Bart Jansen, Utrecht University
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Max Leaf
 Instance: Connected graph G, positive integer k
 Question: Is there a spanning tree for G with at
least k leaves?
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Applications in network design
YES-instance for k ≤ 8
2
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Classical complexity
 Max-SNP complete, so no polynomial-time
approximation scheme (PTAS)
 NP-complete, even for
3-regular graphs
• By P. Lemke, 1988
Planar graphs of maximum degree 4
• By Garey and Johnson, 1979
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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

Classical complexity
 No constant-factor approximation
 NP-complete, even for:
d-regular graphs for d ≥ 4
• By Fusco and Monti, 2007
Planar graphs of maximum degree 4
• By Li and Toulouse, 2006
5
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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?
Leaf weight 16
11
6
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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(nc) 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(2kn) time?
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True for many problems, such as Vertex Cover
Instance of a parameterized problem is <I,k>
 Regular instance and the parameter as a natural number
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If there is an f(k)nc time algorithm for a problem
 Then it is Fixed Parameter Tractable (FPT)
(n is the size of instance I)
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A kernelization algorithm:
 Reduces parameterized instance <I,k> to equivalent <I’,k’>
 Size of I’ does not depend on I but only on k
 Time is poly (|I| + k)
 New parameter k’ is at most k
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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
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Parameterize by the target value k
 Number of leaves, or leaf weight
Max Leaf
• Kernel with 3.75k
vertices
• O(4kk2+p(|V|+|E|))
algorithm
Bipartite Max Leaf
Weighted Max Leaf
• No existing results
• W[1] hard on general
graphs
• No existing results
• Complexity depends
on weight range
• Kernel for restricted
graph classes
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Weight range
General graphs
{1,2,…}
Kernel with 7.5k vertices
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Weight range
General
graphs
{1,2,…}
{0,1,… }
Planar graphs
Genus ≤ g
Degree of
positiveweight ≤ d
Kernel with 7.5k vertices
Hard for W[1]
78k
O(k√g+ g)
O(kd2)
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Weight range
General
graphs
{1,2,…}
{0,1,… }
Q>0
Planar graphs
Genus ≤ g
Degree of
positiveweight ≤ d
Kernel with 7.5k vertices
Hard for W[1]
78k
O(k√g+ g)
O(kd2)
NP-complete for k=1 (not Fixed Parameter Tractable)
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Weight range
General
graphs
{1,2,…}
Q≥1
Planar graphs
Genus ≤ g
Degree of
positiveweight ≤ d
Kernel with 7.5k vertices
{0,1,… }
Hard for W[1]
78k
O(k√g+ g)
O(kd2)
Q≥1 U {0}
Hard for W[1]
O(k)
O(k√g+ g)
O(kd2)
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Terminology and a lemma
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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
<x,v1,v2, .. , vk,y>, s.t.

 x, y have degree ≠ 2
 all vi have degree 2
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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
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Bipartite Max Leaf is hard for W[1]
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We prove that Bipartite
Max Leaf is hard for W[1]
 (Probably) no f(k)nc 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
• ..
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W[i] hardness is proven by parameterized
reduction <I,k>  <I’,k’> 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
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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
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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
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Polynomial time
k’ ≤ g(k) = k
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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
Edges incident on w are not drawn
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A linear kernel for Maximum Leaf Weight Spanning Tree on planar graphs
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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
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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:
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 If no reduction rules apply, there is a spanning tree with
|G|/c leaves of weight ≥ 1 (for some c > 0)
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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
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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
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So we need to avoid:
 A graph can always grow larger without
increasing the optimal leaf weight of a spanning
tree
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All reduction rules are needed to prevent
such situations
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Vertex of positive weight, with arbitrarily
many degree-1 neighbors of weight 0
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Structure:
 Vertex x of degree 1 adjacent to y of degree > 1
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Operation:
 Delete x, decrease k by w(x), set w(y) = 0
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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)
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A connected component of arbitrarily many
vertices of weight 0
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Structure:
 Two adjacent weight-0 vertices x, y
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Operation:
 Contract the edge xy, let w be the merged vertex
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Justification:
 We can always use the edge xy in an optimal tree
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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
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Operation:
 Remove v and its incident edges
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Justification:
 {x, y} forms a cutset
 One of x,y will always be internal in a spanning tree
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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
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Structure:
 a weight-0 degree-2 vertex with neighbors x,y
 a direct edge xy
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Operation:
 remove the edge xy
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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
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Structure:
 Path component <x,v1,v2,..,vp,y> with p ≥ 4
Operation:
 Replace v2,v3, .. , vp-1 by new vertex v*
 Weight of v*:

 Compute maximum of edge endpoint weights on edges (vi,vi+1) for i=1 .. p-1
 Subtract maximum of w(v1) and (vp)
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
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Structure:
 The graph is a simple cycle
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Operation:
 Remove an edge that maximizes the combined weight of its
endpoints
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Justification:
 Any spanning tree for G avoids exactly one edge
 Avoiding an edge with maximum weight of endpoints is optimal
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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) 278k + O(|V| + |E|) algorithm
41

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
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What is it that makes Weighted Max Leaf hard?
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Target leaf weight k
Treewidth w
Positive-weight
vertices p
Zero-weight
vertices x
Parameter k + x
• Not fixed parameter tractable on general graphs
• Hard for W[1] by reduction from k-Independent Set
(Kernel for restricted graph classes)
• Amenable to dynamic programming
• O(ww |V|) time algorithm
• Try all subsets of S positive-weight vertices, check if V \ S is a
Connected Dominating Set
• O(2p(|V|+|E|)) time
• Not fixed parameter tractable
• For x=0 (no zero-weight vertices) we have regular unweighted Max Leaf,
which is NP-complete
• 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
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
Weighted Max Leaf with weight
0 and parameter x + k
Is there a spanning tree of
leaf weight ≥ 13 ?
k = 13, x = 2

Weighted Max Leaf with weight
≥ 1 and parameter k’ = x + k
Is there a spanning tree of
leaf weight ≥ 15
14 ?
45

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.
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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
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