[PPT] - Heuristic and exact approaches to the Quadratic Minimum Spanning PowerPoint Presentation

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7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization May 13-15, 2008 (Gargnano, Italy)

Heuristic and exact approaches to the Quadratic Minimum Spanning Tree Problem

Roberto Cordone Gianluca Passeri

Dipartimento di Tecnologie dell’Informazione Universit` a degli Studi di Milano

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Definition of the QMSTP

Let

G = (V , E) a connected undirected graph (n = |V | and m = |E|)
c : E → Z a linear cost function
q : E × E → Z a quadratic cost function (qee = 0 and qef = qfe)

Find

a spanning tree T = (V , X)

Minimize

the total cost zX =

e∈X

ce +

e,f ∈X

qef

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

An IQP formulation

Objective function min z(x) =

e∈E

ce · xe +

e∈E
f ∈E

qef · xe · xf Constraints Acyclicity :

e∈E(S)

xe ≤ |S| − 1 S ⊆ V , |S| ≥ 2 Cardinality :

e∈E

xe = n − 1 Integrality : xe ∈ {0, 1} e ∈ E

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

An IQP formulation

Objective function min z(x) =

e∈E

ce · xe +

e∈E
f ∈E

qef · xe · xf Constraints Acyclicity :

e∈E(S)

xe ≤ |S| − 1 S ⊆ V , |S| ≥ 2 Cardinality :

e∈E

xe = n − 1 Integrality : xe ∈ {0, 1} e ∈ E

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Computational complexity

Strongly NP-complete: reduction from SAT

1 A vertex xi for each variable, a vertex cl for each clause 2 An edge for each occurrence (xi, cl) of variable xi in clause cl 3 An edge for each pair (xi, xi+1) with i = 1 . . . n − 1

f = (x1 + x2 + x4) · (x1 + x2) · (x4) · (x1 + x3 + x4)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Computational complexity

Strongly NP-complete: reduction from SAT

4 ce = 0 for all e ∈ E 5 qef = 1 when e and f are opposite occurrences of the same variable 6 qef = 0 for all other pairs of edges

f = (x1 + x2 + x4) · (x1 + x2) · (x4) · (x1 + x3 + x4)

q = 1 for (x1, c1, x1, c2) (x1, c2, x1, c4) (x2, c1, x2, c2) (x4, c1, x4, c3) (x4, c1, x4, c4)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Computational complexity

Strongly NP-complete: reduction from SAT

An edge identifies a satisfying occurrence
A spanning tree identifies a satisfying assignment
If zX = 0, tree X employs only zero cost edges
Zero cost edges identify reciprocally consistent occurrences

f = (x1 + x2 + x4) · (x1 + x2) · (x4) · (x1 + x3 + x4)

q = 1 for (x1, c1, x1, c2) (x1, c2, x1, c4) (x2, c1, x2, c2) (x4, c1, x4, c3) (x4, c1, x4, c4)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Approximability and number of solutions

Approximability

Since the optimal cost is z∗ = 0,

the QMSTP is non-approximable unless P = NP Number of solutions

It is exactly given by Kirchoff’s theorem (1847)
For complete graphs (Cayley’s theorem)

|T | = nn−2

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Approximability and number of solutions

Approximability

Since the optimal cost is z∗ = 0,

the QMSTP is non-approximable unless P = NP Number of solutions

It is exactly given by Kirchoff’s theorem (1847)
For complete graphs (Cayley’s theorem)

|T | = nn−2

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Survey on the literature

simple greedy heuristics (Xu, 1995)
genetic algorithm (Zhou and Gen, 1998)
two genetic algorithms for the fuzzy QMSTP (Gao and Lu, 2005)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Algorithmic approaches developed

1 ILP formulation (standard linearization) 2 Heuristic approaches

a) Average contribution method (constructive) b) Minimum contribution method (constructive) c) Sequential fixing method (constructive and adaptive) d) Tabu Search

3 Exact approach

Branch and Bound (combinatorial bound)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Algorithmic approaches developed

1 ILP formulation (standard linearization) 2 Heuristic approaches

a) Average contribution method (constructive) b) Minimum contribution method (constructive) c) Sequential fixing method (constructive and adaptive) d) Tabu Search

3 Exact approach

Branch and Bound (combinatorial bound)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Definition of the QMSTP An IQP formulation Computational complexity Approximability and number of solutions Survey Algorithmic approaches developed

Algorithmic approaches developed

1 ILP formulation (standard linearization) 2 Heuristic approaches

a) Average contribution method (constructive) b) Minimum contribution method (constructive) c) Sequential fixing method (constructive and adaptive) d) Tabu Search

