polynomially solvable cases of np-hard problems
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
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Polynomially solvable cases of NP-hard problems
Vladimir Deineko
Discrete Optimization & OR 2013
Based on joint works with R.Burkard, D.Foster, B.Klinz, R.Rudolf, M.Sviridenko, J.Van der Veen, G. Woeginger
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Outline• Travelling Salesman Problem (TSP)• Four-point (4P) conditions - classification• Euclidean TSP with 4P conditions
– Classification & Recognition
• Summary: Further research opportunities
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The travelling salesman problem (TSP)
city3 city2 city5
city1
city6city4
Find a cyclic permutation that minimizes
n
i
iic1
))(,(
1 2 3 4 5 6
1 30 45 53 58 42
2 30 20 36 50 42
3 45 20 20 35 37
4 53 36 20 16 24
5 58 50 35 16 17
6 42 42 37 24 17
=<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)
11
56
23
4
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Outline• Travelling Salesman Problem (TSP)• Four-point (4P) conditions - classification• Euclidean TSP with 4P conditions
– Classification & Recognition
• Summary: Further research opportunities
of the first version of the talk
Discrete Optimization & OR 2013
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Outline
• Travelling Salesman Problem (TSP)
• Four-point (4P) conditions - classification
• Euclidean TSP with 4P conditions– Classification &
Recognition
So what?
Is it of any use for a wider community?
Discrete Optimization & OR 2013
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Outline
• Travelling Salesman Problem (TSP)
• Four-point (4P) conditions - classification
• Euclidean TSP with 4P conditions– Classification &
Recognition
Discrete Optimization & OR 2013
• 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 TSPReal life OR problems
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cij cik
clk(cmn )= clj
TSP with specially structured matrices
j
i
kl
+
clkcij + clj + cik
cjlcik + cil + cjk
1 2 3 4 5 6
1 30 45 53 58 42
2 30 20 36 50 42
3 45 20 20 35 37
4 53 36 20 16 24
5 58 50 35 16 17
6 42 42 37 24 17
1
2 34
56
<1,2,…,n> is an optimal TSP tour (Kalmanson, 1975)
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1
2
3
4
5
6
1 2 3 4 5 6
- - - --- - - --- - - --- - - --- - - --
Specially structured matrices
c3,4c23 + c2,4 + c3.3Specially Structured Matrices: notations
i
j
k l
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Two-exchange and four-point conditions
i j
lk
We consider the TSP with special matrices (cst)
such that clkcij + clj + cik
All permutations
NP-hard
N-permutationsO(n4)
O(n2)pyramidal tours
arbitrary tour τ
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Pyramidal O(n2)1976
Demidenko conditions
Four Point Conditions for symmetric TSPs: Classification
NP-hardD,W2000
Van der Veen conditions
PyramidalO(n2)1994
Max Demidenko
NP-hardSteiner et al
2005Max Van der Veen
NP-hardD,Tiskin2006
Relaxed KalmansonN-perm O(n4)D, 2004
Relaxed Supnick
Kalmanson O(n)1975
Supnick O(n)1957
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Pyramidal1976
Pyramidal1992
NP-hardD,W1997
NP-hardSteiner et al
2005
N-permO(n4)
NP-hardD,T2006
Demidenko conditions
Four Point Conditions: Classification
Van der Veen conditions
Max Demidenko
Max Van der Veen
Relaxed Kalmanson
Relaxed Supnick
Pyramidal
Similar toSupnick
SpecialMaxKalmanson
NP-hard
New(linear)case
SpecialLawlerO(n2)
Supnick Max,19571971,87
KalmansonMax,1970
Sum Matrixc(i,j)=u(i)+v(j)
Kalmansonsubclass
Kalmanson19701,2,3,…,n
Supnick 19571,3,5,…2,1
Kalmansonsubclass
Similar toSupnick
Sum Matrixc(i,j)=u(i)+v(j)
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Recognition of specially structured matrices
dij dik
dlk(dmn )= dlj
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
34
56
7
8
cij cik
clk
(cmn )= d(m)(n) = clj X
+
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O(n4)1999
UnpublishedD,W
Demidenko conditions
Four Point Conditions for symmetric TSPs: Recognition
Van der Veen conditions ?
