# Geometry of Semidefinite Max-Cut Relaxations via Matrix Ranks

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## 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 \(F_n \) 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 one-dimensional 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 rank-one matrices (corresponding to cuts) which are the vertices of the one-dimensional face of the cut polytope where *X* lies.

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