3 Exact approach

Branch and Bound (combinatorial bound)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Common elements for the constructive heuristics

Starting from the quadratic objective function. . . min z(x) =

e∈E

ce · xe +

e∈E
f ∈E

qef · xe · xf . . . approximate it with a linear one. . . z(x) =

e∈E

xe ·

ce +
f ∈E

qef · xf

≈
e∈E

˜ ce · xe where ˜ ce ≈ ce +

f ∈E

qef · xf e ∈ E . . . and solve the resulting MSTP z(x) = min

x∈T

e∈E

˜ ce · xe

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Average Contribution Method

Objective function z(x) =

e∈E

xe ·

ce +
f ∈E

qef · xf

Approximated objective function

˜ ce = ce + (n − 2) ·

f ∈E

qef m − 1 e ∈ E Complexity Θ(m2 + m log n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Minimum Contribution Method

Objective function z(x) =

e∈E

xe ·

ce +
f ∈E

qef · xf

Approximated objective function

˜ ce = ce +

f ∈E ∗

e

qef e ∈ E E ∗

e includes the n − 2 edges with minimum qef

Complexity Θ(m2 log n + m log n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Sequential Fixing Method

Objective function z(x) =

e∈E

xe ·

ce +
f ∈E

qef · xf

Approximated (adaptive) objective function

Set X := ∅ and F := E Move from F to X the edge with minimum ˜ ce = ce +

f ∈X

qef +(n−2−|X|)·

f ∈F\{e}

qef |F| − 1 e ∈ F Remove from F the edges which close loops with X Complexity Θ(m2n + mnα(m, n))

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Local search

Main elements

Neighbourhood:
ne edge in / one edge out
Feasible edges out (for each edge in):

loop formed by the edge in

Evaluation in Θ(1) time of the
bjective function variation

Move Complexity of a move O((m − n) · n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Local search

Main elements

Neighbourhood:
ne edge in / one edge out
Feasible edges out (for each edge in):

loop formed by the edge in

Evaluation in Θ(1) time of the
bjective function variation

Move Complexity of a move O((m − n) · n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Local search

Main elements

Neighbourhood:
ne edge in / one edge out
Feasible edges out (for each edge in):

loop formed by the edge in

Evaluation in Θ(1) time of the
bjective function variation

Move Complexity of a move O((m − n) · n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Local search

Main elements

Neighbourhood:
ne edge in / one edge out
Feasible edges out (for each edge in):

loop formed by the edge in

Evaluation in Θ(1) time of the
bjective function variation

Move Complexity of a move O((m − n) · n)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

X E \ X

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X De

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X Df

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X −De

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X +Df

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X −qef

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Evaluation of a move

Auxiliary data structure

The contribution of each edge e ∈ E to the total

cost is De = ce +

f ∈X

qef as zX =

e∈X

De

Replace e by f : the total cost z

1 decreases by De 2 increases by Df 3 decreases by qef

zX(new) = zX(old) − De + Df − qef

Finally, update Di for all i ∈ E

Di := Di − qie + qif

e f X E \ X i +qif −qie

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Tabu Search

Main elements

Tabu attribute: last iteration in/out

for each edge

Two independent tabu lists: longer tabu for insertion

than for deletion (E \ X ≫ X)

Adaptive tabu tenure:
increasing when solution worsens
decreasing when solution improves
Stop: maximum number of iterations

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Tabu Search

Main elements

Tabu attribute: last iteration in/out

for each edge

Two independent tabu lists: longer tabu for insertion

than for deletion (E \ X ≫ X)

Adaptive tabu tenure:
increasing when solution worsens
decreasing when solution improves
Stop: maximum number of iterations

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions The general constructive approach Average Contribution Method Minimum Contribution Method Sequential Fixing Method General features of the local search algorithm

Tabu Search

Main elements

Tabu attribute: last iteration in/out

for each edge

Two independent tabu lists: longer tabu for insertion

than for deletion (E \ X ≫ X)

Adaptive tabu tenure:
increasing when solution worsens
decreasing when solution improves
Stop: maximum number of iterations

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Main elements

Lower bound: linear approximation from below →

Kruskal’s algorithm

Upper bound: Kruskal’s solution provides one for free
Branching edge:

cheapest unfixed edge in the relaxed solution

Visit strategy: Best-Lower-Bound first

(or hybrid: Best-Upper-Bound first followed by Best-Lower-Bound first)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Main elements

