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Routing Problems: Edge Disjoint and Fixed-Parameter Tractability

Explore the challenges and solutions in routing problems like Edge Disjoint Problem (EDP), Second Node Disjoint Problem (SNDP), and Fixed-Parameter Tractability (FPT) in directed graphs. Learn about complexities, normal forms, and the motivation behind solving these critical issues in network routing.

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Routing Problems: Edge Disjoint and Fixed-Parameter Tractability

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  1. On Fixed-Parameter Tractability of Some Routing ProblemsHong Lu

  2. Overview • Motivation • Problem Statements • First Edge Disjoint Problem(FEDP) • Special Case • Normal Form • General Case • Complexities

  3. Motivation • Routing Issue: Starvation a Node: computer c Edge: network link 1 2 Path: end to end connection 3 d G b

  4. Problem Statements • Given directed graph G and k source-sink pairs s1t1,s2t2,…sktk, find a routing of k paths p1,p2,…,pk such that • no two paths share the same edge(EDP) • no two paths share the same second node(SNDP) • no two paths share the same first edge(FEDP) • NP-Complete: SAT • Fixed Parameter Tractable (kernelization)

  5. FEDP: Special Case N T1 k1 N T1 k1 N T2 <k k2 small sink p1’ N p1 T2 k k2 S p2 N k3 T3 S N k3 T3 >k big sink p2’ Tm km N Tm km G G’ N N1 bipartite k new nodes Nk P = {p1,p2,…pk} P’ = {p1’,p2’,…pk’}

  6. Normal Form - FEDP 1 n  k m  k k1 k2 K1+k2+…+km = k 2 km T-nodes S- nodes m N-nodes

  7. Normalization I G’ G ST S T Source-Sink Separation

  8. T Normalization II G’ G T Sink Normalization

  9. Normalization III S1 S1 S2 S2 Source Normalization

  10. Normalization IV 1 n  k m  k k1 k2 2 km T-nodes S- nodes m N-nodes

  11. Complexities • FEDP can be solved in O(mk + k5(ek)k) • st-SNDP is FPT • Is general SNDP FPT? Open!

  12. Reference • Aleksandrs Slivkins, Martin Pál, “On Fixed-Parameter Tractability of Some Routing Problems”, Technical Report, PSU

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