1. Bart Jansen, Utrecht University

2. 2  Max Leaf  Instance: Connected graph G, positive integer k  Question: Is there a spanning tree for G with at least k leaves?  Applications in network design  YES-instance for k ≤ 7

3.  Classical complexity  MAX-SNP complete, so no polynomial-time approximation scheme (PTAS)  NP-complete, even for 3 3-regular graphs By P. Lemke, 1988 Planar graphs of maximum degree 4 By Garey and Johnson, 1979

4.  Bipartite Max Leaf  Instance: Connected bipartite graph G with black and white vertices according to the partition, positive integer k  Question: Is there a spanning tree for G with at least k black leaves? 4

5.  Classical complexity  No constant-factor approximation  NP-complete, even for: 5 d-regular graphs for d ≥ 4 By Fusco and Monti, 2007 Planar graphs of maximum degree 4 By Li and Toulouse, 2006

6.  Weighted Max Leaf  Instance: Connected graph G with a non-negative integer weight for each vertex, positive integer k  Question: Is there a spanning tree for G such that its leaves have combined weight at least k? 6 Leaf weight 11Leaf weight 16

7.  Classical complexity  NP-complete by restriction of the previous problems  Hard on all classes of graphs mentioned so far  No constant-factor approximation since it generalizes Bipartite Max Leaf  We consider the fixed parameter complexity 7

8. 8  Technique to deal with problems (presumably) not in P  Asks if the exponential explosion of the running time can be restricted to a “parameter” that measures some characteristic of the instance  An instance of a parameterized problem is:  where k is the parameter of the problem (often integer)  Class of Fixed Parameter Tractable (FPT) problems:  Decision problems that can be solved in f(k) * poly(|I| + k) time  Function f can be arbitrary, so dependency on k may be exponential  For example, the k-Vertex Cover problem is fixed parameter tractable.  “Is there a vertex cover of size k?”  Can be solved in O(n + 2 k k 2 ) (and even faster).

9.  A kernelization algorithm:  Reduces parameterized instance to equivalent  Size of I’ does not depend on I but only on k  Time is poly (|I| + k)  New parameter k’ is at most k  If |I’| is O(g(k)), then g is the size of the kernel  Kernelization algorithm implies fixed parameter tractability  Compute a kernel, analyze it by brute force 9

10.  Existing problems, parameterized by nr. of leaves  Regular Max Leaf has a 3.5k kernel  No FPT results for Bipartite Max Leaf  For Weighted Max Leaf  We take the target weight k as the parameter of the problem  (In)tractable depending on whether weight 0 is allowed  Kernel size depends on class of graphs Complexity of weighted problem General graphsPlanar graphsGenus ≤ g graphs Weights {0,1,.. }W[1] hard84k kernelO(k √g) kernel Weights {1,2,.. }9.5k kernel 10

11. Bipartite Max Leaf is hard for W[1] 11

12.  Unless the Exponential Time Hypothesis is false, being W[1] hard implies:  No f(k)*p(n) algorithm  No polynomial-size kernel  W[2]-hard is assumed to be harder than W[1]-hard  For Weighted Max Leaf:  No proof of membership in W[1]  It might be harder than any problem in W[1]  No hardness proof for W[2] either Fixed parameter tractable Vertex Cover Feedback Vertex Set Maximum Leaf Spanning Tree.. W[1]-complete Independent Set Set Packing.. W[2]-complete Dominating Set.. 12

13.  W[i] hardness is proven by parameterized reduction  from some W[i]- hard problem  Like (Karp) reductions for NP-completeness  Extra condition: new parameter k’ ≤ f(k) for some f  We reduce k-Independent Set (W[1]- complete) to Bipartite Max Leaf 13

14.  k-Independent Set  Instance: Graph G, positive integer k  Question: Does G have an independent set of size at least k? ▪ (i.e. is there a vertex set S of size at least k, such that no vertices in S are connected by an edge in G?)  Parameter: the value k. 14

15.  Given an instance of k-Independent Set, we reduce as follows:  Color all vertices black  Split all edges by a white vertex  Add white vertex w with edges to all black vertices  Set k’ = k  Polynomial time  k’ ≤ f(k) = k 15

16.  If S is a cutset, then at least one vertex of S is internal in a spanning tree  We need to give at least one vertex in S a degree ≥ 2 to connect both sides 16

17. 17  Complement of S is a vertex cover  Build spanning tree:  Take w as root, connect to all blacks  We reach the white vertices from the vertex cover V – S ▪ Since every white vertex used to be an edge Edges incident on w are not drawn

18.  Take the black leaves as the independent set  If there was an edge x,y then they are not both leaves  Since {x,y} is a cutset  By contraposition, black leaves form an independent set 18 Edges incident on w are not drawn

19. A linear kernel for Maximum Leaf Weight Spanning Tree on planar graphs 19

20.  Kernel of size 84k on planar graphs  Strategy:  Give reduction rules ▪ that can be applied in polynomial time ▪ that reduce the instance to an equivalent instance  Prove that after exhaustive application of the rules, either: ▪ the size of the graph is bounded by 84k ▪ or we are sure that the answer is yes ▪ then we output a trivial, constant-sized YES-instance 20

