Better approximations for multi-cost network connectivity problems
Approximation Algorithms for Multi-Criteria Network Design Problems
Data Structures and Algorithms
Summary
Sometimes, when building networks like roads or communication lines, you want a design that is both fair and strong in different ways, measured by several cost rules. This is very hard to do well using old methods. The authors found new ways to build these networks that get much closer to the best possible design. They also showed that their approach is about as good as anyone can do, based on complexity theory assumptions.
What this means in practice
- •For network engineers: Design networks with reliable multiple connectivity that balances different cost criteria more efficiently with guaranteed approximation bounds.
- •For infrastructure planners: Create cost-effective plans to upgrade existing network connectivity by one level using improved approximation techniques.
A theory result. No direct application yet.
Authors
Mehrshad Taziki, Ali Vakilian
Abstract
Min-max multi-criteria optimization asks for a single feasible solution minimizing the maximum of its costs under $\ell$ nonnegative cost functions, a formulation that captures both fairness across groups and robustness across scenarios. It has proved hard to approximate even for basic problems: for robust shortest paths, polylogarithmic approximations are known only in quasi-polynomial time. For network design, the best known guarantee is an $\ell$-approximation, which is tight for the natural LP relaxation. We show that exponentially better and nearly optimal guarantees are achievable for two fundamental network design problems. For multi-criteria connectivity augmentation (MC-CAP), which asks to increase the edge connectivity of a given undirected graph by one, we give a randomized polynomial-time $O(\log \ell)$-approximation algorithm. Applying our augmentation procedure iteratively and analyzing its randomized choices jointly across iterations yields an $O(\log k \log \ell)$-approximation for the multi-criteria $k$-edge-connected spanning subgraph (MC-$k$-ECSS) problem, for every $k \geq 2$. We complement these algorithms with a nearly matching lower bound: for every fixed $\varepsilon > 0$, MC-CAP admits no polynomial-time $O(\log^{1-\varepsilon} \ell)$-approximation, even when the underlying graph is a star, unless $\mathsf{NP}\subseteq\mathsf{DTIME}\bigl(n^{\log^{O(1)}n}\bigr)$. The same hardness extends to the MC-$k$-ECSS problem.