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 ≤ 8

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. 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?

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 non-negative vertex weights; positive number 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 Max Leaf  If weights {0,1} are allowed, no constant-factor approximation since it generalizes Bipartite Max Leaf  We consider the fixed parameter complexity 7

8.  Suppose we encounter a NP-complete problem  No O(n c ) algorithm for fixed c  No efficient algorithm exists?  What happens if we use some information about the instance?  For example: solution size is k, much less than n.  Can we solve it in O(2 k n) time?  True for many problems, such as Vertex Cover  Instance of a parameterized problem is  Regular instance and the parameter as a natural number  If there is an f(k)n c time algorithm for a problem  Then it is Fixed Parameter Tractable (FPT) (n is the size of instance I) 8

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.  Parameterize by the target value k  Number of leaves, or leaf weight 10 Max Leaf Kernel with 3.75k vertices O(4 k k 2 +p(|V|+|E|)) algorithm Bipartite Max Leaf No existing results W[1] hard on general graphs Weighted Max Leaf No existing results Complexity depends on weight range Kernel for restricted graph classes

11. Weight rangeGeneral graphs {1,2,…} Kernel with 7.5k vertices 11

12. Weight range General graphs Planar graphs Genus ≤  Degree of positive- weight ≤  {1,2,…} Kernel with 7.5k vertices {0,1,… } Hard for W[1]78k O(k√  +  )O(k  2 ) 12

13. Weight range General graphs Planar graphs Genus ≤  Degree of positive- weight ≤  {1,2,…} Kernel with 7.5k vertices {0,1,… } Hard for W[1]78k O(k√  +  )O(k  2 ) Q >0 NP-complete for k=1 (not Fixed Parameter Tractable) 13

14. Weight range General graphs Planar graphs Genus ≤  Degree of positive- weight ≤  {1,2,…} Q ≥1 Kernel with 7.5k vertices {0,1,… } Hard for W[1]78k O(k√  +  )O(k  2 ) Q ≥1 U {0} Hard for W[1]O(k) O(k√  +  )O(k  2 ) 14

15. Terminology and a lemma 15

16.  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 16

17.  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 17

18. Bipartite Max Leaf is hard for W[1] 18

19.  We prove that Bipartite Max Leaf is hard for W[1]  (Probably) no f(k)n c algorithm  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.. 19

20.  W[i] hardness is proven by parameterized reduction  from some W[i]- hard problem  Similar to (Karp) reductions for NP-completeness  Reduction in time f(k)*poly(|I|)  New parameter k’ ≤ g(k) for some function g  We reduce k-Independent Set (W[1]-complete) to Bipartite Max Leaf 20

21.  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 21

22.  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’ ≤ g(k) = k 22

23. 23  Complement of S is a vertex cover  Build spanning tree:  Take w as root, connect to all black vertices  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

24.  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 24 Edges incident on w are not drawn

25. A linear kernel for Maximum Leaf Weight Spanning Tree on planar graphs 25

26.  Kernel of size 78k 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 78k ▪ or we are sure that the answer is yes ▪ then we output a trivial, constant-sized YES-instance 26

27.  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  If |G| ≥ ck after kernelization we return YES  If not, the instance is small 27

28.  The reduction rules must enforce:  When we increase the size of the graph, eventually this leads to an increase in optimal leaf weight of a spanning tree  So we need to avoid:  A graph can always grow larger without increasing the optimal leaf weight of a spanning tree  All reduction rules are needed to prevent such situations 28

29.  Vertex of positive weight, with arbitrarily many degree-1 neighbors of weight 0 29

30.  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) 30

31.  A connected component of arbitrarily many vertices of weight 0 31

32.  Structure:  Two adjacent weight-0 vertices x, y  Operation:  Contract the edge xy, let w be the merged vertex  Justification:  We can always use the edge xy in an optimal tree 32

33.  Arbitrarily many weight-0 degree-2 vertices with the same neighborhood 33

34.  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 34

35.  A necklace of arbitrary length  Every pair of positive-weight vertices forms a cutset, so at most 1 leaf of positive weight 35

36.  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 36

37.  Three path components of arbitrary length  At most 4 leaves in any spanning tree 37

38. 38  Structure:  Path component with p ≥ 4  Operation:  Replace v 2,v 3,.., v p-1 by new vertex v*  Weight of v*:  Compute maximum of edge endpoint weights on edges (v i,v i+1 ) for i=1.. p-1  Subtract maximum of w(v 1 ) and (v p )  Justification:  The two spanning trees are equivalent  Suppose a spanning tree avoids an edge inside the path component  We gain at least as much weight by avoiding an edge incident on v*

39.  An arbitrarily long cycle with alternating weighted / zero weight vertices  At most one leaf of positive weight 39

40.  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 40

41.  Reduction rules are necessary and sufficient for the kernelization claim  Rules do not depend on parameter 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 planarity  Reduction rules can be executed in linear time  Yields O(k) 2 78k + O(|V| + |E|) algorithm 41

42.  Kernel for {0,1,…} weights on planar graphs  Current kernel size 78k  Improved analysis may decrease kernel size  New reduction rules needed to go below 31k  Kernel size for {1,2,...} weights  Current kernel size 7.5k  New reduction rules needed to go below 7.5k 42

43. What is it that makes Weighted Max Leaf hard? 43

44. Not fixed parameter tractable on general graphs Hard for W[1] by reduction from k-Independent Set (Kernel for restricted graph classes) Target leaf weight k Amenable to dynamic programming O(w w |V|) time algorithm Treewidth w Try all subsets of S positive-weight vertices, check if V \ S is a Connected Dominating Set O(2 p (|V|+|E|)) time Positive-weight vertices p Not fixed parameter tractable For x=0 (no zero-weight vertices) we have regular unweighted Max Leaf, which is NP-complete Zero-weight vertices x Fixed parameter tractable We reduce (k+x) Weighted Max Leaf with {0,1,…} weights to k’ = k+x Weighted Max Leaf with {1,2,…} weights Parameter k + x 44

45. 45 Is there a spanning tree of leaf weight ≥ 13 ? k = 13, x = 2 Is there a spanning tree of leaf weight ≥ 14 ? Is there a spanning tree of leaf weight ≥ 15 ?  Weighted Max Leaf with weight 0 and parameter x + k  Weighted Max Leaf with weight ≥ 1 and parameter k’ = x + k

46.  Maximum Leaf Weight Spanning tree is a natural generalization of the Maximum Leaf Spanning Tree problem  If weights are ≥ 1:  Kernel with 7.5k vertices  If weights are 0 or ≥ 1:  W[1]-hard on general graphs  Linear kernel when restricted to ▪ planar graphs, ▪ graphs of bounded genus, ▪ graphs in which the degree of positive-weight vertices is bounded. 46

47.  Classifying complexity of general-graph problem  Hardness proof for some W[i] > 1  Membership proof for some W[i]  Investigate connections to approximation algorithms  PTAS on planar graphs using Planar-Separators?  Constant-factor approximation for {0,1} weights 47