Maximum Independent Interval Matching a study on the effectiveness of various parameterized methods J.L.G. Schols & S.C.I. Marin Bachelor Research Project Eindhoven University of Technology Supervised by Dr. B.M.P. Jansen May 15, 2019 J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 1 / 15 Maximum Independent Interval Matching Maximum Independent Interval Matching problem Given a graph G = ( L ∪ R , E ∪ O ) where G B = ( L ∪ R , E ) is a bipartite graph and G I = ( R , O ) is an interval graph. Return a maximum matching M of G B such that the set of vertices in R matched in M are non-overlapping in G I J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 Maximum Independent Interval Matching Maximum Independent Interval Matching problem Given a graph G = ( L ∪ R , E ∪ O ) where G B = ( L ∪ R , E ) is a bipartite graph and G I = ( R , O ) is an interval graph. Return a maximum matching M of G B such that the set of vertices in R matched in M are non-overlapping in G I A set of vertices L (left-hand side) A set of intervals R (right-hand side) J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 Maximum Independent Interval Matching Maximum Independent Interval Matching problem Given a graph G = ( L ∪ R , E ∪ O ) where G B = ( L ∪ R , E ) is a bipartite graph and G I = ( R , O ) is an interval graph. Return a maximum matching M of G B such that the set of vertices in R matched in M are non-overlapping in G I A set of vertices L (left-hand side) A set of intervals R (right-hand side) A set of undirected edges E ⊆ L × R G B = ( L ∪ R , E ) is a bipartite graph J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 Maximum Independent Interval Matching Maximum Independent Interval Matching problem Given a graph G = ( L ∪ R , E ∪ O ) where G B = ( L ∪ R , E ) is a bipartite graph and G I = ( R , O ) is an interval graph. Return a maximum matching M of G B such that the set of vertices in R matched in M are non-overlapping in G I A set of vertices L (left-hand side) A set of intervals R (right-hand side) A set of undirected edges E ⊆ L × R G B = ( L ∪ R , E ) is a bipartite graph A set of undirected edges O ⊆ R × R G I = ( R , O ) is an interval graph J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 Maximum Independent Interval Matching Maximum Independent Interval Matching problem Given a graph G = ( L ∪ R , E ∪ O ) where G B = ( L ∪ R , E ) is a bipartite graph and G I = ( R , O ) is an interval graph. Return a maximum matching M of G B such that the set of vertices in R matched in M are non-overlapping in G I A set of vertices L (left-hand side) A set of intervals R (right-hand side) A set of undirected edges E ⊆ L × R G B = ( L ∪ R , E ) is a bipartite graph A set of undirected edges O ⊆ R × R G I = ( R , O ) is an interval graph Maximum Matching M ⊆ E in G B Find M such that ¬∃ ( `, r ) , ( ` ′ , r ′ ) ∈ M [( r , r ′ ) ∈ O ∧ r 6 = r ′ ] holds J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 l1 R L E l2 l3 l4 l5 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 NP-Hardness of MIIM Consider 3 -cnf-sat formula S : S = c 1 ∧ c 2 ∧ · · · ∧ c n c i = ` i , 1 ∨ ` i , 2 ∨ ` i , 3 ` i , j = b ∨ ` i , j = ¬ b Assign true or false to each boolean b ∈ B such that S = true J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 NP-Hardness of MIIM Consider 3 -cnf-sat formula S : S = c 1 ∧ c 2 ∧ · · · ∧ c n c i = ` i , 1 ∨ ` i , 2 ∨ ` i , 3 ` i , j = b ∨ ` i , j = ¬ b Assign true or false to each boolean b ∈ B such that S = true C1 l1,1 l1,2 l1,3 S L R C2 l2,1 l2,2 l2,3 U Cn ln,1 ln,2 ln,3 b1 b1,t b1,f F B b2 b2,t b2,f bm bm,t bm,f J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 NP-Hardness of MIIM l 1 = b f(b) = false l 2 = b l x = b l' 1 = ¬b f(b) = true l' 2 = ¬b l' y = ¬b ⊆ U ⊆ F C1 l1,1 l1,2 l1,3 S L R C2 l2,1 l2,2 l2,3 U Cn ln,1 ln,2 ln,3 b1 b1,t b1,f F B b2 b2,t b2,f bm bm,t bm,f J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 NP-Hardness of MIIM l 1 = b f(b) = false l 2 = b l x = b l' 1 = ¬b f(b) = true l' 2 = ¬b l' y = ¬b ⊆ U ⊆ F C1 l1,1 l1,2 l1,3 S L R C2 l2,1 l2,2 l2,3 U Cn ln,1 ln,2 ln,3 b1 b1,t b1,f F B b2 b2,t b2,f bm bm,t bm,f Maximum Independent Interval Matching is NP-Hard J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 Parameterized Problems Definition: Parameterized problem A parameterized problem is a language L ⊆ Σ ∗ × N Σ is a fixed, finite alphabet. For a problem instance ( x , k ) ∈ Σ ∗ × N , k is called the parameter. k is a relevant secondary measurement Task is, decide whether ( x , k ) ∈ L J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 4 / 15 Parameterized Problems Definition: Parameterized problem A parameterized problem is a language L ⊆ Σ ∗ × N Σ is a fixed, finite alphabet. For a problem instance ( x , k ) ∈ Σ ∗ × N , k is called the parameter. k is a relevant secondary measurement Task is, decide whether ( x , k ) ∈ L k can for instance be: some aspect of the input instance x some aspect of the output a description of the input structure J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 4 / 15 Fixed-parameter Tractable Problems Definition: Fixed-parameter Tractable problem A parameterized problem L ⊆ Σ ∗ × N is called fixed-parameter tractable (FPT) if these exists an algorithm A , a computable function f : N → N and a constant c such that, given ( x , k ) ∈ Σ ∗ × N , algorithm A correctly decides whether ( x , k ) ∈ L in time bounded by f ( k ) · | ( x , k ) | c Goal, make factor f ( k ) and constant c as small as possible. Not every choice of parameter k leads to a FPT algorithm J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 5 / 15 Fixed-parameter Tractable Problems Definition: Fixed-parameter Tractable problem A parameterized problem L ⊆ Σ ∗ × N is called fixed-parameter tractable (FPT) if these exists an algorithm A , a computable function f : N → N and a constant c such that, given ( x , k ) ∈ Σ ∗ × N , algorithm A correctly decides whether ( x , k ) ∈ L in time bounded by f ( k ) · | ( x , k ) | c Goal, make factor f ( k ) and constant c as small as possible. Not every choice of parameter k leads to a FPT algorithm When we consider k a constant, A runs in O ( | x | c ) time So it is tractable for a known k Feasibility depends on value of k J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 5 / 15 Methods for parameterized algorithms Kernelization Bounded Search Tree Iterative Compression Randomized Methods Subset Dynamic Programming Integer Linear Progranming Feasibility And more ... J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 6 / 15 l1 R L E l2 l3 l4 l5 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 l1 R L E l2 l3 l4 l5 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 MIIM - Reduction Rules Reduction Rule 1: Remove Disconnected Vertices Let vertex v ∈ L ∪ R be independent in G B (i.g. ¬∃ u ∈ L ∪ R [( u , v ) ∈ E ]). Remove v from G J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 7 / 15 l1 R L E l2 l3 l4 r1 r2 r3 r4 r5 r6 r7 r8 r9