On Quadratic Programming with a Ratio Objective

Quadratic Programming (QP) is the well-studied problem of maximizing over {-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture, semidefinite programming techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {-1,0,1}. The specific problems we study are QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2}, and Normalized QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j |a_{ij}| We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an $\tilde{O}(n^{1/3})$ algorithm for QP-ratio. We also obtain an $\tilde{O}(n^{1/4})$ approximation for bipartite graphs, and better algorithms for special cases. As with other problems with ratio objectives (e.g. uniform sparsest cut), it seems difficult to obtain inapproximability results based on P!=NP. We give two results that indicate that QP-Ratio is hard to approximate to within any constant factor. We also give a natural distribution on instances of QP-Ratio for which an n^\epsilon approximation (for \epsilon roughly 1/10) seems out of reach of current techniques

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+n variables. A closed-form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than A dominates at least 2 m+n-2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the domination ratio of this algorithm is at least 14. We then show that for any fixed natural numbers a and b such that η=ab > 1, no polynomial time approximation algorithm exists for BBQP with domination ratio larger than 1-2(1-η)η(m+n), unless P = NP. It is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than A. One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported

Dual Bounds for Redistricting Problems with Non-Convex Objectives

We study optimization models for computational redistricting. We focus nonconvex objectives that estimate expected black voter representation, political representation, and Polsby Popper Compactness. All objectives contain a sum of convolutions with a ratio of variables. The representation objectives are a convolution of a ratio of variables with a cumulative distribution function of a normal distribution, also known as the probit curve, while the compactness objective has a quadratic complication in the ratio. We extend the work of Validi et al. [30], which develops strong optimization models for contiguity constraints and develop mixed integer linear programming models that tightly approximate the nonlinear model, and show that our approach creates tight bounds on these optimization problems. We develop novel mixed integer linear relaxations to these nonconvex objectives and demonstrate the effectiveness of our approaches on county level data

Semidefinite approximation for mixed binary quadratically constrained quadratic programs

Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in $N$-dimensional real or complex Euclidean space, such that $M$ concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. {\color{blue}By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by \cO(Q^2(M-Q+1)+M^2) in the real case and by \cO(M(M-Q+1)) in the complex case.} For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by \cO(\epsilon/\ln(M)) for both the real and the complex cases. Moreover, this ratio is tight up to a constant factor