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Polynomially solvable cases of NP-hard problems

Polynomially solvable cases of NP-hard problems. Based on joint works with R.Burkard , D.Foster , B.Klinz , R.Rudolf , M.Sviridenko , J.Van der Veen , G. Woeginger. Vladimir Deineko. Discrete Optimization & OR 2013. Outline. Travelling Salesman Problem (TSP)

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Polynomially solvable cases of NP-hard problems

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  1. Polynomially solvable cases of NP-hard problems Based on joint works with R.Burkard, D.Foster, B.Klinz, R.Rudolf, M.Sviridenko, J.VanderVeen, G. Woeginger Vladimir Deineko Discrete Optimization & OR 2013

  2. Outline • Travelling Salesman Problem (TSP) • Four-point (4P) conditions - classification • Euclidean TSP with 4P conditions • Classification & Recognition • Summary: Further research opportunities

  3. Find a cyclic permutation  that minimizes 6 5 4 3 2 1 1 The travelling salesman problem (TSP) city3 city2 city5 city1 city4 city6 =<1,5,2,3,4,6,1> c()=c(1,5)+c(5,2)+c(2,3)+c(3,4)+c(4,6)+c(6,1)

  4. Outline of the first version of the talk • Travelling Salesman Problem (TSP) • Four-point (4P) conditions - classification • Euclidean TSP with 4P conditions • Classification & Recognition • Summary: Further research opportunities Discrete Optimization & OR 2013

  5. Outline So what? • Travelling Salesman Problem (TSP) • Four-point (4P) conditions - classification • Euclidean TSP with 4P conditions • Classification & Recognition Is it of any use for a wider community? Discrete Optimization & OR 2013

  6. Outline • Travelling Salesman Problem (TSP) • Four-point (4P) conditions - classification • Euclidean TSP with 4P conditions • Classification & Recognition • Special structures useful in other problems, e.g. • Master Tour problem • Exponential neighbourhoods and solvability conditions: • Optimal implementation of Double-tree algorithm • Techniques useful in other problems • Bipartite TSP • Real life OR problems Discrete Optimization & OR 2013

  7. 3 2 4 j k 5 1 l 6 i cij cik (cmn )= +  clj clk cij cik + + cjl clk   clj cil + + cik cjk TSP with specially structured matrices <1,2,…,n> is an optimal TSP tour (Kalmanson, 1975)

  8. Specially Structured Matrices: notations 1 2 3 4 5 6 1 - - - - - - - - - - - - - - - - - - - - - - - - - c23 + c3,4  c2,4 + c3.3 2 Specially structured matrices 3 4 5 6 i j l k

  9. j i k l O(n2) pyramidal tours NP-hard N-permutations O(n4) cij + clk  clj + cik Two-exchange and four-point conditions We consider the TSP with special matrices (cst) such that All permutations arbitrary tour τ

  10. Four Point Conditions for symmetric TSPs: Classification Demidenko conditions Supnick O(n) 1957 Kalmanson O(n) 1975 Pyramidal O(n2) 1976 NP-hard D,W 2000 Max Demidenko Pyramidal O(n2) 1994 Van derVeen conditions NP-hard Steiner et al 2005 Max Van der Veen NP-hard D,Tiskin 2006 Relaxed Supnick N-perm O(n4) D, 2004 Relaxed Kalmanson

  11. Four Point Conditions: Classification Demidenko conditions Pyramidal 1976 Pyramidal Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Kalmanson 1970 1,2,3,…,n Supnick 1957 1,3,5,…2,1 Pyramidal 1992 Similar to Supnick Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Similar to Supnick Van der Veen conditions NP-hard D,W 1997 NP-hard Supnick Max,1957 1971,87 Kalmanson Max,1970 Max Demidenko NP-hard Steiner et al 2005 New (linear) case Special Max Kalmanson Max Van der Veen N-perm O(n4) Special Lawler O(n2) Relaxed Kalmanson NP-hard D,T 2006 Relaxed Supnick

