The Vickrey-Clarke-Groves (VCG) mechanism offers a general technique for resource allocation with payments, ensuring allocative efficiency while eliciting truthful information about preferences. However, VCG relies on exact computation of an optimal allocation of resources, a problem which is often computationally intractable, and VCG that uses an approximate allocation algorithm no longer guarantees truthful revelation of preferences. We present a series of results for computing or approximating an upper bound on agent incentives to misreport their preferences. Our first key result is an incentive bound that uses information about average (not worst-case) performance of an algorithm, which we illustrate using combinatorial auction data. Our second result offers a simple sampling technique for amplifying the difficulty of computing a utility-improving lie. An important consequence of our analysis is an argument that using state-of-the-art algorithms for solving combinatorial allocation problems essentially eliminates agent incentives to lie. © 2011 Elsevier B.V. All rights reserved.