21.  We want to be sure that the answer is YES if the graph is still big after applying reduction rules  Use a lemma of the following form:  If no reduction rules apply, there is a spanning tree with |G|/c leaves of weight ≥ 1 (for some c > 0)  With such a proof, we obtain:  If |G| ≥ ck then G has a spanning tree with |G|/c≥ck/c=k leaves of weight 1  So a spanning tree with leaf weight ≥ k  So if |G| ≥ ck after kernelization we return YES  Otherwise the instance must be small 21

22.  The reduction rules must avoid the following situation:  We can build an arbitrarily large graph with only constant leaf weight in an optimal spanning tree  All reduction rules are needed to prevent such situations  Reduction rules are motivated by examples of the situations they prevent 22

23.  A set S of vertices is a cutset if their removal splits the graph into multiple connected components  A path component of length k is a path, s.t.  x, y have degree ≠ 2  all v i have degree 2 23

24.  Vertex of positive weight, with arbitrarily many degree-1 neighbors of weight 0 24

25.  Structure:  Vertex x of degree 1 adjacent to y of degree > 1  Operation:  Delete x, decrease k by w(x), set w(y) = 0  Justification:  Vertex x will be a leaf in any spanning tree  The set {y} is a cutset, so y will never be a leaf in a spanning tree k’ = k – w(x) 25

26.  A connected component of arbitrarily many vertices of weight 0 26

27.  Structure:  Two adjacent weight-0 vertices x, y  Operation:  Contract the edge xy, let w be the merged vertex  Justification:  Tree T  Tree T’: ▪ There always is an optimal tree that uses xy ▪ Add xy to tree, remove an edge from resulting cycle ▪ Vertices xy have weight 0 so no loss of leaf weight ▪ Contract the edge xy to obtain T’  Tree T’  Tree T: ▪ Split w into two vertices x, y and connect to neighbors 27

28.  Arbitrarily many weight-0 degree-2 vertices with the same neighborhood 28

29.  Structure:  Two weight-0 degree-2 vertices u,v with equal neighborhoods {x,y}  The remainder of the graph R is not empty  Operation:  Remove v and its incident edges  Justification:  {x, y} forms a cutset  One of x,y will always be internal in a spanning tree 29

30.  A necklace of arbitrary length  Every pair of positive-weight vertices forms a cutset, so at most 1 leaf of positive weight 30

31.  Structure:  a weight-0 degree-2 vertex with neighbors x,y  a direct edge xy  Operation:  remove the edge xy  Justification:  You never need xy  If xy is used, we might as well remove it and connect x and y through z  Since w(z) = 0, leaf weight does not decrease 31

32.  Three path components of arbitrary length  At most 4 leaves in any spanning tree 32

33.  Structure:  Path component with p ≥ 4  Operation:  Replace v 2,v 3,.., v p-1 by new vertex v*  Weight of v* is maximum of edge endpoints – max(w(v 1 ),w(v p ))  Justification:  The two spanning trees are equivalent  If a spanning tree avoids an edge inside the path component, then the optimal leaf weight gained is equal to the leaf weight gained by avoiding an edge incident on v* 33

34.  An arbitrarily long cycle with alternating weighted / zero weight vertices  At most one leaf of positive weight 34

35.  Structure:  The graph is a simple cycle  Operation:  Remove an edge that maximizes the combined weight of its endpoints  Justification:  Any spanning tree for G avoids exactly one edge  Avoiding an edge with maximum weight of endpoints is optimal 35

36.  These reduction rules are necessary and sufficient for the kernelization claim  Reduction rules do not depend on parameter k  Reduced instance is the same, regardless of k  Reduction rules do not depend on planarity of the graph  But the structural proof that every reduced instance has a |G|/c leaf weight spanning tree does depend on k  Reduction rules can be executed in linear time  Planarity is preserved  We only remove and contract edges  Suggests the reduction rules are good preprocessing rules for any instance of Weighted Max Leaf  Even non-planar graphs without given parameter  The structural proof is constructive  When the output of kernelization is YES then we can also find a suitable spanning tree 36

37.  We apply the reduction rules in the given order, until no rule is applicable  Can be done in linear time  Reduced graph is still planar, since all we do is:  Contract an edge, remove an edge, remove a vertex, re-color a vertex.  Reduced instance is highly structured:  White vertices form an independent set  All vertices have degree ≥ 2  No path components of size > 3  … 37

38.  Kernelization yields FPT algorithm  First kernelize, then try all possible leaf sets  Check whether the complement is a connected dominating set  Planar graphs are sparse, so |E| is O(|V|)  Kernelization can be implemented to run in linear time 38

39.  Maximum Leaf Weight Spanning tree is a natural generalization of the Maximum Leaf Spanning Tree problem  It is W[1]-hard on general graphs, so no FPT algorithm  The problem admits a 84k problem kernel on planar graphs  This can be extended to:  O(k √g) kernel on graphs of genus g  O(k d) kernel on graphs on which every vertex of positive weight has at most d neighbors 39

40.  Decreasing constant in the kernel size  Better mathematical analysis of resulting reduced instances  New reduction rules needed to go below 24k kernel  Classifying complexity of general-graph problem  Hardness proof for some W[i] > 1  Membership proof for some W[i]  Determining complexity for real-valued weights  Approximation algorithms  Does (Weighted) Max Leaf have a PTAS on planar graphs? 40