Parameterized Approximation Algorithms for Packing Problems

In the past decade, many parameterized algorithms were developed for packing problems. Our goal is to obtain tradeoffs that improve the running times of these algorithms at the cost of computing approximate solutions. Consider a packing problem for which there is no known algorithm with approximation ratio $\alpha$, and a parameter $k$. If the value of an optimal solution is at least $k$, we seek a solution of value at least $\alpha k$; otherwise, we seek an arbitrary solution. Clearly, if the best known parameterized algorithm that finds a solution of value $t$ runs in time $O^*(f(t))$ for some function $f$, we are interested in running times better than $O^*(f(\alpha k))$. We present tradeoffs between running times and approximation ratios for the $P_2$-Packing, $3$-Set $k$-Packing and $3$-Dimensional $k$-Matching problems. Our tradeoffs are based on combinations of several known results, as well as a computation of "approximate lopsided universal sets."