1 On Fixed-Parameter Tractability of Some Routing Problems Hong Lu

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

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

4 Problem Statements Given directed graph G and k source-sink pairs s 1 t 1,s 2 t 2,…s k t k, find a routing of k paths p 1,p 2,…,p k 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 S N N TmTm T1T1 T2T2 N T3T3 N1N1 NkNk kk >k>k k1k1 k2k2 k3k3 kmkm P = {p 1,p 2,…p k } S N N TmTm T1T1 T2T2 N T3T3 <k k1k1 k2k2 k3k3 kmkm k GG’ new nodes N N bipartite big sink small sink p1p1 p2p2 P’ = {p 1 ’,p 2 ’,…p k ’} p1’p1’ p2’p2’

6 Normal Form - FEDP T-nodes N-nodes S- nodes m  k n  k 1 2 m kmkm k1k1 k2k2 K 1 +k 2 +…+k m = k

7 Normalization I STS T Source-Sink Separation G G’

8 Normalization II T T Sink Normalization G G’

9 Normalization III S2 S1 S2 S1 Source Normalization

10 Normalization IV T-nodes N-nodes S- nodes m  k n  k 1 2 m kmkm k1k1 k2k2

11 Complexities FEDP can be solved in O(mk + k 5 (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