Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work A traveling salesman problem with quadratic cost structure Anja Fischer, Christoph Helmberg Chemnitz University of Technology
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Outline Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Outline Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Problem description of QSTSP Given: • undirected complete 2-graph G = ( V , E ) , V = { 1 , . . . , n } , V 2 := {{ i , j } : i , j ∈ V , i � = j } (write ij ) – arcs V 3 := {� i , j , k � = � k , j , i � : i , j , k ∈ V , |{ i , j , k }| = 3 } (write ijk ) – 2-arcs • cost function c : V 3 → R + with c ijk – costs of path i − j − k , ⇒ quadratic cost structure Goal: find tour T = ( i 1 , . . . , i n , i 1 ) minimizing n − 2 � c i k i k +1 i k +2 + c i n − 1 i n i 1 + c i n i 1 i 2 k =1 4 5 costs c 2 , 1 , 4 1 3 2
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Introduction Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Introduction Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen)
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Introduction Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen) Special case of QSTSP Angular-Metric TSP, see Aggarwal et al. (1997) given points in the plane – find tour minimizing total direction changes applications in robotic
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Introduction 10000 10000 28 10 30 9000 9000 15 29 9 3 26 8000 8000 14 19 7 12 24 27 7000 7000 2 7 4 1 18 5 12 20 6000 23 6000 6 6 13 13 9 17 10 24 5000 5000 16 8 1 16 28 4000 4000 25 25 19 8 3000 3000 4 2 11 11 2000 2000 23 22 17 14 1000 1000 15 26 18 3 30 20 27 21 29 5 21 0 22 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Introduction Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen) Special case of QSTSP Angular-Metric TSP, see Aggarwal et al. (1997) given points in the plane – find tour minimizing total direction changes applications in robotic Complexity NP-complete, even the corresponding cycle cover problem is NP-complete
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Outline Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Integer Linear Program Linearization of the following quadratic integer model � min c ijk x ij x jk ���� ijk ∈ V 3 y ijk � s.t. x ij = 2 , i ∈ V (degree) ij ∈ V 2 � x ij ≥ 2 , S ⊂ V , 2 ≤ | S | ≤ n − 2 (subtour) ij ∈ V 2 i ∈ S , j ∈ V \ S ij ∈ V 2 x ij ∈ { 0 , 1 } ,
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Integer Linear Program Linearization of the following quadratic integer model � min c ijk x ij x jk ���� ijk ∈ V 3 y ijk � s.t. x ij = 2 , i ∈ V (degree) ij ∈ V 2 � x ij ≥ 2 , S ⊂ V , 2 ≤ | S | ≤ n − 2 (subtour) ij ∈ V 2 i ∈ S , j ∈ V \ S ij ∈ V 2 x ij ∈ { 0 , 1 } , � ij ∈ V 2 x ij = y ijk , (flow) ijk ∈ V 3 � ij ∈ V 2 x ij = y kij , (flow) kij ∈ V 3 ijk ∈ V 3 y ijk ∈ [0 , 1] ,
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QSTSP � n � � n � � n � = n 2 variables: 3 + equality constraints: n + 2 3 2 2 Observation The constraint matrix of the QSTSP (degree, flow) has full row rank.
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QSTSP � n � � n � � n � = n 2 variables: 3 + equality constraints: n + 2 3 2 2 Observation The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope P QSCCP n (subtours are allowed). Lemma � n � � n � − n 2 for n ≥ 7 . The dimension of P QSCCP n equals 3 + 3 2
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QSTSP � n � � n � � n � = n 2 variables: 3 + equality constraints: n + 2 3 2 2 Observation The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope P QSCCP n (subtours are allowed). Lemma � n � � n � − n 2 for n ≥ 7 . The dimension of P QSCCP n equals 3 + 3 2 Conjecture � n � � n � − n 2 for n ≥ 7. The dimension of P QSTSP n equals 3 + 3 2
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QSTSP � n � � n � � n � = n 2 variables: 3 + equality constraints: n + 2 3 2 2 Observation The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope P QSCCP n (subtours are allowed). Lemma � n � � n � − n 2 for n ≥ 7 . The dimension of P QSCCP n equals 3 + 3 2 Conjecture � n � � n � − n 2 for n ≥ 7. The dimension of P QSTSP n equals 3 + 3 2 For the STSP: Gr¨ otschel and Padberg used an arc-disjoint Hamiltonian cycle decomposition of the complete graph G n .
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QSTSP Question Is there a 2-arc-disjoint Hamiltonian. cycle decomposition of the complete 2-graph G n , n ≥ 3? Related to an open question (Bailey, Stevens) concerning the decomposition of complete uniform hypergraphs into arc-disjoint Hamiltonian cycles.
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QATSP resp. P QACCP n � � x ( j , i ) = x ( i , j ) = 1 , i ∈ V , ( j , i ) ∈ V 2 ( i , j ) ∈ V 2 � � ( i , j ) ∈ V 2 , x ( i , j ) = y ( i , j , k ) = y ( k , i , j ) , ( i , j , k ) ∈ V 3 ( k , i , j ) ∈ V 3 � x ( i , j ) ≥ 1 , S ⊂ V , 1 ≤ | S | ≤ n − 1 , ( i , j ) ∈ V 2 : i ∈ S , j ∈ V \ S ( i , j ) ∈ V 2 , ( i , j , k ) ∈ V 3 x ( i , j ) ∈ { 0 , 1 } , y ( i , j , k ) ∈ [0 , 1] , V 2 = { ( i , j ): i , j ∈ V , i � = j } , V 3 = { ( i , j , k ): i , j , k ∈ V , |{ i , j , k }| = 3 }
Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work Dimension of the polytope P QATSP resp. P QACCP n � � x ( j , i ) = x ( i , j ) = 1 , i ∈ V , ( j , i ) ∈ V 2 ( i , j ) ∈ V 2 � � ( i , j ) ∈ V 2 , x ( i , j ) = y ( i , j , k ) = y ( k , i , j ) , ( i , j , k ) ∈ V 3 ( k , i , j ) ∈ V 3 � x ( i , j ) ≥ 1 , S ⊂ V , 1 ≤ | S | ≤ n − 1 , ( i , j ) ∈ V 2 : i ∈ S , j ∈ V \ S ( i , j ) ∈ V 2 , ( i , j , k ) ∈ V 3 x ( i , j ) ∈ { 0 , 1 } , y ( i , j , k ) ∈ [0 , 1] , V 2 = { ( i , j ): i , j ∈ V , i � = j } , V 3 = { ( i , j , k ): i , j , k ∈ V , |{ i , j , k }| = 3 } Lemma − (2 n 2 − 1 − n ) The dimension of P QACCP n equals n ( n − 1) 2 for n ≥ 7 . � �� � � �� � # variables rank of constr. matrix Conjecture The dimension of P QATSP n equals n ( n − 1) 2 − (2 n 2 − 1 − n ) for n ≥ 7.
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