Parameterizing edge modification
problems above lower bounds
René van Bevern1
joint work with
Vincent
1 Novosibirsk
Froese2
State University
and Christian Komusiewicz3
2 TU
Berlin
3 Friedrich-Schiller-Universität
Jena
11th International Computer Science Symposium in Russia,
St. Petersburg, June 9-13th, 2016
1 Graph modification problems
Modify a graph to fulfill desired properties by
. adding and deleting a minimum number of vertices,
. adding and deleting a minimum number of edges.
Example 1.1. Vertex Cover, Feedback Vertex Set, Cluster Editing, ...
Problem 1.2 (F -Free Editing).
Instance: A graph G = (V, E) with n vertices and m edges, k ∈ N.
Task: Add/delete ≤ k edges to destroy induced subgraphs isomorphic to F .
The graph F is called the forbidden induced subgraph.
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Example 1.3. Two NP-hard edge modification problems:
K3 -Free Editing (Triangle Deletion),
P3 -Free Editing (Cluster Editing).


y


y
K3 -free
P3 -free
Cluster Editing also known as Correlation Clustering.
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2 Parameterized Complexity
A parameterized problem is a set L ⊆ Σ∗ × N, its instances (x, k) consist of
input x and a parameter k.
Fixed-parameter algorithms or FPT algorithms.
. Decide (x, k) ∈ L in f (k) · nc time for some function f and constant c.
. This is stronger than “polynomial time for fixed k”. Potentially leads
to efficient algorithms if k is small, especially if c = 1.
Problem kernelization.
. In polynomial time, convert an instance (x, k) into an instance (x 0 , k 0 )
with |x 0 | + k 0 ≤ g(k) and (x, k) ∈ L ⇐⇒ (x 0 , k 0 ) ∈ L.
. Provably effective, efficient, and correct preprocessing.
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Known parameterized complexity results
For the parameter k—number of allowed modifications:
Theorem 2.1 (Cai, ILP 1996). F -Free Editing is fixed-parameter tractable
parameterized by k for any finite graph F .
Triangle Deletion:
. O(2.076k + nm)-time solvable
. 6k-vertex problem kernel
Cluster Editing:
. O(1.62k + n + m)-time solvable
. 2k-vertex problem kernel
Problem:
[Wahlström, PhD thesis, 2007]
[Brügmann et al., AGT 09]
[Böcker, JDA 2012]
[Chen & Meng, JCSS 2012]
k ≈ 10n in Cluster Editing for proteine similarity clustering
[Böcker et al., Algorithmica 2009]
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3 Parameterization above lower bounds
In practise, lower bounds are crucial for speeding up
. branching algorithms (branch and bound),
. ILP solvers (by adding valid constraints to relaxation),
empirical major improvements in running time.
Our goals:
. Provable speedups using lower bounds and smaller parameters.
Use lower bound h on number of required edge modifications and
consider parameter ` := k − h instead of k.
Running example in this talk:
René van Bevern
Triangle Deletion.
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3.1 Lower bounds via vertex-disjoint F -Packings
Definition 3.1. Given a graph G, a vertex-disjoint F -packing of G is a
set H = {H1 , . . . , Hh } such that each Hi is an induced F -subgraph in G and
the vertex sets of the Hi are pairwise disjoint.
Example 3.2. At least |H| edge deletions are needed.
or
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3.1.1 Problem kernel for Triangle Deletion
Goal: Data reduction that bounds graph size in ` (edge modifications required
additionally to the lower bound).
Idea: Exploit the packing H for data reduction to bound k ∈ O(`). Then,
apply known kernel for k.
Theorem 3.3. K3 -Free Deletion With K3 -Packing admits a problem kernel
containing at most 18` vertices.
Proof.
. Each packing triangle T ∈ H needs at least one edge deletion.
. If there is an edge e of T whose deletion destroys all triangles that
share an edge with T , then delete e and remove T from H.
This yields |H| ≤ 2` ⇒ k = ` + |H| ≤ 3`. Use known 6k-vertex kernel.
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3.1.2 Simpler and improved algorithm for Triangle Deletion
Using k ≤ 3` and the best known FPT algorithm for Triangle Deletion
. one immediately gets an 2.0763` · nO(1) -time algorithm,
. but our data reduction rule leads to a much simpler and faster algorithm:
Theorem 3.4. K3 -Free Deletion With K3 -Packing is solvable in 5` · nO(1) .
Rule 3.5. If G contains a triangle T and H is empty, then add T to H.
Observation 3.6. If, after all data reduction, G can be made triangle-free
with k edge deletions, then H contains a triangle sharing two edges with two
non-packing triangles.
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Algorithm 3.7 (K3 -Free Deletion With K3 -Packing).
1. Apply data reduction as long as applicable.
2. If not triangle-free, then there is a triangle of the following form.
x
b
a
T
y
d
c
3. Recursively try out the following five possibilities:
. for each e ∈ {a, b, c, d} try to delete e and decrement k.
. try to delete x and y , decrement k by two and remove T from H.
In each case, ` = k − |H| decreases by one and we stop recursion when
k < |H| ⇐⇒ ` < 0. Thus, 5` · nO(1) time.
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3.2 Better Lower Bounds
Instead of packing K3 s, use packing of arbitrary induced subgraphs H that
require at least τ (H) ≥ 1 edge modifications.
Example 3.8. K4 contains
. at most one vertex-disjoint triangle,
. at most one edge-disjoint triangle,
. but requires τ (K4 ) = 2 edge deletions.
Definition 3.9. A vertex-disjoint cost-t packing is a set H = {H1 , . . . , Hh }
of induced subgraphs of G such that 1 ≤ τ (H) ≤ t for each H ∈ H.
P
The packing H yields a lower bound h(H) := H∈H τ (H).
We parameterize by ` := k − h(H).
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Results for Cost-t Packings
Based on the general approach of bounding k ∈ O(t`).
Theorem 3.10. K3 -Free Deletion With Cost-t Packing
i) is solvable in 2O(` log t) · nO(1) time, and
ii) admits a problem kernel with at most (12t + 6)` vertices.
Based on the same approach (but significantly more technical):
Theorem 3.11. Cluster Editing With Cost-t Packing
i) is solvable in 2O(t`) · nO(1) time, and
ii) admits a problem kernel with at most (4t + 2)` vertices.
We also obtain similar results for Feedback Arc Set in Tournaments.
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3.3 Hardness results
General fixed-parameter tractability result for F -Free Editing With VertexDisjoint F -Packing, like for parameter k?
Theorem 3.12. For every fixed q ≥ 6, Kq -Free Deletion With Kq -Packing is
NP-hard for ` = 0.
Better lower bounds via edge-disjoint packings?
Theorem 3.13. K3 -Free Deletion With Edge-Disjoint K3 -Packing is NP-hard
for ` = 0.
What about vertex deletion problems?
Theorem 3.14. For every fixed q ≥ 3, Pq -Free Vertex Deletion With Pq Packing is NP-hard for ` = 0.
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4 Conclusion
Studied edge modification problems: Triangle Deletion, Cluster Editing, and
Feedback Arc Set on Tournaments parameterized above lower bounds derived
from arbitrary vertex-disjoint subgraph packings.
Generic approach for fixed-parameter algorithms and problem kernels.
Explored boundaries of approach: hardness results for other lower bounds and
larger forbidden induced subgraphs.
Cost-t packings: tradeoff between running time and power of data reduction.
needs experimental evaluation.
Open: Cluster Editing parameterized above edge-disjoint P3 -packings? NPhard for ` = 0, XP, FPT?
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Bridges across Ob river in Novosibirsk city center