Master Tour Routing Vladimir Deineko, Warwick Business School

Outline Vehicle routing Master tour problem Travelling Salesman Problem with Kalmanson matrices Quadratic Assignment Problem/ Special Case Summary

3Warwick Business School Vehicle routing problem PostCode 15A15 PostCode 14A14 PostCode 13A13 PostCode 12A12 PostCode 11A11 PostCode 10A10 PostCode 9A9 PostCode 8A8 PostCode 7A7 PostCode 6A6 PostCode 5A5 PostCode 4A4 PostCode 3A3 PostCode 2A2 PostCode 1A1 Given a set of customers Find a tour with the minimal total length PostCode 15A15 PostCode 14A14 PostCode 13A13 PostCode 12A12 PostCode 11A11 PostCode 10A10 PostCode 9A9 PostCode 8A8 PostCode 7A7 PostCode 6A6 PostCode 5A5 PostCode 4A4 PostCode 3A3 PostCode 2A2 PostCode 1A1 Given a set of today’s customers ???

4Warwick Business School Vehicle routing problem PostCode 15A15 PostCode 14A14 PostCode 13A13 PostCode 12A12 PostCode 11A11 PostCode 10A10 PostCode 9A9 PostCode 8A8 PostCode 7A7 PostCode 6A6 PostCode 5A5 PostCode 4A4 PostCode 3A3 PostCode 2A2 PostCode 1A1 Given a set of customers Find a tour with the minimal total length PostCode 15A15 PostCode 14A14 PostCode 13A13 PostCode 12A12 PostCode 11A11 PostCode 10A10 PostCode 9A9 PostCode 8A8 PostCode 7A7 PostCode 6A6 PostCode 5A5 PostCode 4A4 PostCode 3A3 PostCode 2A2 PostCode 1A1 Given a set of today’s customers ???

5Warwick Business School The travelling salesman problem (TSP) city3city2city5 city1 city6city4 Find a cyclic permutation (tour)  that minimizes An optimal TSP tour is called the master tour, if it is an optimal tour and it remains to be an optimal after deleting any subset of cities.

6Warwick Business School c ij c ik c lk (c mn )= c lj TSP with the master tour j i k l  + c lk c ij  + c lj + c ik c jl c ik  + c il + c jk A matrix is called Kalmanson matrix if for all i<j<k<l the inequalities below are satisfied.

7Warwick Business School c ij c ik c lk (c mn )= c lj Specially structured matrices j i k l  + If (c mn ) is a Kalmanson matrix, then is an optimal TSP tour is the master tour for the TSP with (c mn ) then (c mn ) is a Kalmanson matrix

Recognition of specially structured matrices d ij d ik d lk (d mn )= d lj Is there a permutation  to permute rows and columns in the matrix so that the new permuted matrix (c mn ) with c mn = d  (m)  (n) is a Kalmanson matrix? c ij c ik c lk (c mn )= d  (m)  (n) = c lj  X −  K

Recognition of specially structured matrices d ij d ik d lk (d mn )= d lj c ij c ik c lk (c mn )= c lj Permuted Kalmanson matrices can be recognized in O(n 2 ) time  +

Site 1 Site 2 Site 4 Site 3 Related problems: Quadratic Assignment Problem (QAP)

Site 1 Site 2 Site 4 Site 3 Distance matrix d(i,j)=

1212 Warwick Business School Office1 Office2 Office3 Office4 Site1 Site2 Site3 Site4 11 22 33 Distance matrix d(i,j)= Frequency of contacts c(i,j)=

1313 Warwick Business School Quadratic Assignment Problem Find a permutation  that minimizes total distance traveled in the allocation problem above

Quadratic Assignment Problem (QAP) Find a permutation  that minimizes NP-hard  little hopes to find a polynomial algorithm The hardest solved instances  n=22 (30?) Heuristics (approximate algorithms) Identify solvable cases

1515 Warwick Business School Distance matrix d(i,j)= Frequency of contacts c(i,j)= Quadratic assignment problem: Solvable case If the distance matrix d(i,j) is a Kalmanson matrix, and the frequencies c(i,j) are proportional to the distances along a circle, then the identity is an optimal permutation for the QAP

Related specially structured matrices c ij c ik c lk (c mn )= c lj  + Kalmanson matrices   K *= is an optimal TSP tour c ij c ik c lk (c mn )= c lj Demidenko matrices   * is a pyramidal tour  O(n 2 ) time  (c mn )= + Relaxed Kalmanson matrices   S * is in a special set of N-permutations  O(n 4 ) time 1 1 n

Specially Structured Matrices & Heuristics x x x x x ? ?? ?? ? x x x x Know how to solve the TSP with the matrices like 1'1' 2'2' 3'3' 4'4' 5'5' 6'6' x x x x x '1' 2'2' 3'3' 4'4' 5'5' 6'6' x x x x x x x x x + ++ ?? ? 3'3' 4'4' 5'5' 6'6' 3'3' 4'4' 5'5' 6'6' can be transformed to x x x x local search?

1818 Warwick Business School Summary Master tour exists only for the TSP with Kalmanson matrices If distances are calculated along the unique paths in a tree, then the corresponding matrix is the Kalmanson matrix Kalmanson matrices (  the master tour case) can be recognised in O(n 2 ) time