Journal of Combinatorial Optimization
, Volume 6, Issue 3, pp 237270
First online:
Geometry of Semidefinite MaxCut Relaxations via Matrix Ranks
 Miguel F. AnjosAffiliated withDepartment of Combinatorics & Optimization, University of Waterloo
 , Henry WolkowiczAffiliated withDepartment of Combinatorics & Optimization, University of Waterloo
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Semidefinite programming (SDP) relaxations are proving to be a powerful tool for finding tight bounds for hard discrete optimization problems. This is especially true for one of the easier NPhard problems, the MaxCut problem (MC). The wellknown SDP relaxation for MaxCut, here denoted SDP1, can be derived by a first lifting into matrix space and has been shown to be excellent both in theory and in practice.
Recently the present authors have derived a new relaxation using a second lifting. This new relaxation, denoted SDP2, is strictly tighter than the relaxation obtained by adding all the triangle inequalities to the wellknown relaxation. In this paper we present new results that further describe the remarkable tightness of this new relaxation. Let \(F_n \) denote the feasible set of SDP2 for the complete graph with n nodes, let F _{n} denote the appropriately defined projection of \(F_n \) into \(S^n \), the space of real symmetric n × n matrices, and let C _{n} denote the cut polytope in \(S^n \). Further let \(Y \in F_n \) be the matrix variable of the SDP2 relaxation and X ∈ F _{n} be its projection. Then for the complete graph on 3 nodes, F _{3} = C _{3} holds. We prove that the rank of the matrix variable \(Y \in F_3 \) of SDP2 completely characterizes the dimension of the face of the cut polytope in which the corresponding matrix X lies. This shows explicitly the connection between the rank of the variable Y of the second lifting and the possible locations of the projected matrix X within C _{3}. The results we prove for n = 3 cast further light on how SDP2 captures all the structure of C _{3}, and furthermore they are stepping stones for studying the general case n ≥ 4. For this case, we show that the characterization of the vertices of the cut polytope via rank Y = 1 extends to all n ≥ 4. More interestingly, we show that the characterization of the onedimensional faces via rank Y = 2 also holds for n ≥ 4. Furthermore, we prove that if rank Y = 2 for n ≥ 3, then a simple algorithm exhibits the two rankone matrices (corresponding to cuts) which are the vertices of the onedimensional face of the cut polytope where X lies.
 Title
 Geometry of Semidefinite MaxCut Relaxations via Matrix Ranks
 Journal

Journal of Combinatorial Optimization
Volume 6, Issue 3 , pp 237270
 Cover Date
 200209
 DOI
 10.1023/A:1014895808844
 Print ISSN
 13826905
 Online ISSN
 15732886
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 semidefinite programming
 discrete optimization
 Lagrangian relaxation
 MaxCut problem
 Industry Sectors
 Authors

 Miguel F. Anjos ^{(1)}
 Henry Wolkowicz ^{(1)}
 Author Affiliations

 1. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada