Parameterized Complexity of Segment Routing

TL;DR

The paper analyzes the parameterized and structural complexity of Segment Routing (SR), showing that SR is NP-hard even on graphs with small treewidth and remains hard under various parameterizations (e.g., $d$ demands or a small budget $k$) with strong W[1]-hardness results. It develops gadget-based reductions from problems like 2-EDP, 2D1SP, Multicolored Clique, and 3-Partition, and introduces the $\ell$-extended graph $S_\ell(G)$ and triangle chains to establish these hardness results. On the algorithmic side, the authors identify polynomial-time solvable islands: MinUnit SR on undirected cycles and Unit SR on undirected cacti via dynamic programming on the cactus skeleton, while showing that SR remains NP-hard on cacti in general. They also prove NP-hardness under structural constraints such as small vertex cover, via reductions from 3-Edge Coloring and Bin Packing. Practically, these results delimit the feasibility of exact SR algorithms and motivate exploring approximations and topology-tailored methods for traffic engineering in real networks. All mathematical statements are expressed with rigorous notation, including parameters $n,m,d,k$, and complexity classes $P$, NP, W[1].

Abstract

Segment Routing is a recent network technology that helps optimizing network throughput by providing finer control over the routing paths. Instead of routing directly from a source to a target, packets are routed via intermediate waypoints. Between consecutive waypoints, the packets are routed according to traditional shortest path routing protocols. Bottlenecks in the network can be avoided by such rerouting, preventing overloading parts of the network. The associated NP-hard computational problem is Segment Routing: Given a network on $n$ vertices, $d$ traffic demands (vertex pairs), and a (small) number $k$, the task is to find for each demand pair at most $k$ waypoints such that with shortest path routing along these waypoints, all demands are fulfilled without exceeding the capacities of the network. We investigate if special structures of real-world communication networks could be exploited algorithmically. Our results comprise NP-hardness on graphs with constant treewidth even if only one waypoint per demand is allowed. We further exclude (under standard complexity assumptions) algorithms with running time $f(d) n^{g(k)}$ for any functions $f$ and $g$. We complement these lower bounds with polynomial-time solvable special cases.

Figures (13)

• Figure 1: On the left, the data are routed from $a$ to $g$ with traditional shortest path routing. The fraction on each arc indicates how the data split among the multiple shortest paths. On the right, a waypoint in $c$ is introduced, resulting in the data being routed through the bottom part of the network and freeing up capacities in the top part.
• Figure 2: A tricky instance for Segment Routing where edge weights, edge capacities and bandwidth requirements are one. One waypoint per demand is allowed. The demand $(s_3,t_3)$ blocks an edge and forces a waypoint for demand $(s_2,t_2)$ between $s_2$ and $v$. The route of $(s_2, t_2)$ then splits between $v$ and $t_2$. To ensure that capacities are not exceeded on the lower and upper paths from $v$ to $t_2$, the route of demand $(s_1, t_1)$ needs to be split too. The solution is to select a waypoint for $(s_1, t_1)$ on the middle path from $v$ to $t_2$. This means the middle path is used in both directions by the route of demand $(s_1, t_1)$.
• Figure 3: The gadget $\nabla_4$ in a network with unit edge weights and capacities, with two demands $(s_1,t_1)$ and $(s_2,t_2)$ traversing it. Square waypoints are for demand 1 and round waypoints are for demand 2.
• Figure 4: Illustration of the reduction from 2-EDP to Unit Segment Routing: on the left is the reduced instance and on the right its demand graph.
• Figure 5: Illustration of the reduction from 2D1SP to Unit Segment Routing. The reduced instance is at the top and below is its demand graph.

Theorems & Definitions (29)

• proof
• Lemma 2.2: folklore
• proof
• Definition 3.1
• Definition 3.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3