  12. dij dik (dmn )= dlj dlk Recognition of specially structured matrices + cij cik (cmn )= d(m)(n) =  X clj clk Is there a permutation  to permute rows and columns in the matrix so that the new permuted matrix (cmn) with cmn= d(m)(n) is a Relaxed Kalmanson (Kalmanson, Supnick, Demidenko ,…) matrix? 4 5 3 6 7 1 2 8

  13. Four Point Conditions for symmetric TSPs: Recognition Demidenko conditions O(n2log n) 1998 D,R,W O(n2)? C,F,T O(n4) 1999 Unpublished D,W Max Demidenko ? Van derVeen conditions Max Van der Veen ? Relaxed Supnick Relaxed Kalmanson

  14. Four Point Conditions for TSP: Recognition Euclidean Demidenko conditions O(n4) Van derVeen, D., Rudolf, Woeginger n<17 Conjecture: n<7. Max Demidenko O(n4) D., Burkard Van derVeen conditions n<17 Conjecture: n<7. Max Van der Veen n<17 or all are on the line Relaxed Supnick Conjecture: n<7. O(n4) D., Foster Sviridenko Relaxed Kalmanson

  15. Four Point Conditions for TSP: Recognition Euclidean Relaxed Kalmanson > = d(n1,n3 )-d(n2,n3 )≥ d(n1,n4 )-d(n2,n4 ) d(n1,n4 )-d(n2,n4 )= d(n1,n5 )-d(n2,n5 )

  16. Four Point Conditions for TSP: Recognition Euclidean Relaxed Kalmanson (i) Two branches of hyperbolea intersect in no more than 4 points. (ii) Two branches of hyperbolea with a common focal point intersect in no more than 2 points. Xu,Sahni,Rao, 2008 object localisation

  17. Four Point Conditions: Classification Demidenko conditions Pyramidal 1976 Pyramidal Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Kalmanson 1975 1,2,3,…,n Supnick 1957 1,3,5,…2,1 Pyramidal 1992 Similar to Supnick Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Similar to Supnick Van der Veen conditions NP-hard D,W 1997 NP-hard Supnick Max,1957 1971,87 Kalmanson Max,1970 Max Demidenko NP-hard Steiner et al 2005 New (linear) case Special Max Kalmanson Max Van der Veen N-perm O(n4) Special Lawler O(n2) Relaxed Kalmanson NP-hard D,T 2006 Relaxed Supnick

  18. Monge (Supnick) matrices Monge (1781): For an optimal transportation of goods from locations P1 and Q1 to locations P2 and Q2 the routes from P1 and from Q1 must not intersect. GaspardMonge, 1746-1818 P1 P2 d(P1,P2)+d(Q1,Q2)≤ d(P1,Q2)+d(Q1,P2) Monge 1781 - transportation Supnick 1956 - TSP Q1 Q2 Hoffman 1963 – transportation; introduced Monge; Burdjuk,Trofimov 1976 – TSP with permuted Monge matrices D.,Filonenko 1979 – recognition of Monge matrices Burkard, Klinz,Rudolf: Perspectives of Monge Properties in Optimization. Survey(1996) D.,Jonsson, Klasson, Krokhin: The approximability of MAX CSP with fixed-value constraints.) 2008

  19. Four Point Conditions: Classification Demidenko conditions Pyramidal 1976 Pyramidal Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Kalmanson 1975 1,2,3,…,n Supnick 1957 1,3,5,…2,1 Pyramidal 1992 Similar to Supnick Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Similar to Supnick Van der Veen conditions NP-hard D,W 1997 NP-hard Supnick Max,1957 1971,87 Kalmanson Max,1970 Max Demidenko NP-hard Steiner et al 2005 New (linear) case Special Max Kalmanson Max Van der Veen N-perm O(n4) Special Lawler O(n2) Relaxed Kalmanson NP-hard D,T 2006 Relaxed Supnick

  20. j k l i +  KalmansonMatrices:TSPwith the master tour An optimal TSP tour < 1, 2, 3,…, n, 1> is called the master tour, if it is an optimal tour and it remains to be an optimal after deleting any subset of cities.

