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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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