Lower bound: linear approximation from below →

Kruskal’s algorithm

Upper bound: Kruskal’s solution provides one for free
Branching edge:

cheapest unfixed edge in the relaxed solution

Visit strategy: Best-Lower-Bound first

(or hybrid: Best-Upper-Bound first followed by Best-Lower-Bound first)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Main elements

Lower bound: linear approximation from below →

Kruskal’s algorithm

Upper bound: Kruskal’s solution provides one for free
Branching edge:

cheapest unfixed edge in the relaxed solution

Visit strategy: Best-Lower-Bound first

(or hybrid: Best-Upper-Bound first followed by Best-Lower-Bound first)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Main elements

Lower bound: linear approximation from below →

Kruskal’s algorithm

Upper bound: Kruskal’s solution provides one for free
Branching edge:

cheapest unfixed edge in the relaxed solution

Visit strategy: Best-Lower-Bound first

(or hybrid: Best-Upper-Bound first followed by Best-Lower-Bound first)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Lower bounds implemented

Relaxation

z(x) =

e∈E

ce xe +

e∈E
f ∈E

qef xe xf ≥ z1(x), z2(x), z3(x), ∀x ∈ T

Three lower bounds

z1(x) =

e∈F

ce · xe z2(x) =

e∈X

ce +

e∈X
f ∈X

qef +

e∈F

xe ·

ce +
e∈X

qef

z3(x) =
e∈X

ce +

e∈X
f ∈X

qef +

e∈F

xe ·

ce +
e∈X

qef

+

+

e∈F

xe ·

f ∈X ∗

e

qef

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Lower bounds implemented

Relaxation

z(x) =

e∈E

ce xe +

e∈E
f ∈E

qef xe xf ≥ z1(x), z2(x), z3(x), ∀x ∈ T

Three lower bounds

z1(x) =

e∈F

ce · xe z2(x) =

e∈X

ce +

e∈X
f ∈X

qef +

e∈F

xe ·

ce +
e∈X

qef

z3(x) =
e∈X

ce +

e∈X
f ∈X

qef +

e∈F

xe ·

ce +
e∈X

qef

+

+

e∈F

xe ·

f ∈X ∗

e

qef

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Branch and bound algorithm

Comparison between three lower bounds

Comparison between the three lower bounds. The graph considered has n = 10, 67% density and c, q ∈ [1; 100]

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions System employed Instances Heuristic algorithms ILP solver vs B&B

Machine features

Machine features Processor 2 Dual Core AMD Opteron Processor 275 2.2GHz RAM 3 Gb HD 250 Gb Operating system Linux Language ANSI-C Compiler gcc

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions System employed Instances Heuristic algorithms ILP solver vs B&B

Instance features

Graph properties

Number of vertices n
Density ρ = 2m/n (n − 1)
Linear costs c uniformly random
Quadratic q uniformly random

Values

5 classes: 10, 15, 20, 25, 30
3 classes: 33%, 67%, 100%
2 classes: 1-10, 1-100
2 classes: 1-10, 1-100

Total number of instances 5 · 3 · 2 · 2 = 60

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions System employed Instances Heuristic algorithms ILP solver vs B&B

Average gap of the heuristic algorithms

Average gap wrt the best known result Vertici ACM MCM SFM TS(SFM) 10 24,70% 27,50% 3,68% 0,00% 15 21,20% 28,74% 4,66% 0,00% 20 22,14% 28,74% 5,47% 0,05% 25 23,31% 30,40% 4,42% 0,15% 30 26,16% 31,21% 4,09% 0,21% Computational time: < 1 sec for the constructive algorithms, < 1 min for Tabu Search (100000 iterations)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions System employed Instances Heuristic algorithms ILP solver vs B&B

ILP solver and Branch and Bound compared

Average gap between upper and lower bound after 2 hours (branch-and-bound initialized by Tabu Search, and this by SFM) Density 33% 67% 100% n Solver B&B Solver B&B Solver B&B 10 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 15 0.00% 0.00% 56.60% 10.39% 95.93% 20.40% 20 59.12% 16.98% 92.58% 37.98% 98.40% 41.54% 25

34.30%
54.07%
57.15%

30

46.37%
60.67%
69.11%

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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SLIDE 48

The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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SLIDE 49

The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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The Problem Heuristic approaches Exact approach Computational experiments Conclusions Conclusions and future developments

Conclusions and future developments

Conclusions

ILP formulation impractical
Constructive algorithms fast, but quite ineffective
Local search fast and effective (not always optimal)
Branch and bound algorithm: viable for small instances (n ≤ 15)

Future developments

Tighter lower bounds (combinatorial, Lagrangean, semidefinite)
Quadratic Minimum Spanning Arborescence (QMSAP)

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