Packing under Convex Quadratic Constraints

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three algorithms for problem instances arising in the context of real-world gas transport networks.