The Central Tree Problem

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Heuristics for Central Tree Problem
Yury Nikulin
Department of Mathematics, University of Turku

The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The Central Tree Problem
Heuristics for the Central Tree Problem Yury Nikulin - p. 2/33

Distance Measure
The Central Tree
Problem
Let G = (V, E) be a connected graph and let
T G := {T | T = (V, E T )} represent the set of all spanning trees
of graph G.
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 3/33

Distance Measure
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Let G = (V, E) be a connected graph and let
T G := {T | T = (V, E T )} represent the set of all spanning trees
of graph G.
Definition. The distance between two spanning trees T 1 and T 2
of a graph is defined as a half of cardinality of the symmetric
difference between set of edges E T1 and E T2 :
Dist(T 1 , T 2 ) = 1 ˙|T 1 △ T 2 |= 1 1 ∪ T 2 )\(T 1 ∩ T 2 ) |
2 2˙|(T
Heuristics for the Central Tree Problem Yury Nikulin - p. 3/33

Distance Measure
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Let G = (V, E) be a connected graph and let
T G := {T | T = (V, E T )} represent the set of all spanning trees
of graph G.
Definition. The distance between two spanning trees T 1 and T 2
of a graph is defined as a half of cardinality of the symmetric
difference between set of edges E T1 and E T2 :
Dist(T 1 , T 2 ) = 1 ˙|T 1 △ T 2 |= 1 1 ∪ T 2 )\(T 1 ∩ T 2 ) |
2 2˙|(T
Definition. The distance between two spanning trees T 1 and T 2
of a graph is defined as the number of edges presented in one
tree but not in the other:
Dist(T 1 , T 2 ) =| T 1 \T 2 |=| T 2 \T 1 | .
Heuristics for the Central Tree Problem Yury Nikulin - p. 3/33

The Central Tree Problem
The Central Tree
Problem
Heuristics for the
CTP
Recall that the cospanning tree ¯T of a spanning tree T is the
edge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .
Also, if a graph G has n vertices and c connected components,
then the rank ρ(G) of graph G is n − c.
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The Central Tree Problem
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Recall that the cospanning tree ¯T of a spanning tree T is the
edge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .
Also, if a graph G has n vertices and c connected components,
then the rank ρ(G) of graph G is n − c.
Definition. [Deo, 1966]
The Central Tree Problem (shortly CTP) consists in finding central
tree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ is
minimal, i.e.
ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .
Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The Central Tree Problem
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Recall that the cospanning tree ¯T of a spanning tree T is the
edge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .
Also, if a graph G has n vertices and c connected components,
then the rank ρ(G) of graph G is n − c.
Definition. [Deo, 1966]
The Central Tree Problem (shortly CTP) consists in finding central
tree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ is
minimal, i.e.
ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .
Thus, a central tree is a spanning tree whose edge removal
leads to splitting graph G into the maximum number of
components.
Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The CTP vs. MDTP
The Central Tree
Problem
The Maximally Distant Tree Problem (MDTP) consists in
finding a pair of extremal trees with the largest distance.
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 5/33

The CTP vs. MDTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The Maximally Distant Tree Problem (MDTP) consists in
finding a pair of extremal trees with the largest distance.
Definition. The goal of the CTP is to find a spanning tree such
that the distance to maximally distant tree is minimized:
min
T ∈T G
max
T ′ ∈T G
Dist(T, T ′ ) (1)
Heuristics for the Central Tree Problem Yury Nikulin - p. 5/33

The CTP vs. MDTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The Maximally Distant Tree Problem (MDTP) consists in
finding a pair of extremal trees with the largest distance.
Definition. The goal of the CTP is to find a spanning tree such
that the distance to maximally distant tree is minimized:
min
T ∈T G
max
T ′ ∈T G
Dist(T, T ′ ) (1)
Thus, CTP can be considered as dual to MDSTP
Heuristics for the Central Tree Problem Yury Nikulin - p. 5/33

Short Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Figure 1: Initial graph G and extremal spanning trees
Heuristics for the Central Tree Problem Yury Nikulin - p. 6/33

Short Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Figure 1: Initial graph G and extremal spanning trees
Figure 2: Initial graph G, the central tree and the tree which is
maximally distant to the central tree
Heuristics for the Central Tree Problem Yury Nikulin - p. 6/33

