Complexity and Exact Algorithms for Mulitcut

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionComplexity and Exact Algorithms for MulticutJ. Guo 1 , F. Hüffner 1 , E. Kenar 2 , R. Niedermeier 1 , J. Uhlmann 21 Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany2 Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, GermanySOFSEM 2006

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionMulticutDefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × STask: Find a minimum set of edges or vertices whose removaldisconnects each pair of H.edge deletionMultiterminal Cut (MTC)Edge Multicut (EMC)vertex deletionUnrestricted Vertex Multicut (UVMC)Restricted Vertex Multicut (RVMC)

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionMulticutDefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × STask: Find a minimum set of edges or vertices whose removaldisconnects each pair of H.edge deletionMultiterminal Cut (MTC)Edge Multicut (EMC)vertex deletionUnrestricted Vertex Multicut (UVMC)Restricted Vertex Multicut (RVMC)

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRestricted Vertex Multicut (RVMC)Figure: RVMC instance G = (V , E)with H = {(1, 3), (3, 4)}, k = 2DefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × SInteger k ≥ 0Task: Find a subset V ′ of V with |V ′ | ≤ kthat contains no terminal and whoseremoval disconnects each pair of H.1 2 34 5 6Note: No solution forH = {(1, 3), (1, 6), (3, 4)} !

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRestricted Vertex Multicut (RVMC)Figure: RVMC instance G = (V , E)with H = {(1, 3), (3, 4)}, k = 2DefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × SInteger k ≥ 0Task: Find a subset V ′ of V with |V ′ | ≤ kthat contains no terminal and whoseremoval disconnects each pair of H.1 2 34 5 6

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRestricted Vertex Multicut (RVMC)Figure: RVMC instance G = (V , E)with H = {(1, 3), (3, 4)}, k = 2DefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × SInteger k ≥ 0Task: Find a subset V ′ of V with |V ′ | ≤ kthat contains no terminal and whoseremoval disconnects each pair of H.1 2 34 5 6

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRestricted Vertex Multicut (RVMC)Figure: RVMC instance G = (V , E)with H = {(1, 3), (3, 4)}, k = 2DefinitionInput:Undirected graph G = (V , E)Set of terminals S ⊆ VSet of pairs of terminals H ⊆ S × SInteger k ≥ 0Task: Find a subset V ′ of V with |V ′ | ≤ kthat contains no terminal and whoseremoval disconnects each pair of H.1 2 34 5 6Note: No solution forH = {(1, 3), (1, 6), (3, 4)} !

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFixed-parameter tractabilityIdeaRestrict the seemingly inherent combinatorial explosion of hardproblems to some problem-specific parameters.nknkDefinition (fixed-parameter tractable)Problem P is fixed-parameter tractable⇐⇒P is solvable in O(f (k) · n c ) time

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFixed-parameter tractabilityIdeaRestrict the seemingly inherent combinatorial explosion of hardproblems to some problem-specific parameters.nknkDefinition (fixed-parameter tractable)Problem P is fixed-parameter tractable⇐⇒P is solvable in O(f (k) · n c ) time

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC in trees

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC hardness results for treesPaths: RVMC in trees with maximum vertex degree two issolvable in polynomial time: O(|V | · |H|).Trees: NP-completeness has been shown for RVMC in treeswith maximum vertex degree four. [Călinescu et al., Journalof Algorithms, 2003]In comparison: Unrestricted Vertex Multicut in treescan easily be solved in O(|V | · |H|) time (least commonancestor).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC hardness results for treesPaths: RVMC in trees with maximum vertex degree two issolvable in polynomial time: O(|V | · |H|).TheoremRestricted Vertex Multicut in trees with maximum vertexdegree three is NP-complete.Trees: NP-completeness has been shown for RVMC in treeswith maximum vertex degree four. [Călinescu et al., Journalof Algorithms, 2003]In comparison: Unrestricted Vertex Multicut in treescan easily be solved in O(|V | · |H|) time (least commonancestor).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC hardness results for treesPaths: RVMC in trees with maximum vertex degree two issolvable in polynomial time: O(|V | · |H|).TheoremRestricted Vertex Multicut in trees with maximum vertexdegree three is NP-complete.Trees: NP-completeness has been shown for RVMC in treeswith maximum vertex degree four. [Călinescu et al., Journalof Algorithms, 2003]In comparison: Unrestricted Vertex Multicut in treescan easily be solved in O(|V | · |H|) time (least commonancestor).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithmTheoremRVMC in trees can be solved in O(2 k · |V | · |H|) time, where k isthe number of allowed vertex removals.FPT-algorithm is based on a depth-bounded search tree

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithmTheoremRVMC in trees can be solved in O(2 k · |V | · |H|) time, where k isthe number of allowed vertex removals.FPT-algorithm is based on a depth-bounded search treePreprocessing of the instance T = (V , E) with S and H:112 6 72, 3,46 7, 93 4 5 8 958Figure: H = {(2, 9), (3, 6), (4, 7)}. Edges with both endpoints beingterminals are contracted.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — search tree1) Compute the least common ancestor (lca) for each terminal pairand sort them by decreasing depth in a list L.2ndlca( )lca( )3rdlca( )5thlca( )4thlca( )1st

