Tutte Colloquium - Matthias Mnich

Title: New algorithms for maximum disjoint paths based on tree-likeness

Abstract:

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is $2^{Ω(√log n)}, assuming $\mbox{NP}\subsetneq \mbox{DTIME}(n^{O(log n)})$. This constitutes a significant gap to the best known approximation upper bound of O(√n) due to Chekuri et al. (2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n/log log n) paths.

We strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:

(1) For MaxEDP, we give an O(√rlog(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio O(√n) due to Chekuri et al., as r≤n.

2) Further, we show how to route Ω(OPT*) pairs with congestion bounded by O(log(kr)/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.

(3) For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)^O(r)n. This is a substantial improvement on the run time of 2^kr^O(r)n, , which can be obtained via an algorithm by Scheffler.

We complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless FPT=W[1]and that our approximability results are relevant even for very small constant values of r.

Appeared in Math. Programming, September 2018. Joint work with Joachim Spoerhase (Aalto) and Krzysztof Fleszar (Warsaw).