approximate algorithm for combinatorial optimization with two parameters

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approximate algorithm for combinatorial optimization with two parameters

David Blokh,

Dept. of Industrial Engineering and LU'~U~~""'-'"""'L'V"""V,

Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.

Gregory Gutin*

Dept. of Mathematics and Statistics,

BruneI University, Uxbridge, Middlesex UB8 3PH, U.K.

and Dept. of Mathematics and Computer IJ~Jl'-'Uv"',

Odense University, DK-5230, Odense, Denmark.

Abstract

We call a minimum cost restricted time combinatorial optimization (MCRT) problem any problem that has a finite set P, finite family S of subsets of P, non-negative threshold h, and two non-negative real-valued functions y : (say, cost) and x :

p...-?

R+ (say, time). One seeks a solution F* E S with y(F*) min{y(F): F E S,

x(F):S

h}, where x(G)

L:gEGx(g), y(G) = L:gEGy(g)

and G E S. We also assume that for the corresponding minimum cost problem there is an efficient exact or approximate algorithm. We describe a very simple approximate algorithm for any MCRT problem. Though our algorithm is not polynomial in general, we provide some evidence that the algorithm may be fairly fast in many cases.

1 Introduction

It is well known that there is a large variety of different real-world optimization problems (d. [2, 8, 14]). Thus, along with development of very effective algorithms solving some important optimization problems, it seems to be a right strategy to design algorithms for quite general optimization problems. In this paper, we propose

* Corresponding author.

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a very UL>,~V.LLUL"LLH for a wide of different combinatorial V P U H l U L I < k U J . V H

Problems from this are often appear in both of combinatorial (d. [1,3, 4,5,6,9, 11, 13, our is not polynomial in (which seems to be

HLL/--,",-"HOJL'" to expect in such we some evidence that the

algorithm may be fast in many

We call a minimum cost restricted time combinatorial optimization

any n r { , h l p T r l that has a finite set finite of subsets of

threshold h, and two non-negative real-valued functions y time). One seeks a solution F* E x(F') h}, where x(G) x(g), y(G)

that for the C01Te,;po,ndmg is an efficient or approximate alj1(Oflthm"

assume that the last A well known ex<unPle

This n r { " , h l p r n

Others de vel-

o , L I " , \ J U l U H 1 U " may bound for the ratio of a found solution that ae]:>erlas on both instance of a MCRT problem and solution. It is difficult to a prac-

tical algorithm with performance ratio for any

MCRT an a ratio may better evaluate

ratio of the found solution than a "uniform" guaranteed one.

2

We first an informal description of our We have assumed that there is effective algorithm

A

for the minimum cost problem.

A

we can find two solution F and

H

such that

y(F) =

^min{y(T):

^T ^S}

^and

x( Ii) min{ x(T): T E If

:s;

h, then F is an optimal solution. If

x( H)

>

h, then there is no solution for our problem. we may assume

that

x(H) :s;

h

< x(F).

The main idea of the algorithm is to apply

A

to a "linear combination" of F and H in order to obtain a feasible solution which is better than

H.

Represent

F

and

H

as points with coordinates

(x(F),y(F))

and

(x(H),y(H))

in a rectangular Cartesian coordinate system. We shall use the equation of the straight

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3

If x( F) h, then If x(H) h, then and c x(P)y(H) - x(H)y(F).

A find G so that w( 0)

=

.;:r.I"1T:H,n when

xC

^G)

^:s;

^hand^H^{is a}

for the

hand F := G,

of our <hLr.VL""U.J..ll

with coordinates and let

R

be another

:s;

y(F)

:s;

y(R)

:s;

y(H) and ax(R)

+

< c} where a, band ^care the algorithm. Let G

I-

be the solution f01.md in 3 the algorithm F and H above and let the line that is parallel to the line F H and contains intersect the lines H Rand FR in the

M and respectively Then

x( H)

:s;

<

P:roof: Since M and N are and

:s;

x(R) inequalities

x(H) :s; x(M),

x(M)

:s;

x(G)

:s;

x(N).

x(G) y(G)

(1) (2) belonging to the segments [H R] and [F R], re- x(F), the x-coordinates of M and N satisfy the

:s;

x(F). G lies on the segment Therefore, (1) holds. Analogously, we can prove (2). D.

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y

F

x Fig. 1

Theorem 3.2 After every performance of Step 5 the value of x(F) +y(H) decreases by positive number.

Proof:

Suppose that Step 3 of the algorithm is performed T

+

1 times, T Let F_{i ,} Hi and Gi be the current values of F, Hand G in the i-th performance of Step 3 (i 1,2, ... , T). Ri , Mi and Ni are defined recursively as follows. Rl denotes the point with coordinates X(Rl) X(Hl) and y(Rl) y(FI)' For i

=

1,2, ... , T, Mi and Ni denote the points where the line that is parallel to the line FJ1i and contains Gi

intersects the lines Ht~ and FiRi. For i

=

2,3, ... , T, ~

=

if X(Gi-l ) > hand Rt Nz-1 if x( Gi -I ) h.

We first prove by induction on i that

x ( Hi) ::; X ( Mi) ::; X ( G i) ::; X ( Ni ) ~ x ( Fi ) ,

y(Fi) y(Ni ) ~ y(Gi ) ::; y(Mi) ~ y(Hi) for every i = 1,2, ... , ^T.