Complexity of the CTP
The Central Tree
Problem
Theorem. [ Bezrukov et al., 1996]
The Central Tree Problem is NP-hard
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Theorem. [ Bezrukov et al., 1996]
The Central Tree Problem is NP-hard
To prove this fact the authors used transformation from Exact
Cover by 3-set problem, which is known to be NP-Complete
Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Theorem. [ Bezrukov et al., 1996]
The Central Tree Problem is NP-hard
To prove this fact the authors used transformation from Exact
Cover by 3-set problem, which is known to be NP-Complete
Thus, there is no polynomial algorithm for the CTP unless
P = NP .
Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Theorem. [ Bezrukov et al., 1996]
The Central Tree Problem is NP-hard
To prove this fact the authors used transformation from Exact
Cover by 3-set problem, which is known to be NP-Complete
Thus, there is no polynomial algorithm for the CTP unless
P = NP .
A non-trivial class of graphs for which the CTP is known to
have polynomial algorithm is a class of complete graphs.
Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Applications
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The CTP can find its application in many real-life models. For example,
in computer networking, spanning tree topologies can be
efficiently used to model network configurations.
Similar network problems have many applications in telecommunications,
circuit analysis, facility allocation, and related domains
in electric, gas, and sewer networks.
Heuristics for the Central Tree Problem Yury Nikulin - p. 8/33

Equivalence between 0 − 1 RSTP and CTP
The Central Tree
Problem
The following theorem establishes equivalence between 0 − 1
RSTP and CTP.
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 9/33

Equivalence between 0 − 1 RSTP and CTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The following theorem establishes equivalence between 0 − 1
RSTP and CTP.
Theorem. [Aron and Van Hentenryck, 2004] A spanning tree T ∈
T G is a robust spanning tree of zero-one robust spanning tree
problem if and only if it is a central tree of G.
Heuristics for the Central Tree Problem Yury Nikulin - p. 9/33

One more evidence of NP-hardness
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Averbakh and Lebedev proved in 2004 that the RSTP is
NP-Hard even if the intervals of uncertainty are all equal to
[0, 1]. They also proved that the RSTP is polynomially solvable
if the number of edges with uncertain lengths is fixed or
bounded by the logarithm of a polynomial function of the total
number of edges.
Heuristics for the Central Tree Problem Yury Nikulin - p. 10/33

Robust deviation for the central tree
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Thus for a central tree T ∗ , the robust deviation is expressed as:
dev s T ∗
T ∗
= max
T ∈T G
Dist(T ∗ , T).
Let T ∗ be a central tree and N T ∗
comp be the number of
components of G − E T ∗. Then the robust deviation dev s T ∗
T ∗
N T ∗
comp are related as follows:
and
dev s T ∗
T ∗
+ N T ∗
comp = |E T ∗| + 1 = |V |
Heuristics for the Central Tree Problem Yury Nikulin - p. 11/33

MILP Formulation
Corollary from the result by Yaman et al., 2001
The Central Tree
Problem
Heuristics for the
CTP
min ∑
ij∈E
x ij −
∑
k∈V, k≠1
[α k k − α k 1] − (n − 1)µ (2)
Final Remarks
subject to
σij k ≥ αj k − αi k ∀{ij} ∈ A ∀k ∈ V \{1} (3)
∑
σij k + µ ≤ x ij ∀ij ∈ E (4)
k≠1
∑
σji k + µ ≤ x ij ∀ij ∈ E (5)
k≠1
Heuristics for the Central Tree Problem Yury Nikulin - p. 12/33

MILP Formulation
The Central Tree
Problem
∑
{ij}∈A
f ij −
∑
{ji}∈A
f ji =
{
n-1, if i = 1
-1, if i ∈ V \{1}
(6)
Heuristics for the
CTP
f ij ≤ (n − 1)x ij ∀ij ∈ E (7)
Final Remarks
f ji ≤ (n − 1)x ij ∀ij ∈ E (8)
∑
ij∈E
x ij = n − 1 (9)
f, σ, α ≥ 0, µ − unrestricted (10)
x ij ∈ {0, 1} ∀ij ∈ E. (11)
Heuristics for the Central Tree Problem Yury Nikulin - p. 13/33

Approximation for the CTP
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
For the CTP, any spanning tree represents an approximated solution
with approximation ratio 2 in terms of the robust deviation,
i.e.
∀T ∈ T G dev s T
T
≤ 2dev s T 0
T
, 0
where T 0 is a central tree.
Heuristics for the Central Tree Problem Yury Nikulin - p. 14/33

Case of two disjoint spanning trees
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Theorem 1. Let T 1 and T 2 be two edge-disjoint spanning trees in the graph
G = (V, E). Then dev s T ∗
T
≥ ⌈ ∗
|V |−1
2
⌉, where T ∗ is a central tree.
In other words, theorem 1 states that if the graph contains two
edge-disjoint spanning trees, then the central tree has at most
|V |−1
⌊
2
⌋ edges in common with the maximally distant tree, or
|V |−1
equivalently, the robust spanning tree has at most ⌊
2
⌋
edges in common with the corresponding minimum spanning
tree in the worst-case scenario associated with T ∗ .
Heuristics for the Central Tree Problem Yury Nikulin - p. 15/33

