It is well known that \(\operatorname {NP}\) -hard problems can be solved efficiently on instances that are sufficiently well-structured. Many admit exact parameterized algorithms with running time \(f(k)\cdot n^{\mathcal {O}(1)}\) where k is some structural measure such as treewidth. For small k, this is comparable to a polynomial-time exact algorithm and outperforms exact exponential-time algorithms for a large range of k. In this work, we are interested instead in leveraging instance structure for polynomial-time approximation algorithms. Concretely, we aim for polynomial-time algorithms that produce a solution of value at most/least \(c {{\,\textrm{OPT}\,}}\pm f(k)\) where c is a constant. Unlike for standard parameterized algorithms, we do not assume that any structural information is provided with the input. Ideally, we can obtain algorithms with small additive error only, i.e., \(c=1\) and f(k) is polynomial or even linear in k. For small k, this is similarly comparable to a polynomial-time exact algorithm and it will beat general case approximation for a large range of k. We study (Connected) Vertex Cover, Chromatic Number, and Triangle Packing. As parameters we consider the size of minimum modulators to graph classes on which the respective problem is tractable. For most cases we give efficient algorithms that compute a solution of size at least/most \({{\,\textrm{OPT}\,}}\pm k\) . For Vertex Cover, most of our algorithms are tight under the Unique Games Conjecture and have considerably better approximation guarantees than standard 2-approximations whenever the modulator is smaller than the optimum solution.