Max Demidenko
Max Van der Veen
?
Relaxed Kalmanson
Relaxed Supnick
O(n2log n)1998 D,R,W
O(n2)?C,F,T
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O(n4)Van der Veen,
D., Rudolf,Woeginger
Demidenko conditions
Four Point Conditions for TSP: Recognition
n<17
Van der Veen conditions
O(n4) D.,
Burkard
Max Demidenko
n<17Max Van der Veen
n<17 orall are on the line
Relaxed Kalmanson O(n4)D., FosterSviridenko
Relaxed Supnick
Euclidean
Conjecture: n<7.
Conjecture: n<7.
Conjecture: n<7.
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Four Point Conditions for TSP: Recognition
Relaxed Kalmanson
Euclideann1 n2 n3 n4 n5
n1
n2
n3
n4
n5
d(n1,n3 )-d(n2,n3 )≥ d(n1,n4 )-d(n2,n4 )
> =
d(n1,n4 )-d(n2,n4 )= d(n1,n5 )-d(n2,n5 )
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Four Point Conditions for TSP: Recognition
Relaxed Kalmanson
Euclidean
(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
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Pyramidal1976
Pyramidal1992
NP-hardD,W1997
NP-hardSteiner et al
2005
N-permO(n4)
NP-hardD,T2006
Demidenko conditions
Four Point Conditions: Classification
Van der Veen conditions
Max Demidenko
Max Van der Veen
Relaxed Kalmanson
Relaxed Supnick
Pyramidal
Similar toSupnick
SpecialMaxKalmanson
NP-hard
New(linear)case
SpecialLawlerO(n2)
Supnick Max,19571971,87
KalmansonMax,1970
Sum Matrixc(i,j)=u(i)+v(j)
Kalmansonsubclass
Kalmanson19751,2,3,…,n
Supnick 19571,3,5,…2,1
Kalmansonsubclass
Similar toSupnick
Sum Matrixc(i,j)=u(i)+v(j)
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Gaspard Monge, 1746-1818
Monge (Supnick) matricesMonge (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
Q2Q1
d(P1,P2) d(P1,Q2)
d(Q1,P2) d(Q1,Q2)
d(P1,P2)+d(Q1,Q2)≤ d(P1,Q2)+d(Q1,P2)
Burkard, Klinz,Rudolf: Perspectives of Monge Properties in Optimization. Survey(1996)
Monge 1781 - transportation Supnick 1956 - TSPHoffman 1963 – transportation; introduced Monge;
D.,Filonenko 1979 – recognition of Monge matrices
D., Jonsson, Klasson, Krokhin: The approximability of MAX CSP with fixed-value constraints.) 2008
Burdjuk,Trofimov 1976 – TSP with permuted Monge matrices
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Pyramidal1976
Pyramidal1992
NP-hardD,W1997
NP-hardSteiner et al
2005
N-permO(n4)
NP-hardD,T2006
Demidenko conditions
Four Point Conditions: Classification
Van der Veen conditions
Max Demidenko
Max Van der Veen
Relaxed Kalmanson
Relaxed Supnick
Pyramidal
Similar toSupnick
SpecialMaxKalmanson
NP-hard
New(linear)case
SpecialLawlerO(n2)
Supnick Max,19571971,87
KalmansonMax,1970
Sum Matrixc(i,j)=u(i)+v(j)
Kalmansonsubclass
Kalmanson19751,2,3,…,n
Supnick 19571,3,5,…2,1
Kalmansonsubclass
Similar toSupnick
Sum Matrixc(i,j)=u(i)+v(j)
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Kalmanson Matrices:TSP with the master tourj
i
kl 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.
+
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Illustration to Master Tour 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 1A1Given a set of customers
TSP: 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 1A1Given a set of today’s customers
???