  21. Given a set of customers Given a set of today’s customers A1 A1 PostCode 1 PostCode 1 A2 A2 PostCode 2 PostCode 2 A3 A3 PostCode 3 PostCode 3 A4 A4 PostCode 4 PostCode 4 A5 A5 PostCode 5 PostCode 5 A6 A6 PostCode 6 PostCode 6 A7 A7 PostCode 7 PostCode 7 A8 A8 PostCode 8 PostCode 8 A9 A9 PostCode 9 PostCode 9 A10 A10 PostCode 10 PostCode 10 A11 A11 PostCode 11 PostCode 11 A12 A12 PostCode 12 PostCode 12 A13 A13 PostCode 13 PostCode 13 A14 A14 PostCode 14 PostCode 14 A15 A15 PostCode 15 PostCode 15 Illustration to Master Tour problem TSP: Find a tour with the minimal total length ???

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

  23. j k l i Kalmanson Matrices: TSP with the master tour An optimal TSP tour < 1, 2, 3,…, n, 1> is called the master tour, if it is an optimal tour and it remains to be an optimal after deleting any subset of cities. Conjecture (Papadimitriou, 1983) The master tour problem is ∑2P-complete. Sometimes travelling is easy: D., Rudolf, Woeginger, 1998 For a distance matrix C, a tour < 1, 2, 3,…, n, 1> is the master tour, if and only if C is a Kalmanson matrix. Permuted Kalmansonmatrices can be recognized in O(n2) time

  24. Four Point Conditions: Classification Demidenko conditions Pyramidal 1976 Pyramidal Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Kalmanson 1975 1,2,3,…,n Supnick 1957 1,3,5,…2,1 Pyramidal 1992 Similar to Supnick Sum Matrix c(i,j)=u(i)+v(j) Kalmanson subclass Similar to Supnick Van der Veen conditions NP-hard D,W 1997 NP-hard Supnick Max,1957 1971,87 Kalmanson Max,1970 Max Demidenko NP-hard Steiner et al 2005 New (linear) case Special Max Kalmanson Max Van der Veen N-perm O(n4) Special Lawler O(n2) Relaxed Kalmanson NP-hard D,T 2006 Relaxed Supnick

  25. n n-1 6 5 4 3 2 1 1 cij cij cik cik n (cmn )= (cmn )= (cmn )= Supnick (1957) matrices  S*=<1,3,5,7…,n,…,6,4,2> is an optimal TSP tour    +  8 clj clj clk clk 7 6 5 4 2 3 1 1 Demidenko (1976)  * is a pyramidal tour Specially structured matrices & Exponential neighbourhoods Kalmanson (1970)  K*=<1,2,…,n> is an optimal TSP tour

  26. Structure of dynamic programming recursions: P1 P2 P1 P1 P2 Demidenko TSP: Demidenko,1979: An optimal TSP tour can be found among 2n-2 pyramidal tours in O(n2) time n 2 5 6 7 8 9 3 4 P1(i,j)=min{ci,j+1+P2(j,j+1), cj+1,j+P1(i,j+1)} P2(i,j)=min{cj,j+1+P2(i,j+1), cj+1,i+P1(j,j+1)} P1(i,n)=cj,n P2(i,n)=cj,n 1 1

  27. begin Compute the minimum spanning tree; Double every edge in the tree to get an Eulerian graph; Find an Eulerian circuit and transform the circuit into a TSP tour by shortcutting: for every city remove all but one of its occurrence in the Eulerian circuit. end A C B F(9,6) E A(5,8) D H C(2,7) B(5,6) E(3,4) G(9,4) I D(0,4) A H(6,2) C I(6,0) F B G E D H I F G Double Tree algorithm & Exponential neighbourhoods <IHEBABECEDEHGFGH I> <IHEBA C D GF I>

