Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Aleksandrs SlivkinsCornell University

ESA 2003Budapest, Hungary

ESA 2003 2A.Slivkins. Edge-disjoint paths on DAGs

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 si to ti (for all

i) that do not share edges

s t

s s

tt

ESA 2003 3A.Slivkins. Edge-disjoint paths on DAGs

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)

We prove: IMPOSSIBLE! (modulo complexity-theoretic assumptions)

ESA 2003 4A.Slivkins. Edge-disjoint paths on DAGs

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))Ptime O(f(k) |G|

c)

ESA 2003 5A.Slivkins. Edge-disjoint paths on DAGs

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 6A.Slivkins. Edge-disjoint paths on DAGs

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 7A.Slivkins. Edge-disjoint paths on DAGs

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

L1L2 cannot go back to L1

siti

L1L2

row i

ESA 2003 8A.Slivkins. Edge-disjoint paths on DAGs

Construction (3/4)

Path sijtij (“verifier”) pair i<j of rows verifies edge vivj in G enters at row i, exits at row j gadgets vivj are connected

iff edge vivj is in G

vi

vj

sij

tij

row i

row j

ESA 2003 9A.Slivkins. Edge-disjoint paths on DAGs

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

ESA 2003 10A.Slivkins. Edge-disjoint paths on DAGs

Algorithmic results

demand graph H same vertex set pair siti add 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 11A.Slivkins. Edge-disjoint paths on DAGs

Alg: G DAG, G+H Eulerian

Fix sources, permute sinks find all perm’s s.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)

s1s2s3

vs1

s2

s3

t1

t2

t3

t4

s4

ESA 2003 12A.Slivkins. Edge-disjoint paths on DAGs

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 13A.Slivkins. Edge-disjoint paths on DAGs

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!Thanks!