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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 1A1Given 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 1A1Given a set of today’s customers
???
Illustration to Master Tour problem
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Kalmanson Matrices: TSP with the master tour
j
i
kl
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 Kalmanson matrices can be recognized in O(n2) time
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Pyramidal1976
Pyramidal1992
NP-hardD,W1997
NP-hardSteiner et al
2005
N-permO(n4)
NP-hardD,T2006
Demidenko conditions
Four Point Conditions: Classification
Van der Veen conditions
Max Demidenko
Max Van der Veen
Relaxed Kalmanson
Relaxed Supnick
Pyramidal
Similar toSupnick
SpecialMaxKalmanson
NP-hard
New(linear)case
SpecialLawlerO(n2)
Supnick Max,19571971,87
KalmansonMax,1970
Sum Matrixc(i,j)=u(i)+v(j)
Kalmansonsubclass
Kalmanson19751,2,3,…,n
Supnick 19571,3,5,…2,1
Kalmansonsubclass
Similar toSupnick
Sum Matrixc(i,j)=u(i)+v(j)
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Specially structured matrices & Exponential neighbourhoods
Kalmanson (1970)
K*= <1,2,…,n> is an optimal TSP tour
cij cik
clk(cmn )= clj
Demidenko (1976) * is a pyramidal tour
(cmn )=
Supnick (1957) matrices
S*= <1,3,5,7…,n,…,6,4,2>
is an optimal TSP tour
cij cik
clk(cmn )= clj
+
1 1
n
2 3
4 5
67
8
1 1
n
234
56
n-1
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Demidenko TSP:
Demidenko,1979: An optimal TSP tour can be found among 2n-2 pyramidal tours in O(n2) time
1 1
n
2 3 4 5 6 7 8 9
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
P2
P1
P1 P2
P1
Structure of dynamic programming recursions:
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Double Tree algorithm & Exponential neighbourhoods
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
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)
<IHEBABECEDEHGFGH I>
<IHEBA C D GF I>
D
I
H E
B A
C F
G
HI
FB
ED
AC
G
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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
~2n+2n(1- )
~2n+2
4 5
6
1 2
m
3
4 5
6
2 3
n
1
1 tree
opt
Good N
ews
Bad News
GB
H
F
I
ED
AC
F G D
A
I H
E B C
I H
F E D
B A
C
G F
D
I
H E
B A
C
2 is the tight bound for the Double Tree Algorithm
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Is it possible to find in polynomial time the best tour among all tours constructed by the Double-Tree Algorithm?
GB
H
F
I
ED
AC
F G D
A
I H
E B C
I H
F E D
B A
C
G F
D
I
H E
B A
CF G D
A
I
H E B
C
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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.
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Dynamic programming for the TSP
F G DA I E C
n n-1 n-2 ... 2 1
Pyramidal Tours
O(n2) algorithm +Special solvable
cases
F G D I
H
E C
A
Set of all n! tours
O(n2 2n) well known
DP algorithm +exact solution
GB
H
F
I
ED
AC
GB
H
F
I
ED
AC
O(2dn2) algorithm
+”good” heuristic
DTd
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The best tour constructing heuristic
x
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The bipartite travelling salesman problem (BTSP)
city3 city2 city5
city1
city6city4
“black” and “white” points have to alternate in a feasible tour
Shoe-lace problem (Halton,1995; Misiurewicz, 1996)
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The bipartite travelling salesman problem (BTSP)
point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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point1
item1
item2item3
point2 point3
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set of permutations
Transformation techniqueArbitrary permutation 0
1
c(0) c(1)
2c(1) c(2)special
subset
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Warwick Business School
