SDP Relaxations for Some Combinatorial Optimization Problems

Abstract

In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem. NotationThe space of p × q real matrices is denoted by R p×q , the space of k × k symmetric matrices is denoted by S k , and the space of k×k symmetric positive semidefinite matrices by S + k . We will sometimes also use the notation X 0 instead of X ∈ S + k , if the order of the matrix is clear from the context. For index sets α, β ⊂ {1, . . . , n}, we denote the submatrix that contains the rows of A indexed by α and the columns indexed by β as A(α, β). If α = β, the principal submatrix A(α, α) of A is abbreviated as A(α). The ith column of a matrix C is denoted by C :,i .We use I n to denote the identity matrix of order n, and e i to denote the ith standard basis vector. Similarly, J n and u n denote the n×n all-ones matrix and all-ones n-vector respectively, and 0 n×n is the zero matrix of order n. We will omit subscripts if the order is clear from the context. The set of n × n permutation matrices is denoted by Π n . We set E ij = e i e T j . The 'vec' operator stacks the columns of a matrix, while the 'diag' operator maps an n × n matrix to the n-vector given by its diagonal. The adjoint operator of 'diag' we denote by 'Diag'. The trace operator is denoted by 'tr'.The Kronecker product A ⊗ B of matrices A ∈ R p×q and B ∈ R r×s is defined as the pr × qs matrix composed of pq blocks of size r × s, with block ij given by a ij B, i = 1, . . . , p, j = 1, . . . , q.

An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

“…Therefore, we suggest the well known SDP relaxation of the QAP [48] for a relaxation of the GPP. The same relaxation was used as the SDP relaxation of the GEP in [11,45].…”

“…This problem is studied in detail in [45]. Here, we relate the known relaxations for the GEP with GPP RS .…”

“…In [45] it is proven that, when restricted to the GEP case, relaxations GPP ZWN , GPP QAP , and k-GEP KR are equivalent. We show here that also k-GEP KR and GPP RS are equivalent.…”

“…By collecting all results from this paper and [45], it follows that for the graph equipartition problem the following relaxations are equivalent: k-GEP KR , GPP RS , GPP FJ , GPP ZWN , and GPP QAP .…”

“…This bound can be computed efficiently for larger graphs that have suitable algebraic symmetry. For a comparison of the SDP bounds for the GEP, see [45].…”