foundations of Computer Science, FOCS 1984

Y2 - 24 October 1984 through 26 October 1984

ER -