1. Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs Aleksandrs Slivkins Cornell University ESA 2003 Budapest, Hungary

2. A.Slivkins. Edge-disjoint paths on DAGs ESA 20032 The Edge-Disjoint Paths Problem (EDP) Given: graph G, pairs of terminals s 1 t 1... s k t k Several terms can lie in one node Find: paths from s i to t i (for all i) that do not share edges s t s s t t

3. A.Slivkins. Edge-disjoint paths on DAGs ESA 20033 Background Parameter k = #terminal pairs Undirected NP-complete (Karp 75) k=2 polynomial (Shiloach 80) O(f(k) n 3 ), huge f(k) (Robertson & Seymour 95) Directed NP-complete for k=2 (Fortune, Hopcroft, Wyllie 80) Directed acyclic NP-complete O(kmn k ) (FHW 80) How about O(f(k) n c ) ??? We prove: IMPOSSIBLE! (modulo complexity-theoretic assumptions)

4. A.Slivkins. Edge-disjoint paths on DAGs ESA 20034 Background: Fixed- Parameter Tractability (FPT) Parameterized problem instance (x, k) FPT if alg O(f(k) |x| c ) k-Clique not believed FPT (Downey and Fellows 92) Parameterized reduction f,g recursive fns, c constant P not likely FPT call P W[1]-hard (G,k) k-clique (x, g(k)) P time O(f(k) |G| c )

5. A.Slivkins. Edge-disjoint paths on DAGs ESA 20035 Our results EDP on DAGs is W[1]-hard even if 2 source/ 2 sink nodes.. also for node-disjoint version Unsplittable Flow Problem EDP w/ capacities and demands sharper hardness results Algorithmic results efficient (FPT) algs for NP-complete special cases of EDP and Unsplittable Flows on DAGs.

6. A.Slivkins. Edge-disjoint paths on DAGs ESA 20036 EDP on DAGs is W[1]-hard Sketch of the pf (4 slides) reduce from k-clique problem instance (G,k) G undirected n-node graph does G contain a k-clique? array of identical gadgets k rows, n columns k copies of V(G) select & verify k-clique

7. A.Slivkins. Edge-disjoint paths on DAGs ESA 20037 Construction (2/4) Path s i t i (selector) goes through row i visits all gadgets but one, hence selects a vertex of G row has two levels L1, L2 selector starts at L1 to skip a gadget must go L1 L2 cannot go back to L1 sisi titi L1 L2 row i

8. A.Slivkins. Edge-disjoint paths on DAGs ESA 20038 Construction (3/4) Path s ij t ij (verifier) pair i<j of rows verifies edge v i v j in G enters at row i, exits at row j gadgets v i v j are connected iff edge v i v j is in G vivi vjvj s ij t ij row i row j

9. A.Slivkins. Edge-disjoint paths on DAGs ESA 20039 Construction (4/4) a gadget k-1 wires for verifiers two levels for the selector jump edge from L1 to L2 selector blocks verifiers see paper for complete proof... even if 2 distinct source nodes and 2 distinct sink nodes L1 L2

10. A.Slivkins. Edge-disjoint paths on DAGs ESA 200310 Algorithmic results demand graph H same vertex set pair s i t i add edge t i s i s i t i path in G cycle in G+H EDP = cycle packing in G+H standard restriction: G+H Eulerian G acyclic, G+H Eulerian NP-complete (Vygen 95) Our alg: O(k!n+m) extends to nearly Eulerian capacities and demands

11. A.Slivkins. Edge-disjoint paths on DAGs ESA 200311 Alg: G DAG, G+H Eulerian Fix sources, permute sinks find all perms s.t. EDP has sol'n Outline of the alg pick v s.t. deg in (v)=0 v: #sources = #nbrs sol'n on G remains valid if: move sources from v to nbrs delete v recurse on G-v (use dynam progr) s1s1 s2s2 s3s3 v s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 t4t4 s4s4

12. A.Slivkins. Edge-disjoint paths on DAGs ESA 200312 Unsplittable Flow Problem UFP: EDP w/caps and demands (x,y)-UFP x source nodes, y sink nodes (1,1)-UFP on DAGs is W[1]-hard If all caps 1, all demands ½ standard restriction for approx algs undirected UFP is fixed-parameter tractable (Kleinberg 98) our results for DAGs: (1,1)-UFP fixed-param tractable (1,3)- and (2,2)-UFP W[1]-hard (1,2)-UFP ???

13. A.Slivkins. Edge-disjoint paths on DAGs ESA 200313 Open problems Fixed-param tractable? W[1]-hard? EDP, G acyclic and planar NP-complete but poly-time if G+H is planar (Frank 81, Vygen 95) no node-disjoint version Directed planar EDP NP-complete even if G+H is planar (Vygen 95) node-disjoint: n O(k) (Schrijver 94) very complicated alg no edge-disjoint version Thanks!