We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a non-negative vector cost $c_e \in \mathbb{R}^{\ell}_{\ge 0}$. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the $\ell_p$-norm of the obtained cost vector (we assume that $p \ge 1$ is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.

翻译：我们针对具有向量成本的短路径、群组斯坦纳树和群组ATSP问题的变体，提出了多对数近似算法。在这些问题中，每条边e具有非负向量成本$c_e \in \mathbb{R}^{\ell}_{\ge 0}$。对于可行解（分别为路径、子树或环游），我们计算解中所有边的总向量成本，然后对所得成本向量取$\ell_p$-范数（我们假设$p \ge 1$为整数）。针对串并联图的算法在多项式时间内运行，而针对任意图的算法在拟多项式时间内运行。为得到这些结果，我们引入并使用了基于流的新和平方松弛方法。此外，我们还获得了若干不可近似性结果。