Approximation Algorithms for $\ell_p$-Shortest Path and $\ell_p$-Group Steiner Tree

TL;DR

This paper develops polylogarithmic approximation algorithms for vector-cost variants of fundamental network problems, notably the $\ell_p$-Shortest Path, and extends the techniques to $\ell_p$-Group ATSP and $\ell_p$-Group Steiner Tree. Central to the approach is a novel flow-based Sum-of-Squares (SoS) relaxation of degree $2p$ and a rounding scheme that leverages majorization inequalities to control higher-moment costs across series-parallel decompositions and layered graph reductions. The authors derive tight bounds depending on the graph structure: a series-parallel depth $d$ yields an $O(p d^{1-1/p})$-type approximation, while arbitrary graphs admit an $O(p\log^{1-1/d} n)$-approximation with quasi-polynomial running time; the group variants achieve $O(c^2 p \log^{2-1/p} n \log k)$ factors with time $m^{O(p)+ce^{O(1/c)} \log n}$. Hardness results show limitations when costs can be negative and establish connections to Congestion Minimization and Closest Vector problems, underscoring the necessity of non-negativity and the depth of the approach. Overall, the work provides a novel SoS-based framework for vector-cost network design and situates it within a broader MOCO context, opening avenues for robust, multi-criteria optimization in graphs.

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

Figures (6)

• Figure 1: The structure of the reduction for the proof of Theorem \ref{['thm:CVP-reduction']}. On the left, we illustrate the structure of the $\ell_p$-Shortest Path instance graph $G$ constructed. On the right we show the $B$-block used to construct $G$.
• Figure 2: A diagram of the construction for the proof of Theorem \ref{['thm:minimum-congestion-reduction']}.
• Figure 3: The construction of $G^{(1)}_N$.
• Figure 4: The construction of $G^{(h)}_N$
• Figure 5: The construction for the discussion below. At the top, we illustrate the structure of the instance graph. At the bottom, we highlight the structure of the path $P_2$ as an example.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.9
• Theorem 1.10
• Definition 2.1
• Lemma 2.1