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 GB = (L ∪ R, E ) is a bipartite graph and GI = (R, O) is an interval graph. Return a maximum matching M of GB such that the set of vertices in R matched in M are nonoverlapping in GI . 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 GB = (L ∪ R, E ) is a bipartite graph and GI = (R, O) is an interval graph. Return a maximum matching M of GB such that the set of vertices in R matched in M are nonoverlapping in GI . A set of vertices L (lefthand side) A set of intervals R (righthand 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 GB = (L ∪ R, E ) is a bipartite graph and GI = (R, O) is an interval graph. Return a maximum matching M of GB such that the set of vertices in R matched in M are nonoverlapping in GI . A set of vertices L (lefthand side) A set of intervals R (righthand side) A set of undirected edges E ⊆ L × R GB = (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 GB = (L ∪ R, E ) is a bipartite graph and GI = (R, O) is an interval graph. Return a maximum matching M of GB such that the set of vertices in R matched in M are nonoverlapping in GI . A set of vertices L (lefthand side) A set of intervals R (righthand side) A set of undirected edges E ⊆ L × R GB = (L ∪ R, E ) is a bipartite graph A set of undirected edges O ⊆ R × R GI = (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 GB = (L ∪ R, E ) is a bipartite graph and GI = (R, O) is an interval graph. Return a maximum matching M of GB such that the set of vertices in R matched in M are nonoverlapping in GI . A set of vertices L (lefthand side) A set of intervals R (righthand side) A set of undirected edges E ⊆ L × R GB = (L ∪ R, E ) is a bipartite graph A set of undirected edges O ⊆ R × R GI = (R, O) is an interval graph Maximum Matching M ⊆ E in GB Find M such that ¬∃(`,r ),(`0 ,r 0 )∈M [(r , r 0 ) ∈ O ∧ r 6= r 0 ] holds J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 2 / 15 L E R r1 l1 r2 r3 l2 r4 r5 l3 r6 r7 l4 r8 r9 l5 r10 NPHardness of MIIM Consider 3cnfsat formula S: S = c1 ∧ c2 ∧ · · · ∧ cn ci = `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 NPHardness of MIIM Consider 3cnfsat formula S: S = c1 ∧ c2 ∧ · · · ∧ cn ci = `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 S B L C1 C2 Cn b1 b2 bm U F R l1,1 l1,2 l1,3 l2,1 l2,2 l2,3 ln,1 ln,2 ln,3 b1,t b1,f b2,t b2,f bm,t bm,f J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 NPHardness of MIIM l1 = b l2 = b lx = b l'1 = ¬b l'2 = ¬b l'y = ¬b ⊆U f(b) = false f(b) = true ⊆F S B L C1 C2 Cn b1 b2 bm U F R l1,1 l1,2 l1,3 l2,1 l2,2 l2,3 ln,1 ln,2 ln,3 b1,t b1,f b2,t b2,f bm,t bm,f J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 3 / 15 NPHardness of MIIM l1 = b l2 = b lx = b l'1 = ¬b l'2 = ¬b l'y = ¬b ⊆U f(b) = false f(b) = true ⊆F S B L C1 C2 Cn b1 b2 bm U F R l1,1 l1,2 l1,3 l2,1 l2,2 l2,3 ln,1 ln,2 ln,3 b1,t b1,f b2,t b2,f bm,t bm,f Maximum Independent Interval Matching is NPHard 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 Fixedparameter Tractable Problems Definition: Fixedparameter Tractable problem A parameterized problem L ⊆ Σ∗ × N is called fixedparameter 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 Fixedparameter Tractable Problems Definition: Fixedparameter Tractable problem A parameterized problem L ⊆ Σ∗ × N is called fixedparameter 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(xc ) 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 L E R r1 l1 r2 r3 l2 r4 r5 l3 r6 r7 l4 r8 r9 l5 r10 L E R r1 l1 r2 r3 l2 r4 r5 l3 r6 r7 l4 r8 r9 l5 r10 MIIM  Reduction Rules Reduction Rule 1: Remove Disconnected Vertices Let vertex v ∈ L ∪ R be independent in GB (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 L E R r1 l1 r2 r3 l2 r4 r5 l3 r6 r7 l4 r8 r9 L E R r1 l1 r2 r3 l2 r4 r5 l3 r6 r7 l4 r8 r9 MIIM  Reduction Rules Let m be the size of a maximum matching in GB Reduction Rule 2: Too Much Overlap Let interval r ∈ R overlap more than R − m intervals (i.g. N (r , GI ) > R − m). Remove r from G . A miim that matches r , can’t match any interval r 0 ∈ N (r , GI ) Matching r denies more than R − m intervals to be matched For any matching of size m it holds m ≤ R Contradiction, r can’t be matched J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 8 / 15 L E R r1 l1 r2 r3 l2 r5 l3 r6 r7 l4 r8 r9 L E R r1 l1 r2 r3 l2 r5 l3 r6 r7 l4 r8 r9 L E R r1 l1 r2 r3 l2 l3 r' l4 r9 MIIM  Reduction Rules Reduction Rule 3: Disjoint Clique Merge Let C ⊆ R be a clique in GI