  28. Theorem (Folklore). A tour tree found by the Double Tree Algorithm is guaranteed to have no more than twice the length of the optimal tour opt for the TSP. 1 4 A A 2 C C B F B F 5 6 3 E E G G D D 1 1 H H tree n I I A A C C B B F F E E G D ~2n+2n(1- ) D H H opt I I 1 m 4 2 ~2n+2 5 6 3 2 is the tight bound for the Double Tree Algorithm Bad News Good News

  29. A A C C B F B F E E G G D D H H I I A A C C B B F F E E G D D H H I I A C F B E G D H I Is it possible to find in polynomial time the best tour among all tours constructed by the Double-Tree Algorithm?

  30. Double tree for metric TSP: Optimal implementation Burkard, D.,Weginger, 1999, TSP & PQ-trees; O(n3)time, O(n2)space D., Tiskin, 2009, An optimal tour amongst of all those constructed by the tree algorithm can be found in O(2dn2)time and O(22dn)space, where d is a maximum vertex degree in the spanning tree. “Conjecture” (Papadimitiou, Vazirani, 1986) The problem of finding the best tour among all tours constructed by the Double-Tree Algorithm is NP-hard.

  31. A Pyramidal Tours C E A C B F D I G F E D E I G C G D F H A H I Set of all n! tours O(2dn2) algorithm +”good” heuristic O(n2 2n)well known DP algorithm +exact solution O(n2)algorithm +Special solvable cases A C B F E G D H ... I n n-1 n-2 2 1 Dynamic programming for the TSP DTd

  32. The best tour constructing heuristic x

  33. The bipartite travelling salesman problem (BTSP) city3 city2 city5 “black” and “white” points have to alternate in a feasible tour city1 city4 city6 Shoe-lace problem (Halton,1995; Misiurewicz, 1996)

  34. The bipartite travelling salesman problem (BTSP) point2 point1 point3 item1 item3 item2

  35. point2 point1 point3 item1 item3 item2

  36. point2 point1 point3 item1 item3 item2

  37. point2 point1 point3 item1 item3 item2

  38. point2 point1 point3 item1 item3 item2

  39. point2 point1 point3 item1 item3 item2

  40. point2 point1 point3 item1 item3 item2

  41. point2 point1 point3 item1 item3 item2

  42. Transformation technique Arbitrary permutation 0 c(0) c(1) 1 set of permutations c(1) c(2) 2 special subset

  43. Bipartite Travelling Salesman, or ShoeLace Problem for very old shoes V.D.,G.Woeginger

  44. 7 7 8 8 9 9 10 10 12 12 1 1 -5.6 -0.2 -0.5 0.2 -0.0 -6.0 -10.6 1.5 -1.1 -0.5 2 2 -8.5 -13.5 -0.2 -5.4 +0.0 -3.0 2.1 -4.2 1.4 1.2 -5.7 -0.2 -0.6 -11.9 -0.8 -0.5 -5.4 0.8 -0.8 -0.3 4 4 -0.7 -5.5 -0.1 -9.4 -4.0 -5.7 0.1 -12.3 11.3 -0.0 5 5 -0.2 0.1 -0.4 -2.7 -4.4 5.2 -0.6 7.7 -13.8 -2.5 6 6

  45. Summary “Byproducts” • Travelling Salesman Problem (TSP) • Four-point (4P) conditions - classification • Euclidean TSP with 4P conditions • Classification & Recognition • Special structures useful in other problems, e.g. • Master Tour problem • Exponential neighbourhoods and solvability conditions, • Optimal implementation of Double-tree algorithm • Techniques useful in other problems • Bipartite TSP • Real life OR problems

  46. Transformation technique for Bipartite TSP

  47. Transformation technique

  48. Recognition of special cases

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