2,408 research outputs found
The computational complexity of convex bodies
We discuss how well a given convex body B in a real d-dimensional vector
space V can be approximated by a set X for which the membership question:
``given an x in V, does x belong to X?'' can be answered efficiently (in time
polynomial in d). We discuss approximations of a convex body by an ellipsoid,
by an algebraic hypersurface, by a projection of a polytope with a controlled
number of facets, and by a section of the cone of positive semidefinite
quadratic forms. We illustrate some of the results on the Traveling Salesman
Polytope, an example of a complicated convex body studied in combinatorial
optimization.Comment: 24 page
A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope
For a convex body B in a vector space V, we construct its approximation P_k,
k=1, 2, . . . using an intersection of a cone of positive semidefinite
quadratic forms with an affine subspace. We show that P_k is contained in B for
each k. When B is the Symmetric Traveling Salesman Polytope on n cities T_n, we
show that the scaling of P_k by n/k+ O(1/n) contains T_n for k no more than
n/2. Membership for P_k is computable in time polynomial in n (of degree linear
in k).
We discuss facets of T_n that lie on the boundary of P_k. We introduce a new
measure on each facet defining inequality for T_n in terms of the eigenvalues
of a quadratic form. Using these eigenvalues of facets, we show that the
scaling of P_1 by n^(1/2) has all of the facets of T_n defined by the subtour
elimination constraints either in its interior or lying on its boundary.Comment: 25 page
On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)
AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;semidefinite programming;quadratic as- signment problem
The Quadratic Cycle Cover Problem: special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of
node-disjoint cycles visiting all the nodes such that the total sum of
interaction costs between consecutive arcs is minimized. In this paper we study
the linearization problem for the quadratic cycle cover problem and related
lower bounds.
In particular, we derive various sufficient conditions for the quadratic cost
matrix to be linearizable, and use these conditions to compute bounds. We also
show how to use a sufficient condition for linearizability within an iterative
bounding procedure. In each step, our algorithm computes the best equivalent
representation of the quadratic cost matrix and its optimal linearizable matrix
with respect to the given sufficient condition for linearizability. Further, we
show that the classical Gilmore-Lawler type bound belongs to the family of
linearization based bounds, and therefore apply the above mentioned iterative
reformulation technique. We also prove that the linearization vectors resulting
from this iterative approach satisfy the constant value property.
The best among here introduced bounds outperform existing lower bounds when
taking both quality and efficiency into account
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