Data Reduction for Maximum weight Matchings in Real World Graphs Pitta Venkat Indian Institute of Technology Madras cs16b017@smail.iitm.ac.in November 1, 2019 Pitta Venkat (IITM) Data Reduction November 1, 2019 1 / 12
Definitions and Notations Feedback Edge Set A set of Edges X in a Graph G is called feedback edge set if G − X is a forest. Feedback Edge Number Minimum Cardinality of all Feedback Edge Sets of Graph G is called Feedback Edge Number of G. Notations w ( G ):- Denotes weight of maximum weight Matching in Graph G . w ( M ):- Denotes weight of Matching M . Maximum weight Matching Problem :- Input:- weighted undirected Graph G and S . Output:- Yes iff ∃ Matching M in G s.t w ( M ) ≥ S . Pitta Venkat (IITM) Data Reduction November 1, 2019 2 / 12
Definitions and Notations Parameterized problem A Set of Instances ( I , k ) where I ∈ Σ ∗ and K ∈ N is a parameter. Equivalance of Instances Instance ( I , K ) and Instance ( I 1 , K 1 ) of problem P are equivalent if ( I , K ) is an YES instance of P iff ( I 1 , K 1 ) is an YES instance of P . Kernelization and Data Reduction Rules Kernelization is an Algorithm that given an Instance ( I , K ) of parameterized problem P , computes in polynomial time equivalent Instance ( I 1 , K 1 ) s.t | I 1 | + K 1 ≤ f ( K ) for some computable function f . If f ( K ) ∈ K O (1) then we say P admits a polynomial Kernel. Kernelization is generally acheived by applying poly-time executable Data reduction Rules. A Rule R is correct if on applying R on ( I , K ) results ( I 1 , K 1 ) equivalent to ( I , K ) . Pitta Venkat (IITM) Data Reduction November 1, 2019 3 / 12
Kernelization for Maximum cardinality Matching in unweighted graphs Reduction Rule 1 Let v ∈ V . If deg ( v ) = 0 then remove v from G . If deg ( v ) = 1 then remove v and it’s neighbour and decrease S by 1. Reduction Rule 2 Let v ∈ V . If deg ( v ) = 2 and u , w are neighbour’s of v then remove v and merge u , w and decrease S by 1. Theorm1 Maximum cardinality Matching admits linear-time computable linear-size Kernel with respective to parameter as feedback edge number. Result from Theorm1 Total time taken to compute Maximum cardinality Matching is O ( m + n + K 1 . 5 ). Pitta Venkat (IITM) Data Reduction November 1, 2019 4 / 12
Kernelization for Maximum weight Matching Reduction Rule 1 Let v ∈ V . If deg ( v ) = 0 then remove v . Or Let e ∈ E If w ( e ) = 0 then remove e . Reduction Rule 2 Let v ∈ V . If deg ( v ) = 1 and u is neighbour of v then ∀ e ∈ E s.t u incident on e set weight of edge as max { 0 , w ( e ) − w ( uv ) } , remove v and decrease S by w ( uv ) Lemma1 Rule 1 and Rule 2 are correct. Lemma2 Rule 1 and Rule 2 can be applied exhaustively in linear time. Pitta Venkat (IITM) Data Reduction November 1, 2019 5 / 12
Proof of Lemma1 Correctness of Rule 1 :- is simple to see because by applying Rule 1 Matchings doesnot effected. Correctness of Rule 2 :- Let us consider v be with deg ( v ) = 1 and uv ∈ E , G 1 is the graph obtained by applying Rule 2. If ∃ Matching M s.t w ( M ) ≥ S . Let X be set of all edges incident on u , v and M 1 = M − X . M 1 exists in G 1 with same weight. if uv ∈ M then w ( M 1 ) ≥ S − w ( uv ). if e ∈ X − uv ∈ M then w G 1 ( M 1 + e ) ≥ S − w ( e ) + Max ≥ S − w ( uv ). Hence ∃ Matching M 11 s.t w G 1 ( M 11 ) ≥ S − w ( uv ). We can also proove other way also by following similar procedure. Hence Rule 2 is correct. Pitta Venkat (IITM) Data Reduction November 1, 2019 6 / 12
