1. Polynomially solvable cases of NP-hard problems
Vladimir Deineko
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
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. c()=c(1,5)+c(5,2)+c(2,3)+c(3,4)+c(4,6)+c(6,1)
The travelling salesman problem (TSP) 1 2 3 4 5 6 30 45 53 58 42 20 36 50 35 37 16 24 17
city3
city2
city5
city1
city4
city6
Find a cyclic permutation  that minimizes
=<1,5,2,3,4,6,1> 1 5 6 2 3 4
c()=c(1,5)+c(5,2)+c(2,3)+c(3,4)+c(4,6)+c(6,1)

5. of the first version of the talk
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

6. Outline So what? Is it of any use for a wider community?
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

7. Outline 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
Travelling Salesman Problem (TSP)
Four-point (4P) conditions - classification
Euclidean TSP with 4P conditions
Classification & Recognition
Discrete Optimization & OR 2013

8. TSP with specially structured matrices1 2 3 4 5 6 1 2 3 4 5 6 30 45 53 58 42 20 36 50 35 37 16 24 17 j i k l
cjl
cik  +
cil
cjk
clk
cij  +
clj
cik
cij
cik
clk
(cmn )=
clj + 
<1,2,…,n> is an optimal TSP tour (Kalmanson, 1975)

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

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

11. Four Point Conditions for symmetric TSPs: Classification
Demidenko conditions
Pyramidal
O(n2)
1976
Supnick
O(n)
1957
Kalmanson
O(n)
1975
NP-hard
D,W
2000
Max Demidenko
Pyramidal
O(n2)
1994
Van der Veen 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

12. 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
Van der Veen conditions
Pyramidal
1992
Similar to
Supnick
Sum Matrix
c(i,j)=u(i)+v(j)
Kalmanson
subclass
Similar to
Supnick
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

13. Recognition of specially structured matrices
dij
dik
dlk
(dmn )=
dlj
cij
cik
clk
(cmn )= d(m)(n) =
clj
 X +
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? 1 2 3 4 5 6 7 8

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Unpublished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

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

16. 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 )

17. 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

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
Van der Veen conditions
Pyramidal
1992
Similar to
Supnick
Sum Matrix
c(i,j)=u(i)+v(j)
Kalmanson
subclass
Similar to
Supnick
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. 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. P1 P2 Q2 Q1
d(P1,P2)
d(P1,Q2)
d(Q1,P2)
d(Q1,Q2)
Gaspard Monge,
d(P1,P2)+d(Q1,Q2)≤ d(P1,Q2)+d(Q1,P2)
Monge transportation
Supnick TSP
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

21. 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
Van der Veen conditions
Pyramidal
1992
Similar to
Supnick
Sum Matrix
c(i,j)=u(i)+v(j)
Kalmanson
subclass
Similar to
Supnick
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

22. Kalmanson Matrices:TSP with the master tour +
Kalmanson Matrices:TSP with the master tour j i k l
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.

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

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

25. Kalmanson Matrices: TSP with the master tourj i k l
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, 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 Kalmanson matrices can be recognized in O(n2) time

26. 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
Van der Veen conditions
Pyramidal
1992
Similar to
Supnick
Sum Matrix
c(i,j)=u(i)+v(j)
Kalmanson
subclass
Similar to
Supnick
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

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

28. Demidenko TSP:
Demidenko,1979: An optimal TSP tour can be found among 2n-2 pyramidal tours in O(n2) time n P2 P1
Structure of dynamic programming recursions: 2 3 4 5 6 7 8 9 1 1
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

29. Double Tree algorithm & Exponential neighbourhoods
F(9,6)
H(6,2)
B(5,6)
I(6,0)
E(3,4)
D(0,4)
A(5,8)
C(2,7)
G(9,4)
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 H I F B E D A C G
<IHEBABECEDEHGFGH I>
<IHEBA C D GF I> D I H E B A C F G

30. 2 is the tight bound for the Double Tree Algorithm
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.
2 is the tight bound for the Double Tree Algorithm
Bad News
Good News 1
~2n+2n(1- )
~2n+2 4 5 6 2 m 3 n
tree
opt G B H F I E D A C

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

32. Double tree for metric TSP: Optimal implementation
“Conjecture” (Papadimitiou, Vazirani, 1986)
The problem of finding the best tour among all tours constructed by the Double-Tree Algorithm is NP-hard.
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.

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

34. The best tour constructing heuristicx

35. 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)

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

43. point2
point1
point3
item1
item3
item2

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

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

46. 1 2 4 5 6 7 8 9 10 12 1 2 4 5 6 7 8 9 10 12

47. Summary “Byproducts” 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
Travelling Salesman Problem (TSP)
Four-point (4P) conditions - classification
Euclidean TSP with 4P conditions
Classification & Recognition

48. Transformation technique for Bipartite TSP

49. Transformation technique

50. Recognition of special cases

51. Recognising Relaxed Supnick / Kalmanson matrices1 1
0-1 RK matrix before and after permuting rows/columns

52. Hard Instances for the Optimal Double-Tree Algorithm
Optimal TSP tour
 6A
Minimal Spanning Tree
Optimal double-tree tour
 6A+√3 A
The approximation ratio is (6A+√3 A)/6A  1.622

53. Relaxed Supnick TSP: dynamic programming recursions− } ≥h
h=2
linear time algorithm P2 P1
Compare with DP for pyramidal tours

54. Three-exchange and six-point conditionsj i l k m n
Three-exchange and six-point conditions i j l k m n
<i, j, k, l, m, n>
5!*4=480 cases l i j k m n
add 480*479/2= cases
if consider pairs of the conditions i j l k m n i j l k m n

55. Two dimensional bin packing3 X 1 2 4 5 6 Y 1 2 3 4 5 6

56. Binary vector packing problem
Motivation: Allocation of Students to Working Teams
We want groups with the same number of
male/female
maths / non-maths
leaders
collaborators
mature students …
Student A being allocated to Group X should not be disadvantaged (compared with student B being allocated to Group Y)

57. Vehicle Routing Problem (CCC case study)
Given a set of customers (& demand), a set of vehicles (& capacity), SERVE all customers satisfying the demand and not exceeding the capacity.

58. Current partition of customers for one of the collection services in CCC
Partition of customers found by the new algorithms (up to 20% savings in transportation costs)