that has no overlapping intervals outside the clique. Then let N (C ) ⊆ L be the set of vertices from neighbouring at least 1 node r ∈ C in GB . Remove C from G , add a new interval r 0 to G adjacent to each vertex N (C ). J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 9 / 15 MIIM  Reduction Rules Reduction Rule 3: Disjoint Clique Merge Let C ⊆ R be a clique in GI that has no overlapping intervals outside the clique. Then let N (C ) ⊆ L be the set of vertices from neighbouring at least 1 node r ∈ C in GB . Remove C from G , add a new interval r 0 to G adjacent to each vertex N (C ). Any miim M in G can be transformed to a miim M 0 in G 0 Any miim M 0 in G 0 can be transformed to a miim M in G J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 9 / 15 MIIM  Reduction Rules Reduction Rule 3: Disjoint Clique Merge Let C ⊆ R be a clique in GI that has no overlapping intervals outside the clique. Then let N (C ) ⊆ L be the set of vertices from neighbouring at least 1 node r ∈ C in GB . Remove C from G , add a new interval r 0 to G adjacent to each vertex N (C ). Any miim M in G can be transformed to a miim M 0 in G 0 Any miim M 0 in G 0 can be transformed to a miim M in G A disjoint clique is a scc C that is a clique Finding all scc takes linear time1 A scc is a clique iff ∀r ∈C [degree(r ) = C  − 1] Hence, found/verified/reduced in linear time 1 Tarjan’s algorithm or Kosaraju’s algorithm J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 9 / 15 L E R r1 l1 r2 r3 l2 l3 r' l4 r9 L E R r1 l1 r2 r3 l2 l3 r' l4 r9 MIIM  Reduction Rules Reduction Rule 4: Remove Non Maximum Matchable Edges Let e ∈ E be an edge not included in any maximum matching in GB . Remove e from G . ∀M⊆E [M is a maximum matching =⇒ e ∈ / M] J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 10 / 15 MIIM  Reduction Rules Reduction Rule 4: Remove Non Maximum Matchable Edges Let e ∈ E be an edge not included in any maximum matching in GB . Remove e from G . ∀M⊆E [M is a maximum matching =⇒ e ∈ / M] Every miim M is a maximum matching in GB Every edge not maximum matchable in GB can not be in M J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 10 / 15 MIIM  Reduction Rules Reduction Rule 4: Remove Non Maximum Matchable Edges Let e ∈ E be an edge not included in any maximum matching in GB . Remove e from G . ∀M⊆E [M is a maximum matching =⇒ e ∈ / M] Every miim M is a maximum matching in GB Every edge not maximum matchable in GB can not be in M Finding all maximum matchable edges takes O(GB )2 2 T.Tassa. Finding all maximallymatchable edges in a bipartite graph. Theoretical Computer Science, 423:5058, 2012 J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 10 / 15 L E R r1 l1 r2 r3 l2 l3 r' l4 r9 L E R r1 l1 r2 r3 l2 l3 r' l4 r9 MIIM  Reduction Rules Reduction Rule 5: Remove Overlapping Essential Let S ∈ R be the set of intervals included in every maximum matching in GB . Let N (S) be the set of intervals overlapping at least 1 interval r ∈ S in GI . If S ∩ N (S) 6= ∅, G is a Noinstance. Else remove N (S) from G . ∀M⊆E ∧r ∈S [M is a maximum matching =⇒ (`, r ) ∈ M] J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 11 / 15 MIIM  Reduction Rules Reduction Rule 5: Remove Overlapping Essential Let S ∈ R be the set of intervals included in every maximum matching in GB . Let N (S) be the set of intervals overlapping at least 1 interval r ∈ S in GI . If S ∩ N (S) 6= ∅, G is a Noinstance. Else remove N (S) from G . ∀M⊆E ∧r ∈S [M is a maximum matching =⇒ (`, r ) ∈ M] Removing a node r ∈ S from G reduces maximum matching size by 1 A node r 0 ∈ N (r , GI ) matched in miim M implies r is not matched No node r 0 ∈ N (S) can be matched in a miim J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 11 / 15 MIIM  Reduction Rules Reduction Rule 5: Remove Overlapping Essential Let S ∈ R be the set of intervals included in every maximum matching in GB . Let N (S) be the set of intervals overlapping at least 1 interval r ∈ S in GI . If S ∩ N (S) 6= ∅, G is a Noinstance. Else remove N (S) from G . ∀M⊆E ∧r ∈S [M is a maximum matching =⇒ (`, r ) ∈ M] Removing a node r ∈ S from G reduces maximum matching size by 1 A node r 0 ∈ N (r , GI ) matched in miim M implies r is not matched No node r 0 ∈ N (S) can be matched in a miim Finding set S takes O(GB ) J.L.G. Schols & S.C.I. Marin (TU/e) Maximum Independent Interval Matching May 15, 2019 11 / 15 L E R l1 r3 l2 l3 r' l4 r9
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