Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

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

2. t t s t s s The Edge-Disjoint Paths Problem (EDP) • Given: graph G, pairs of terminals s1t1 ... sktk • Several term’s can lie in one node • Find: paths from sitoti(for all i) that do not share edges ESA 2003

3. Background Parameter k = #terminal pairs • Undirected • NP-complete (Karp ’75) • k=2 polynomial (Shiloach ’80) • O(f(k) n3), huge f(k) (Robertson & Seymour ’95) • Directed • NP-complete for k=2(Fortune, Hopcroft, Wyllie ’80) • Directed acyclic • NP-complete • O(kmnk) (FHW ’80) • How about O(f(k) nc) ??? We prove: IMPOSSIBLE! (modulo complexity-theoretic assumptions) ESA 2003

4. (G,k) k-clique (x, g(k)) P time O(f(k) |G| c) 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 ESA 2003

5. 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. ESA 2003

6. 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 ESA 2003

7. row i L1 ti si L2 Construction (2/4) • Path siti(“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 ESA 2003

8. Construction (3/4) • Path sijtij (“verifier”) •  pair i<j of rows • verifies edge vivjin G • enters at row i, exits at row j • gadgets vivj are connectediff edge vivjis in G sij vi row i row j tij vj ESA 2003

9. L1 L2 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 ESA 2003

10. Algorithmic results • demand graph H • same vertex set •  pair sitiadd edge tisi • siti 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 ESA 2003

11. t1 t4 s1 s1 v t2 s2 s2 s3 s3 s4 t3 Alg: G DAG, G+H Eulerian • Fix sources, permute sinks • find all perm’ss.t. EDP has sol'n • Outline of the alg • pick v s.t. degin(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) ESA 2003

12. 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 ??? ESA 2003

13. 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: nO(k) (Schrijver ’94) • very complicated alg • no edge-disjoint version Thanks! ESA 2003