Proof of Lemma2 Rule 1 can be exauhstively applied in linear time by collecting all vertices of 0 degree and all edges of 0 weight in one reading. We will give an Algorithm to show Rule 2 can be exauhstively applied in linear time as follows. There are counters for each vertex initialised to 0. weight of edge xy at any iteration is w ( xy ) − c ( x ) − c ( y ). Each time when the Rule 2 is applied Let deg ( x ) = 1 and y is neighbour of x we will decrease S by max { 0 , w ( xy ) − c ( x ) − c ( y ) } and c ( y ) = c ( y ) + max { 0 , w ( xy ) − c ( x ) − c ( y ) } . So we will first collect all vertices of degree 1 in O ( n + m ) time. We will go through each vertex we collected and apply Rule 2. After that change all weights of edges as w ( xy ) = w ( xy ) + c ( x ) + c ( y ). correctness of Algorithm can be easily varified. Pitta Venkat (IITM) Data Reduction November 1, 2019 7 / 12
Definitions and Reduction Rules of degree 2 vertices Maximal path Let G be a Graph. A path P = v 0 v 1 .. v l is a Maximal path in G if l ≥ 3. and deg ( V 1 ) = deg ( v 2 ) = .. = deg ( v l − 1 ) = 2, deg ( v 0 ) � = 2, deg ( v l ) � = 2. Pending cycle Let G be the Graph. A Cycle C = c 0 c 1 .. c l is a pending cycle if atmost one vertex in cycle has degree greater than 2. Reduction Rule 3 Let G be a graph and C be a pending cycle in G where u ∈ C with deg ( u ) ≥ 3.Then replace the Cycle by an edge uz with w ( uz ) = w ( C ) − w ( C − u ) and decrease S by w ( C − u ). Pitta Venkat (IITM) Data Reduction November 1, 2019 8 / 12
Reduction Rules and Lemma’s Reduction Rule 4 Let G be a graph. P be a maximal path with end points u , v then remove all vertices in P execpt u , v and add vertex z . s.t w ( uz ) = w ( P − v ) − w ( P − u − v ), w ( vz ) = w ( P − u ) − w ( p − u − v ) and w ( uv ) = max { w ( uv ) , w ( P ) − w ( P − u − v ) } . and decrease S by w ( P − u − v ) Lemma 3 Maximal paths in a Graph is atmost feedback edge number. Lemma 4 Reduction Rules 3 doesnot increase feedback edge number and Reduction Rule 4 can increase atmost double. Lemma 5 Rules 3,4 are correct and can be applied exhaustively in linear time. Pitta Venkat (IITM) Data Reduction November 1, 2019 9 / 12
Kernel Size for Maximum weight Matching Kernelization for Maximum weight Matching is exhaustively applying Rule 1 , Rule 3 , Rule 4 and Rule 2 in Order. Let G 1 be the final graph obtained K 1 ≤ 2 ∗ K because feedback edge number doesnot increase in any rule. Let X is minimum size feedback edge set in G 1 . Consider the graph G 1 − X . Devide the vertices into V 1 , V 2 , V 3 represents vertices of degree 1 , degree 2 and degree ≥ 3 respectively in G 1 − X . V 3 < V 1 (2 V 1 + 2 V 2 + 2 V 3 − 2 ≤ V 1 + 2 V 2 + 3 V 3 ). in G 1 there are no degree 1 vertices so in G 1 − X degree 1 vertices are obtained by removing X so V 1 < 4 ∗ K . degree 2 vertices in G are incident to X or V 3 + 4 ∗ K so V 2 < 12 ∗ K . Total vertices < 4 ∗ K + 4 ∗ K + 12 ∗ K = 20 ∗ K and Total Edges < 20 ∗ K + 2 ∗ K in G 1 . So the Kernel as atmost 20 ∗ K vertices and atmost 22 ∗ K edges. Pitta Venkat (IITM) Data Reduction November 1, 2019 10 / 12
Correctness of Rules 3 and 4 First we consider the Instances before and After applying Reduction Rules From Matchings in G we will remove edges which are different from both instances and add some other edges We will Obtain relation ship like w ( G 1 ) ≥ w ( G ) − w ( C − u ) and w ( G 1 ) ≥ w ( G ) − w ( P − u − v ). From Matchings in G 1 we will remove edges which are different from both instances and add some other edges We will Obtain relation ship like w ( G ) ≥ w ( G 1 ) + w ( C − u ) and w ( G ) ≥ w ( G 1 ) + w ( P − u − v ). Hence we can conclude equivalance of 2 Instances. Pitta Venkat (IITM) Data Reduction November 1, 2019 11 / 12
Conclusion Comparing Kolmogorov’s Algorithm with and without data Reductions the average speedup is 3800% in case of unweighted graphs. Comparing Kolmogorov’s Algorithm with and without data Reductions the average speedup is 30% in case of weighted graphs. because weighted graphs can reduce only 50% size on average where unweighted graphs can reduce 80% size. Hence weighted graph Kernelization is not as good as unweighted Graph Kernelization Pitta Venkat (IITM) Data Reduction November 1, 2019 12 / 12
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