The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the CTP
Heuristics for the Central Tree Problem Yury Nikulin - p. 16/33

MINCUT
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Algorithm MINCUT(G = (E, V ))
Input: A connected graph G = (E, V )
Output: A cut of minimum weight
1. Let v 1 be any vertex of G and S = {v 1 }
2. for i = 2 to |V |
3. do v i = arg max {d(v, S)}
v∈V \S
4. S = S ∪ {v i }
5. if |V | = 2 then return δ({v |V | })
6. else
7. G ′ is obtained from G by contracting {v |V |−1 , v |V | }
in G
8. C is returned by MINCUT(G ′ = (E ′ , V ′ ))
9. Among C and δ(v |V | ) return the smallest cut in
terms of weight
Heuristics for the Central Tree Problem Yury Nikulin - p. 17/33

Greedy Construction Heuristic
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Algorithm Greedy Construction Heuristic(G = (E, V ))
Input: A connected graph G = (E, V )
Output: The edges E 0 of a potential central tree T 0
1. Initialize G ′ = G, E 0 = {}, LC = [G]
2. while |E 0 | < |V | − 1
3. do Take an arbitrary element C from LC
4. Apply MINCUT(C) and return the set of edges
MINCUTSET(C)
5. for all edges e ∈ MINCUTSET(C)
6. do if E 0 + {e} does not contain cycles
7. then E 0 = E 0 + {e}
8. else do nothing
9. Update LC by removing C and adding those
components of C ′ = C − MINCUTSET(C) which
are not an isolated vertex
10. return E 0
Heuristics for the Central Tree Problem Yury Nikulin - p. 18/33

Complexity of Greedy strategy
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
While running of the greedy strategy we perform the MINCUT
procedure at most |V | − 1 times, so the complexity of our construction
heuristic is O(|E||V | 2 + |V | 3 log|V |). Notice also that the
actual running time of the algorithm may also depend on the way
how the components are ordered in the list LC.
Heuristics for the Central Tree Problem Yury Nikulin - p. 18/33

Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Example
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Open Problem
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Given graph G = (V, E). Is it true that if there exists
a spanning tree T 0 and a sequence of MINCUTSETs
MCS 1 , MCS 2 , ..., MCS k with |MCS 1 | + |MCS 2 | + |MCS k | =
|V | − 1 where k is a number of iterations in GCH, such that T 0
can be constructed as a union over all i of all edges e ∈ MCS i ,
then T 0 is a central tree? In other words: suppose that when we
execute GCH all the edges from the MINCUTSET are included in
the final tree T 0 ; is it then true that T 0 is always a central tree?
Heuristics for the Central Tree Problem Yury Nikulin - p. 20/33

