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

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The Edge-Disjoint Paths Problem (EDP)

• Given graph G, pairs of terminals s1t1 ...
  sktk
• Several terms can lie in one node
• Find paths from si to ti (for all i) that do not
  share edges

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Background

• Parameter k terminal pairs
• Undirected
• NP-complete (Karp 75)
• k2 polynomial (Shiloach 80)
• O(f(k) n3), huge f(k) (Robertson Seymour 95)
• Directed
• NP-complete for k2(Fortune, Hopcroft, Wyllie
  80)
• Directed acyclic
• NP-complete
• O(kmnk) (FHW 80)
• How about O(f(k) nc) ???

We prove IMPOSSIBLE! (modulo complexity-theoreti
c assumptions)
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Background Fixed-Parameter Tractability (FPT)

• Parameterized problem
• instance (x, k)
• FPT if alg O(f(k) xc)
• k-Clique not believed FPT
• (Downey and Fellows 92)
• Parameterized reduction
• f,g recursive fns, c constant
• P not likely FPT
• call P W1-hard

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Our results

• EDP on DAGs is W1-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.

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EDP on DAGs is W1-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

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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

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Construction (3/4)

• Path sijtij (verifier)
• ? pair iltj 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

sij
vi
row i
row j
vj
tij
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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

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Algorithmic results

• demand graph H
• same vertex set
• ? pair siti add edge tisi
• siti path in G ? cycle in GH
• EDP cycle packing in GH
• standard restriction GH Eulerian
• G acyclic, GH Eulerian
• NP-complete (Vygen 95)
• Our alg O(k!nm)
• extends to
• nearly Eulerian
• capacities and demands

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Alg G DAG, GH Eulerian

• Fix sources, permute sinks
• find all perms 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)

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Unsplittable Flow Problem

• UFP EDP w/caps and demands
• (x,y)-UFP
• x source nodes, y sink nodes
• (1,1)-UFP on DAGs is W1-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 W1-hard
• (1,2)-UFP ???

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Open problems

• Fixed-param tractable? W1-hard?
• EDP, G acyclic and planar
• NP-complete but poly-time if GH is planar (Frank
  81, Vygen 95)
• no node-disjoint version
• Directed planar EDP
• NP-complete even if GH is planar (Vygen 95)
• node-disjoint nO(k) (Schrijver 94)
• very complicated alg
• no edge-disjoint version

Thanks!