A Bound on the Sum of Weighted Pairwise Distances of Points Constrained to Balls

w(i, j)d(p i , p j ) subject to    p i ∈ IR n-1 (i = 1, .., n);

Here each w(i, j) ≥ 0 and each ℓ(i) ≥ 0 is fixed, d(p, q) denotes the Euclidean distance between points p and q, and ||p|| denotes the Euclidean length (distance from the origin) of point p.

We derive the following dual problem D(n, w, ℓ):

Throughout the paper, 0 0 is defined to be 0. We show that the value of the maximization problem is at most the value of the minimization problem. We use a physical interpretation of the two problems to show that the values are equal provided the maximization problem admits a set of points {p i } that is both affinely independent and stationary (i.e., the gradient of the objective function is a nonnegative combination of the gradients of the active constraints, a necessary condition at any local maximizer of P (n, w, ℓ)).

We sketch how a near-optimal solution to the problem can be found in polynomial time via the ellipsoid method.

The case w(i, j) = ℓ(i) = 1 (in which the optimal points are given by the vertices of the regular n-simplex, achieving a value of n n

2 ) was previously considered by [3]. Our Lemma 1 generalizes a bound in that paper.

Specific instances of P (n, w, ℓ) were studied to obtain geometric inequalities that were used to analyze approximation algorithms for finding low-degree, low-weight spanning trees in Euclidean spaces [2].

Goemans and Williamson [1] consider related problems with applications to approximating the maximum cut in a graph and to maximizing the number of satisfied clauses in a CNF formula. We modify their approach to solving their problems to obtain a polynomial time algorithm for ours.

Lemma 1 For any n, w, and ℓ, the value of the maximization problem P (n, w, ℓ) is at most the value of the minimization problem D(n, w, ℓ).

Proof: Fix any n, w, and ℓ. Fix any set of points {p i } and values {x i } meeting the constraints of P (n, w, ℓ) and D(n, w, ℓ), respectively. Let A(i, j) = w(i,j)

Then, by the Cauchy-Schwartz inequality A • B ≤ A × B (where A and B are interpreted as n 2 -dimensional vectors, and • denotes the dot product):

It remains only to show

Expanding the left-hand side,

Lemma 2 Fix any n, w, and ℓ. Suppose the maximization problem P (n, w, ℓ) admits a set of points {p i } that is both stationary and affinely independent. Then the values of the two problems are equal. Further, there exists {x i } such that

(where x i = 0 in case p i < ℓ i , and w(i, j) = w(j, i) and w(i, i) = 0), and {p i } and {x i } are global optima for the two problems.

Proof: Fix any n, w, and ℓ. Consider the objective function Φ({p i }) = ij w(i, j)d(p i , p j ) of P (n, w, ℓ). That {p i } is stationary means that the gradient of Φ is a nonnegative combination of the gradients of the constraints of P (n, w, ℓ) active at {p i }. By elementary calculus, the gradient of Φ consists of a vector f i for each point p i , with each f i equal to the right-hand side of (4). The only constraint on p i is p i ≤ ℓ(i), whose gradient (again by elementary calculus) is a nonnegative multiple of p i . Thus, for each i, there exists an x i ≥ 0 such that (4) holds. Note that if p i < ℓ(i), then the constraint is not active, so that f i must be the zero vector. In this case we take x i = 0. We will show that each inequality in Lemma 1 is tight for these {p i } and {x i }. Inequality (3) is tight because, by (4), i x i p i is the zero vector. Inequality (2) is tight because

Inequality (1) is tight provided the vector A (in the proof of Lemma 1) is a scalar multiple of B. Assume {p i } is affinely independent. Then, considering {x i } and {p i } fixed and {w(i, j)} as the set of unknowns (i.e., reversing their roles), (4) uniquely determines each w(i, j). Since

is consistent with (4) (check this by substitution for w(i, j) in (4)), it follows that (5) necessarily holds. Thus, A is a scalar multiple of B and Inequality (1) is tight.

A physical model for the quantities involved is as follows. Consider a physical system of n points {p i }. Each point p i is constrained to a ball of radius ℓ(i) centered at the origin. For each pair of points (p i , p j ), p i repels p j (and vice versa) with a force of magnitude w(i, j).

Under this interpretation, each vector f i in the proof corresponds to the force on p i , and x i is the magnitude of this force, divided by p i .

If the instance of P (n, w, ℓ) is small or has a high degree of symmetry, the dual problem D(n, w, ℓ) might yield a function that can be minimized directly by symbolic methods. In general, it is possible to solve P (n, w, ℓ) (to any given degree of precision) in polynomial time using semi-definite programming, following the approach in [1].

Those authors consider a related problem GW (w, n):

The authors show how to solve this problem in polynomial time by formulating it as a semi-definite program, and how to round a (near-)optimal set of points {p i } to obtain an app

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