The Parameterized Complexity Landscape of the Unsplittable Flow Problem

TL;DR

Novel algorithms and lower bounds are developed which result in a full classification of the parameterized complexity of the problem with respect to natural structural parameterizations for the problem -- notably maximum capacity, treewidth, maximum degree, and maximum flow length.

Abstract

We study the well-established problem of finding an optimal routing of unsplittable flows in a graph. While by now there is an extensive body of work targeting the problem on graph classes such as paths and trees, we aim at using the parameterized paradigm to identify its boundaries of tractability on general graphs. We develop novel algorithms and lower bounds which result in a full classification of the parameterized complexity of the problem with respect to natural structural parameterizations for the problem -- notably maximum capacity, treewidth, maximum degree, and maximum flow length. In particular, we obtain a fixed-parameter algorithm for the problem when parameterized by all four of these parameters, establish XP-tractability as well as W[1]-hardness with respect to the former three and latter three parameters, and all remaining cases remain paraNP-hard.

Figures (3)

• Figure 1: The complexity landscape of Unsplittable Flow under structural parameterizations. Here $\textnormal{tw}$, $\Delta$, $c$ and $\ell$ denote the treewidth, maximum degree, maximum capacity and maximum length of an admissible route, respectively. A discussion of the (non-)applicability of other major structural parameters is provided in Section \ref{['sec:bounded_l']}.
• Figure 2: A join node $t$ with two children $t_1$, $t_2$, where two paths between the red vertices---one in each of the partial solutions for $t_1$ and $t_2$---create a cycle. By combining a pair of tuples $(\Lambda_1,\Theta_1,\Omega_1)\in R(t_1)$ and $(\Lambda_2,\Theta_2,\Omega_2)\in R(t_2)$ which represent partial solutions containing such paths, the algorithm could produce a tuple which does not represent any partial solution at $t$.
• Figure 3: An illustration of the construction used in the proof of Theorem \ref{['thm:Wctwd']}. On the left, we depict an instance of Multi-Colored Clique with a solution highlighted in bold. On the right, we show the corresponding instance of Unsplittable Flow on the path (the horizontal edges), where the tasks corresponding to the highlighted clique are depicted via bold curves. For ease of presentation, we omit the tasks not chosen in this particular solution from the figure.

Theorems & Definitions (16)

• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
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• theorem thmcountertheorem
• proposition thmcounterproposition: Korhonen21
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof