- Geometry of Semidefinite Max-Cut Relaxations via Ranks Miguel F. Anjos (anjosstanfordalumni.org) Henry Wolkowicz (hwolkowiorion.math.uwaterloo.ca) Abstract: 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 NP-hard problems, the Max-Cut problem (MC). The well-known SDP relaxation for Max-Cut, 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 well-known relaxation. In this paper we present new results that further describe the remarkable tightness of this new relaxation. Let ${\cal F}_n$ denote the feasible set of SDP2 for the complete graph with $n$ nodes, let $F_n$ denote the appropriately defined projection of ${\cal F}_n$ into $\Sn$, the space of real symmetric $n \times n$ matrices, and let $C_n$ denote the cut polytope in $\Sn$. Further let $Y \in {\cal F}_n$ be the matrix variable of the SDP2 relaxation and $X \in F_n$ 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 $Y \in {\cal F}_3$ 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 $n \geq 4$. For this case, we show that the characterization of the vertices of the cut polytope via $\rank Y = 1$ extends to all $n \geq 4$. More interestingly, we show that the characterization of the one-dimensional faces via $\rank Y = 2$ also holds for $n \geq 4$. Furthermore, we prove that if $\rank Y = 2$ for $n \geq 3$, then a simple algorithm exhibits the two rank-one matrices (corresponding to cuts) which are the vertices of the one-dimensional face of the cut polytope where $X$ lies. Keywords: Semidefinite Programming, Discrete Optimization,Lagrangian Relaxation, Max-Cut Problem. Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 2: Combinatorial Optimization Citation: Journal of Combinatorial Optimization, Vol. 6 (3), 2002, 237-270. Download: Entry Submitted: 06/14/2001Entry Accepted: 06/14/2001Entry Last Modified: 06/13/2002Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.