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — search tree2) While L ≠ ∅, consider the first element u of L (least commonancestor of pair (v, w)):ulca(v, w)vwCase 1:u /∈ S (nonterminal)→ remove u

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — search tree2) While L ≠ ∅, consider the first element u of L (least commonancestor of pair (v, w)):ulca(v, w)uu’lca(v, w)u = vvwwCase 1:u /∈ S (nonterminal)→ remove uCase 2a:u ∈ S, u = v or u = w→ remove u ′ ∈ P

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — search tree2) While L ≠ ∅, consider the first element u of L (least commonancestor of pair (v, w)):ulca(v, w)uu’lca(v, w)u = vuv’ w’lca(v, w)uv’ w’vwwvwvwCase 1:u /∈ S (nonterminal)→ remove uCase 2a:u ∈ S, u = v or u = w→ remove u ′ ∈ PCase 2b: (branching)u ∈ S, u ≠ v and u ≠ w→ remove u ′ or v ′

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — search tree2) While L ≠ ∅, consider the first element u of L (least commonancestor of pair (v, w)):ulca(v, w)uu’lca(v, w)u = vuv’ w’lca(v, w)uv’ w’vwwvwvwCase 1:u /∈ S (nonterminal)→ remove uCase 2a:u ∈ S, u = v or u = w→ remove u ′ ∈ PCase 2b: (branching)u ∈ S, u ≠ v and u ≠ w→ remove u ′ or v ′Termination: If L = ∅ or k nodes have been removed.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs Conclusion¤¥¤¥¦§¦§¨©¨©FPT-algorithm — run timesearch tree¢£¢£Depth: parameter kSize: O(2 k )Update step: O(|V | · |H|) time→ Run time: O(2 k · |V | · |H|)¡¡depth kpossible solutions

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC in interval graphs

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionInterval graphA graph is an interval graph if we can label its vertices byintervals of the real line such that there is an edge betweentwo vertices if and only if their intervals intersect.Figure: Example for an interval graph and its correspondingintervals on the real line.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionComplexity resultsTheoremRestricted Vertex Multicut in interval graphs can besolved in polynomial time.→ Dynamic programming algorithm with run time O(|V | 2 · |H| 2 )UVMC RVMCTrees P NP-cInterval graphs NP-c P

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionRVMC in general graphs

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionHardness results for general graphsRestricted Vertex Multicut is NP-complete if thereare at least six terminals. [Dahlhaus et al., SIAM Journal onComputing, 1994]There is no FPT-algorithm with respect to the parameter“number of terminals”.RVMC is NP-complete for trees with bounded vertex degreeand bounded pathwidth → no FPT-algorithm with path- ortreewidth as parameter.IdeaA combination of both parameters “number of terminals” andtreewidth leads to an FPT-algorithm.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionHardness results for general graphsRestricted Vertex Multicut is NP-complete if thereare at least six terminals. [Dahlhaus et al., SIAM Journal onComputing, 1994]There is no FPT-algorithm with respect to the parameter“number of terminals”.RVMC is NP-complete for trees with bounded vertex degreeand bounded pathwidth → no FPT-algorithm with path- ortreewidth as parameter.IdeaA combination of both parameters “number of terminals” andtreewidth leads to an FPT-algorithm.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — basic ideaKey observation: Any solution of RVMC divides the input graphinto at least two connected components → the two terminals of aterminal pair are in distinct connected components!

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — basic ideaKey observation: Any solution of RVMC divides the input graphinto at least two connected components → the two terminals of aterminal pair are in distinct connected components!

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage one: Enumerate all possible colorings of the terminal set Swith colors C such that for each terminal pair the two terminalsare differently colored (pre-coloring of S).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage one: Enumerate all possible colorings of the terminal set Swith colors C such that for each terminal pair the two terminalsare differently colored (pre-coloring of S).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage one: Enumerate all possible colorings of the terminal set Swith colors C such that for each terminal pair the two terminalsare differently colored (pre-coloring of S).illegal coloring!

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage two: Extend the legal pre-colorings of stage one with theprevious colors of C plus an additional color (e.g. yellow).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage two: Extend the legal pre-colorings of stage one with theprevious colors of C plus an additional color (e.g. yellow).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — the two stagesStage two: Extend the legal pre-colorings of stage one with theprevious colors of C plus an additional color (e.g. yellow).

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionFPT-algorithm — run timeThe color extension can be computed by dynamicprogramming on the tree decomposition of the input graphwith treewidth ω.TheoremGiven an undirected graph G = (V , E) with a tree decompositionof width ω, Restricted Vertex Multicut can be solved inO(|S| |S|+ω+1 · (|V | + |E|)) time, where S is the terminal set.Note: This FPT-algorithm also works for UnrestrictedVertex Multicut in general graphs.

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionConclusion

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionSummaryGraph class Parameter EMC UVMC RVMCInterval graphs NP-c NP-c PTrees NP-c P NP-ck FPT P FPTGeneral graphs NP-c NP-c NP-ck open open open|S| NP-c FPT NP-cω NP-c NP-c NP-c|S| and ω FPT FPT FPTTable: Complexity of Multicut problems for several graph classes.|S|: number of terminals, k: number of deletions, ω: treewidth of theinput graph

Introduction RVMC in Trees RVMC in Interval Graphs RVMC in General Graphs ConclusionThank you for listening!