V.D.,G.Woeginger
Bipartite Travelling Salesman, or ShoeLace Problem for very old shoes
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1
2
4
5
6
7 8 9 10 12
-6.0 -10.6 1.5 -1.1 -0.5
-3.0 2.1 -4.2 1.4 1.2
-0.5 -5.4 0.8 -0.8 -0.3
-0.7 -5.5 -0.1 -9.4 -4.0
-0.2 0.1 -0.4 -2.7 -4.4
1
2
4
5
6
7 8 9 10 12
-5.6 -0.2 -0.5 0.2 -0.0
-8.5 -13.5 -0.2 -5.4 +0.0
-5.7 -0.2 -0.6 -11.9 -0.8
-5.7 0.1 -12.3 11.3 -0.0
5.2 -0.6 7.7 -13.8 -2.5
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Summary
• Travelling Salesman Problem (TSP)
• Four-point (4P) conditions - classification
• Euclidean TSP with 4P conditions– Classification &
Recognition
“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 TSPReal life OR problems
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Transformation technique for Bipartite TSP
28
Transformation steps to transform arbitrary tour
1
7 i2 i6
j3 1
i3 8 i5
2 j4 j5 j6
1
7 9 8
4 1
11 12 10
2 6 5 3
into specially structured tour
O(n2 ) inequalities tobe satisfied
1
2
4
5
6
7 8 9 10 12
- - - --- - - --- - - --- - - --- - - --
3
![Page 49: Polynomially solvable cases of NP-hard problems](https://reader036.vdocuments.us/reader036/viewer/2022062814/568168a4550346895ddf3a41/html5/thumbnails/49.jpg)
Transformation technique
28
Transformation steps to transform arbitrary tour
1
7 i2 i6
j3 1
i3 8 i5
2 j4 j5 j6
1
7 9 8
4 1
11 12 10
2 6 5 3
into specially structured tour
O(n2 ) inequalities tobe satisfied
1
2
4
5
6
7 8 9 10 12
- - - --- - - --- - - --- - - --- - - --
3
![Page 50: Polynomially solvable cases of NP-hard problems](https://reader036.vdocuments.us/reader036/viewer/2022062814/568168a4550346895ddf3a41/html5/thumbnails/50.jpg)
Recognition of special cases
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Recognising Relaxed Supnick / Kalmanson matrices
1 1
1 1 1 1 1
1 1 1
1 1 1 1
1 1 1
1 1 1 1
1 1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1 1
1 1 1
1 1 1
1 1 1
0-1 RK matrix before and after permuting rows/columns
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52
Minimal Spanning TreeOptimal double-tree tour
6A+√3 A
The approximation ratio is (6A+√3 A)/6A 1.622
Optimal TSP tour
6A
Hard Instances for the Optimal Double-Tree
Algorithm
![Page 53: Polynomially solvable cases of NP-hard problems](https://reader036.vdocuments.us/reader036/viewer/2022062814/568168a4550346895ddf3a41/html5/thumbnails/53.jpg)
Relaxed Supnick TSP: dynamic programming recursions
−
− } ≥hh=2 linear time algorithm
P2
P1
P1 P2
P1
Compare with DP for pyramidal tours
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Three-exchange and six-point conditionsi j
l
k
mn
ji
l
k
mn
l
i jk
mni j
l
k
mn
i j
l
k
mn
<i, j, k, l, m, n>
5!*4=480 cases
add 480*479/2=114960 cases
if consider pairs of the conditions
![Page 55: Polynomially solvable cases of NP-hard problems](https://reader036.vdocuments.us/reader036/viewer/2022062814/568168a4550346895ddf3a41/html5/thumbnails/55.jpg)
Warwick Business School
Two dimensional bin packing3
X1 2
4 5 6
Y
1 2
3
4 5 6
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Warwick Business School
Binary vector packing problem
Student A being allocated to Group X should not be disadvantaged (compared with student B being allocated to Group Y)
We want groups with the same number of• male/female• maths / non-maths• leaders • collaborators• mature students …
Motivation: Allocation of Students to Working Teams
![Page 57: Polynomially solvable cases of NP-hard problems](https://reader036.vdocuments.us/reader036/viewer/2022062814/568168a4550346895ddf3a41/html5/thumbnails/57.jpg)
Warwick Business School
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.
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Warwick Business School
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)