CORC REPORT 2003-09 Tree-width and the Sherali–Adams operator.pdf

CORC REPORT 2003-09 Tree-width and the Sherali–Adams operator.pdf

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CORC REPORT 2003-09 Tree-width and the Sherali–Adams operator

CORC REPORT 2003-09 Tree-width and the Sherali–Adams operator? Daniel Bienstock and Nuri Ozbay Columbia University New York, NY 10027 November 11, 2003 version 2004-March-23 Abstract We describe a connection between the tree-width of graphs and the Sherali–Adams reformulation procedure for 0/1 integer programs. For the case of vertex packing problems, our main result can be restated as follows: let G be a graph, let k ≥ 1 and let x? ∈ RV (G) be a feasible vector for the formulation produced by applying the level-k Sherali–Adams algorithm to the edge formulation for STAB(G). Then for any subgraph H of G, of tree-width at most k, the restriction of x? to RV (H) is a convex combination of stable sets of H. 1 Introduction A 0/1 packing set is a feasible region of the form P bA = {x ∈ {0, 1}n : Ax ≤ b}, where A is a nonnegative, m × n matrix, and b ∈ n+. Given such a matrix A we can define its clique graph, which is the graph GA with a vertex corresponding to each column of A and an edge between two vertices j1 and j2 if there exists some row i with ai,j1 0 and ai,j2 0. Given a vector α ∈ n, denote by suppt(α) the support of α, i.e. the set {j : αj 6= 0}. We will use the notation GA[α] to abbreviate GA[suppt(α)], that is, the subgraph of GA induced by suppt(α). In this note we consider the relationship between valid inequalities αTx ≤ β that are “simple”, as measured by the tree-width (defined below) of an appropriate subgraph GA[α] and the strength of the relaxation provided by the Sherali–Adams operator (also defined below). Given a set of rows R of a matrix A, we denote by A(R) the corresponding submatrix. Definition 1.1 Consider a 0/1 packing set P bA. The tree-width of a valid inequality α Tx ≤ β is the mini- mum, over all subset R of rows of A such that αTx ≤ β is valid for P b(R)A(R), of the tree-width of GA(R)[α]. Our main result is: Theorem 1.2 Consider a 0/1 packing set P bA. Let k ≥ 1, and suppose that a vector x? ∈ n satisfies the constraints imposed by the

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