Geometry of Semidefinite Max-Cut Relaxations via Matrix Ranks
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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 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 Fn denote the appropriately defined projection of \(F_n \) into \(S^n \), the space of real symmetric n × n matrices, and let Cn denote the cut polytope in \(S^n \). Further let \(Y \in F_n \) be the matrix variable of the SDP2 relaxation and X ∈ Fn be its projection. Then for the complete graph on 3 nodes, F3 = C3 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 C3. The results we prove for n = 3 cast further light on how SDP2 captures all the structure of C3, 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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