Bart Jansen
Kernelization for Maximum Leaf Spanning Tree
with Positive Vertex Weights
1
CIAC 2010, Rome
CLASSICAL PROBLEM AND
ITS COMPLEXITY
2
Maximum Leaf Spanning Tree

Max Leaf




Input:
Connected graph G, integer k
Question: Does G have a spanning tree with at least k leaves?
Applications in Network Design, Facility Location
Strongly related to Connected Dominating Set

Spanning tree ≥ k leaves  Connected dominating set ≤ n – k
NP-complete
• Planar graphs
max. degree 4
• 3-regular graphs
3
Approximability
• 2-approximation
[Solis-oba]
• No PTAS unless
P=NP
Fixed-Parameter Tractability



Solve NP-hard problems exactly
Measure the structure of an instance through a parameter,
and determine whether combinatorial explosion can be
restricted to the parameter
Example: Vertex Cover




General case:


4
“Does graph G have a vertex cover of size k?”
Use the requested size k as the parameter
Problem can be solved in O(2kn2) time
instance of a parameterized problem is a tuple (x,k) where x
is the “classical” instance and k measures some property of x
Problem is Fixed Parameter Tractable (FPT) if it can be solved
in f(k)|x|c time for a computable function f and constant c
Kernelization


Kernelization is a technique to obtain FPT algorithms, and is
useful in a broad context for preprocessing
A kernelization algorithm (or kernel) for a parameterized
problem is a polynomial-time algorithm that transforms (x,k)
into (x’,k’) such that:



The function f is the size of the kernel

Intuitively: a kernelization uses polynomial time to obtain an
equivalent instance whose size depends only on the old
parameter k, but not on the old instance size
Example: Vertex Cover



5
(x,k) and (x’,k’) are equivalent instances of the same problem
|x’|, k’ ≤ f(k) for some computable function f
We can transform (G,k) into (G’,k’) such that |V’| ≤ 2k
Size of kernel is O(k2)
Parameterized Complexity of Max Leaf



Use the target number of leaves k as the parameter
Long history of improved running times and kernel sizes
Current best:
FPT algorithm
• O(4knc) time
[Kneis et al.
ISAAC 2008]


• 3.75k vertices
[Estivill-Castro et al.
ACiD 2005]
Max Leaf is tractable from the parameterized perspective
We should see which generalizations are tractable!

6
Kernel
Introduce vertex weights
WEIGHTED PROBLEM AND
ITS KERNEL
7
Weighted Maximum Leaf Spanning Tree

Weighted Max Leaf





Input:
Connected graph G with vertex weights, value k
Question: Does G have a spanning tree whose leaves have
combined weight at least k?
Parameter:k
Complexity of the problem depends on allowed weights
Our results:
Rational
weights > 0
• NP-complete
for k=1
8
Weights {0,1}
• W[1]-hard
Weight 11
Rational
weights ≥ 1
Weight 16
• Kernel with
5.5k vertices
Leafy Spanning Trees

Let G be a connected graph on n vertices such that:
1. The graph is not a simple cycle
2. No degree-1 vertex is adjacent to a degree-2 vertex
3. There is no path of ≥ 4 consecutive degree-2 vertices

Then G has a spanning tree with n/7.5 leaves

9
Every leaf has weight ≥ 1  G has a S.T. with weight n/7.5
Kernelization strategy


Transform G into a graph G’ that satisfies the given 3
properties, such that
MaxWeight(G) = MaxWeight(G’)
Then:



Kernelization algorithm transforms (G,k) into (G’,k) in
polynomial time


10
MaxWeight(G’) ≥ MaxLeaves(G’) ≥ |V’|/7.5
|V’| ≤ 7.5 ∙ MaxWeight(G’)
If |V’| ≥ 7.5k then output YES
Otherwise |V’| < 7.5k, output (G,k’)
Rule 1: Solve cycles


11
A spanning tree for a cycle avoids exactly 1 edge
Choose an edge with maximum weight of its endpoints and
delete it
Rule 2: Shrink pendant paths


12
Any spanning tree must use all edges on a degree-2 path
to a leaf
We can delete the degree-2 vertices and re-attach the leaf
Rule 3: Shrink large path components



Consider a path component of degree-2 vertices between
x,y of degree ≠ 2
A spanning tree avoids at most one edge in this
component
When avoiding an edge not incident on x or y, it’s optimal
to avoid one with maximum weight of its endpoints


13
Reduce by merging all degree-2 vertices except the first and
last into one
Set weight to: maximum edge weight – max(first,last)
Better Bounds

If additionally:



Then G has a spanning tree with n/5.5 leaves

14
Distinct degree-2 vertices have distinct open neighborhoods
Every triangle {x,y,z} contains a vertex of degree ≥ 4
Gives kernel with 5.5k vertices
Proof of the extremal result

Extends a technique of Griggs, Kleitman and Shastri who
showed:



Amortized analysis by “keeping track of dead leaves”
Constructive proof






15
every cubic graph on n vertices has a spanning tree with n/4
leaves
Initialize a tree T subgraph as a star around 1 vertex
As long as T does not span G: show we can augment to a bigger
tree T’
Show that the increase in # vertices is balanced against the
increase in # leaves
Prove that as long as the tree does not span G, some
augmentation is possible
Extensive case analysis to get to n/5.5
Factor 5.5 is best-possible for these graphs
Kernelization result

[Decision version] Given an instance (G,k) of Weighted
Max Leaf with weights ≥ 1, we can compute in O(n+m)
time an equivalent instance (G’,k’) such that:




[Optimization version] Given an instance G of Weighted
Max Leaf with weights ≥ 1, we can compute in O(n+m)
time an instance G’ and constant c such that:



16
G has a spanning tree with weight ≥ k  G’ has a spanning
tree with weight ≥ k’,
|V’| ≤ 5.5k’,
k’ ≤ k.
opt(G) = opt(G’) + c,
|V’| ≤ 5.5 opt(G’)
c ≥ 0.
APPROXIMABILITY AND
CONCLUSION
17
Approximability

Through reduction from Independent Set we obtain results
on approximability of Weighted Max Leaf with weights ≥ 1


No n1/2-ε approximation
No OPT1/3-ε approximation
in polynomial time for any ε > 0 unless P=NP

Surprising behavior


18
linear-vertex kernel but no constant-factor approximation
For weights in the range [1..c] there is a 2c approximation
Conclusion




We should study generalizations of FPT problems
Eliminating long degree-2 paths guarantees
MaxLeaves ≥ |V|/c
Weighted Max Leaf with weights ≥ 1 has a kernel with
5.5k vertices by using weight-aware reduction rules
Other results


Open problems



19
Weighted Max Leaf with all weights 0 or ≥ 1 has a linearvertex kernel on bounded-genus graphs
Is there a fast FPT algorithm for Weighted Max Leaf?
Are there other FPT problems that remain tractable when
allowing vertex weights?
What is the effect of vertex weights on kernelizability?