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 1 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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 ( q ee = 0 and q ef = q fe ) Find • a spanning tree T = ( V , X ) Minimize • the total cost z X = � c e + � q ef e ∈ X e , f ∈ X 2 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions Algorithmic approaches developed An IQP formulation Objective function � � � c e · x e + q ef · x e · x f min z ( x ) = e ∈ E e ∈ E f ∈ E Constraints � Acyclicity : x e ≤ | S | − 1 S ⊆ V , | S | ≥ 2 e ∈ E ( S ) � Cardinality : x e = n − 1 e ∈ E Integrality : x e ∈ { 0 , 1 } e ∈ E 3 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions Algorithmic approaches developed An IQP formulation Objective function � � � c e · x e + q ef · x e · x f min z ( x ) = e ∈ E e ∈ E f ∈ E Constraints � Acyclicity : x e ≤ | S | − 1 S ⊆ V , | S | ≥ 2 e ∈ E ( S ) � Cardinality : x e = n − 1 e ∈ E Integrality : x e ∈ { 0 , 1 } e ∈ E 3 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions Algorithmic approaches developed Computational complexity Strongly NP -complete: reduction from SAT 1 A vertex x i for each variable, a vertex c l for each clause 2 An edge for each occurrence ( x i , c l ) of variable x i in clause c l 3 An edge for each pair ( x i , x i +1 ) with i = 1 . . . n − 1 f = ( x 1 + x 2 + x 4 ) · ( x 1 + x 2 ) · ( x 4 ) · ( x 1 + x 3 + x 4 ) 4 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions Algorithmic approaches developed Computational complexity Strongly NP -complete: reduction from SAT 4 c e = 0 for all e ∈ E 5 q ef = 1 when e and f are opposite occurrences of the same variable 6 q ef = 0 for all other pairs of edges f = ( x 1 + x 2 + x 4 ) · ( x 1 + x 2 ) · ( x 4 ) · ( x 1 + x 3 + x 4 ) q = 1 for ( x 1 , c 1 , x 1 , c 2 ) ( x 1 , c 2 , x 1 , c 4 ) ( x 2 , c 1 , x 2 , c 2 ) ( x 4 , c 1 , x 4 , c 3 ) ( x 4 , c 1 , x 4 , c 4 ) 5 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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 z X = 0, tree X employs only zero cost edges • Zero cost edges identify reciprocally consistent occurrences f = ( x 1 + x 2 + x 4 ) · ( x 1 + x 2 ) · ( x 4 ) · ( x 1 + x 3 + x 4 ) q = 1 for ( x 1 , c 1 , x 1 , c 2 ) ( x 1 , c 2 , x 1 , c 4 ) ( x 2 , c 1 , x 2 , c 2 ) ( x 4 , c 1 , x 4 , c 3 ) ( x 4 , c 1 , x 4 , c 4 ) 6 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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 | = n n − 2 7 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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 | = n n − 2 7 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions Algorithmic approaches developed Survey on the literature 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) 8 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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) 9 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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) 9 / 25
Definition of the QMSTP The Problem An IQP formulation Heuristic approaches Computational complexity Exact approach Approximability and number of solutions Computational experiments Survey Conclusions 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) 9 / 25
The Problem The general constructive approach Heuristic approaches Average Contribution Method Exact approach Minimum Contribution Method Computational experiments Sequential Fixing Method Conclusions General features of the local search algorithm Common elements for the constructive heuristics Starting from the quadratic objective function. . . � � � min z ( x ) = c e · x e + q ef · x e · x f e ∈ E e ∈ E f ∈ E . . . approximate it with a linear one. . . � � � � � x e · q ef · x f ≈ c e · x e z ( x ) = c e + ˜ e ∈ E f ∈ E e ∈ E � c e ≈ c e + q ef · x f e ∈ E where ˜ f ∈ E . . . and solve the resulting MSTP � z ( x ) = min ˜ c e · x e x ∈T e ∈ E 10 / 25
The Problem The general constructive approach Heuristic approaches Average Contribution Method Exact approach Minimum Contribution Method Computational experiments Sequential Fixing Method Conclusions General features of the local search algorithm Average Contribution Method Objective function � � � � z ( x ) = x e · c e + q ef · x f e ∈ E f ∈ E Complexity Approximated objective function Θ( m 2 + m log n ) � q ef f ∈ E c e = c e + ( n − 2) · e ∈ E ˜ m − 1 11 / 25
The Problem The general constructive approach Heuristic approaches Average Contribution Method Exact approach Minimum Contribution Method Computational experiments Sequential Fixing Method Conclusions General features of the local search algorithm Minimum Contribution Method Objective function � � � � z ( x ) = x e · c e + q ef · x f e ∈ E f ∈ E Complexity Approximated objective function Θ( m 2 log n + m log n ) � e ∈ E ˜ c e = c e + q ef f ∈ E ∗ e E ∗ e includes the n − 2 edges with minimum q ef 12 / 25
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