1 / 13

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs. Aleksandrs Slivkins Cornell University ESA 2003 Budapest, Hungary. t. t. s. t. s. s. The Edge-Disjoint Paths Problem (EDP). Given: graph G , pairs of terminals s 1 t 1 ... s k t k

emily
Download Presentation

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related