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Bart Jansen, Utrecht University

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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

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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

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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?

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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Terminology and a lemma 15

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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

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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

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Bipartite Max Leaf is hard for W[1] 18

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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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

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A connected component of arbitrarily many vertices of weight 0 31

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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

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

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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

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

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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

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

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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*

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

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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

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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

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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

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What is it that makes Weighted Max Leaf hard? 43

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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

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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

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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

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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

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