The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Fast Local Search Improvement
Algorithm Fast Local Search Improvement(G = (E, V ), T )
Input: A connected graph G = (E, V ), a spanning tree T
Output: An improved potential central tree T 0
1. Initialize T 0 = T
2. Find a spanning tree ˆT which is maximally distant to T by solving
minimum spanning tree problem for the worst-case 0 − 1 scenario
associated with T
3. for all edges e ∈ E ˆT
\E T
4. do Let V 1 and V 2 be vertex sets of two components of ˆT − e
5. if ∃ ẽ ∈ E\{E T ∪ E ˆT
} such that one endpoint of ẽ belongs to V 1
and the other endpoint belongs to V 2 :
6. then choose another edge e ′ ∈ E ˆT
\E T
7. else Find the fundamental cycle C of T ∪ {e}
8. for all edges ē ∈ E C , ē ≠ e
9. do Find the fundamental cycle ¯C of ˆT ∪ {ē}
10. if ∃ e ∗ ∈ E ¯C, e ∗ ≠ ē such that e ∗ ∈ E T :
11. then choose another ē
12. else construct a new spanning tree T 0 by
replacing edge ē with e
13. break both cycles
14. return T 0
Heuristics for the Central Tree Problem Yury Nikulin - p. 21/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
L
Instance A 30% 30%
Instance B 30% 70%
Instance C 70% 30%
Instance D 70% 70%
d
Table 1: Instance generating parameters L and d.
Heuristics for the Central Tree Problem Yury Nikulin - p. 22/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
|V | 10 15 20 40
Instance A 17 24 46 148
Instance B 11 21 35 106
Instance C 21 50 82 297
Instance D 16 27 47 158
Table 2: Number of edges.
Heuristics for the Central Tree Problem Yury Nikulin - p. 23/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
GCH 10 15 20 40
Distance 6 6 11 33
Running time (in sec.) 0.10 0.16 0.46 8.46
GCH+FLSI 10 15 20 40
Distance 5 6 11 31
Running time (in sec.) 0.11 0.17 0.48 8.92
SLS 10 15 20 40
Distance 6 7 13 36
Running time (in sec.) 0.32 0.94 30.64 1175.0
ES 10 15 20 40
Distance 5 6 - -
Running time (in sec.) 14.92 486.0 - -
Table 3: Instance sample A.
Heuristics for the Central Tree Problem Yury Nikulin - p. 24/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
GCH 10 15 20 40
Distance 6 12 17 87
Running time (in sec.) 0.07 0.34 0.82 38.35
GCH+FLSI 10 15 20 40
Distance 6 12 17 37
Running time (in sec.) 0.07 0.37 0.88 38.72
SLS 10 15 20 40
Distance 7 13 19 39
Running time (in sec.) 0.46 0.62 52.33 2888.0
ES 10 15 20 40
Distance 6 - - -
Running time (in sec.) 195.0 - - -
Table 4: Instance sample B.
Heuristics for the Central Tree Problem Yury Nikulin - p. 25/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
GCH 10 15 20 40
Distance 2 7 12 29
Running time (in sec.) 0.02 0.17 0.31 6.17
GCH+FLSI 10 15 20 40
Distance 2 6 12 26
Running time (in sec.) 0.03 0.19 0.32 6.54
SLS 10 15 20 40
Distance 2 7 12 27
Running time (in sec.) 0.07 0.59 3.79 453.9
ES 10 15 20 40
Distance 2 6 - -
Running time (in sec.) 0.28 279.0 - -
Table 5: Instance sample C.
Heuristics for the Central Tree Problem Yury Nikulin - p. 26/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
GCH 10 15 20 40
Distance 6 8 15 34
Running time (in sec.) 0.08 0.19 0.47 15.9
GCH+FLSI 10 15 20 40
Distance 5 8 13 33
Running time (in sec.) 0.09 0.24 0.57 16.3
SLS 10 15 20 40
Distance 6 8 15 38
Running time (in sec.) 0.31 2.25 28.46 2742.7
ES 10 15 20 40
Distance 5 - - -
Running time (in sec.) 12.69 - - -
Table 6: Instance sample D.
Heuristics for the Central Tree Problem Yury Nikulin - p. 27/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
|V | = 10 Instance A Instance B Instance C Instance D
GCH 75 90 80 73
GCH+FLSI 92 95 94 90
Table 7: Average (over 100 runs) performance of GCH
and.GCH+FLSI on instances with 10 vertices
|V | = 15 Instance A Instance B Instance C Instance D
GCH 7 9 8 7
GCH+FLSI 9 9 9 8
Table 8: Average (over 50 runs) performance of GCH and
GCH+FLSI on instances with 15 vertices
Heuristics for the Central Tree Problem Yury Nikulin - p. 28/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
|V | 50 100 150 200
Instance A 15sec 1h 2h 6h
Instance B 40sec 3h 6h 19h
Instance C 13sec 50min 2h 6h
Instance D 25sec 2h30min 7h 20h
Table 9: Average (over 10 runs) running time of GCH+FLSI on
large instances.
Heuristics for the Central Tree Problem Yury Nikulin - p. 29/33

Computational Experiments
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
|V | 50 100 150 200
GCH 30 59 89 120
GCH+FLSI 29 56 81 115
Lower Bound 25 50 75 100
Running time 6sec 110sec 10min 40min
Table 10: Large instances with graphs containing the union of
two edge-disjoint spanning trees.
Heuristics for the Central Tree Problem Yury Nikulin - p. 30/33

The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
Final Remarks
Heuristics for the Central Tree Problem Yury Nikulin - p. 31/33

What to do next
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
■ Find new mechanisms which can be incorporated into the
greedy strategy to make it deterministic and faster.
■ Specify classes of graphs for which we can guarantee the
quality of the solution found by the greedy construction
heuristic (e.g. being at most a factor of 1.5 from the
optimum). Here we measure in terms of the number of
connected components in G − E T versus the number of
connected components in G − E T ∗, where T ∗ is a central
tree
■ Develop new approximation algorithms with guaranteed
approximation ratio better than 2 in terms of robust deviation.
■ What is the worst case performance of the greedy
construction heuristic in terms of the number of connected
components in G − E T versus the number of connected
components in G − E T ∗?
Heuristics for the Central Tree Problem Yury Nikulin - p. 32/33

Conclusion
The Central Tree
Problem
Heuristics for the
CTP
Final Remarks
The results represent a very first attempt to attack the CTP with
heuristics. The proposed approach can be efficiently used to
solve zero-one RSTP. So, it may be considered as a first step
towards general intention to create a powerful methodological arsenal
to deal with interval RSTP of larger sizes.
Heuristics for the Central Tree Problem Yury Nikulin - p. 33/33