Approximation Algorithms for ${\ell }_p$-Shortest Path and ${\ell }_p$-Group Steiner Tree

Abstract

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 $c_e\in {\mathbb{R}}_{\ge 0}^{\ell }$. 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.