(3) (4)

The case i = 1 follows from Lemma 3.1 and the definition of

R

I . Let i 2: 2. Then (3, and (4) follow from the definition of

R

i ) induction hypothesis, and Lemma 3.1.

To prove the theorem, it is sufficient to show that for every i

=

1,2, "0' T

if x(Gi )::::; h, then y(Gi )

<

y(Gj ), where j = max{k:

-1:::;

k

<

i,x(Gk )::; h}, (5)

and

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if h, where m

o k i,x(Ch)

h}, (6)

j and m are defined in (5) and (6).

(4), y(G

i )

, S-point if there C

An is minimal if there is no S-point (x",

y")

such that x"

x"+-

y"

x' +

y'. Let m(S) denote the number of minimal S-points.

every solution obtained in 3, y( G)) a minimal from Theorem that after 5 we obtain distinct elements G of

1)

Step 3 of the algorithm is performed at most m( S)

+

1 times.

2) The algorithm

+

time} where is the time

f:mmVI.f?T.:1.r1] of

If the functions x and y then we can obtain a bound the number of iterations of 3 to be still performed

terminated fitVhere

~"u"',uu~u"'"~""U of the ~""'.~L". VH""H

every ) each of the functions

be the current values of points P and H in an iter- Then the number of iterations of 3 to be still

- y(Fo),

V f J " H J . 1 U J solution of and G be a solution found

algorithm w.r.t G the performance ratio

fl.l.lIlJr".I.lI.""II. A find solutions. If our algorithm has

a Vf?1rtorTrUl/rl.f:f' Step

4

and G is the found solution} then r( 0) S

r(

G)

y(

G) /

(y

(G)

+ ---~---c---:-""-:----.

Proof: Let the line G F intersect the line x h in the point B and let A have coordinates (h,

y(

G)). Then the triangle G BA contains all optimal solutions. Indeed, no feasible solution can be to the right of the line AB, no optimal solution can be above the line AG, and no solution can belong to a line parallel to

GF

and "below"

GF

such solution

G'

would have

ax(G') + by(G')

< c).

Hence, for an optimal solution

G, y(B)*

~

y(G).*

Now it is easy to check that

r( G) y( G)/y(B) = r( G).

D.

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4 Computational

We have tested the behavior of our algorithm for the minimum restricted time

as~ngnrrlerlt problem: given two n X n-matrices C and T with non-

negative entries and a non-negative real h, find a permutation ^7r E Sn so that

Ci1r(t) min{Lf=l Cicr(i) . a E Sn, h}, where Sn is the set of

all permutation on {I, 2, ... , n}. We generated 60 instances of the mini- mum cost constrained time assignment problem with n 6,8 and 10. The entries of matrices varied from 0 to 99. The value of h was calculated by the formula h = max{30n, nJ-Lj2}, where J-L is random integer from the interval [0,99]. For only five of the instances, Step 3 was not performed at all: two instances with n 6 did not have any solution, an instance with n = 8 and two instances with n 10 only needed 1 to find an optimal solution. Some results obtained for the rest 55 are reflected in Table 1: we show the number of instances with specified real ratio of the found solution (i.e.

r(

G)) and specified quality of the bound r(G) (i.e. q(G)

=

r(G)jr(G)). Note that (for the 55 instances) the number of i, of Step 3 varied from 2 to 5 i = 5 only for two instances).

Table 1: Number of instances with specified rand q.

n 1 1 < r ~ 1.1 T 1.1 q 1.1 1.1 < q 1.2 q > 1.2

6 14 3 1 11 7 0

8 8 8 3 9 7 3

10 7 7 4 16 2 0

We have also tested the behavior of our algorithm and Gvozdev's algorithm [3J for the two parameters minimum spanning tree (in complete graph) problem: given two symmetric n x n-matrices C [Cij] and T

=

[tij] with non-negative entries and a non-negative real

h,

find a spanning tree

F = (V, E)

in the complete graph

Kn

on vertices

{1,2, ...

,n} such that Cij = min{LijEE1Cij:

E'

E E,LijEE,tij ~

h},

where E is the family of all edge sets of spanning trees in Kn.

The matrices C and T were generated analogously to the corresponding matrices in the assignment problem above. They have orders n=20,30,40,50 and 60. The value of h was calculated by the formula h = max{25n, O.4nfL}, where J-L is a random integer from the interval [0,99].

Table 2 reflects, in a sense, the quality of solutions found by our algorithm. Not knowing optimal solutions in most of cases, we measure the quality by the bound if (see Proposition 3.5). One can see that the quality of the solutions steadily increases while the order n grows.

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Table Number of instances with

Table 3 shows some data which can be compare our algorithm with In Table 3, ml is the number of instances when the costs of solutions found by our algorithm and Gvozdev's coincide, m2

(m3,

is the number when solutions obtained by our v"LF,~.LLULCLLL

time Gvozdev's ULF,V.LLUU.H.L.

siderably more time than our COL!',VL."H.UH

~), in Gvozdev's algorithm (before time COJD.S11IDLln;g.

Table 3: Comparison of our

about half of requires con- C01TI]::mtatllon of the parameters .:\ and

iterations the procedure) is quite

and Gvozdev's algorithm.

~3

!---+~~---

20 17 1 2 7.2 9 13.4 30 10 6 4 8.7 10 16.2 40 6 7 7 9.6 11 18.5 50 6 6 8 10.3 11 21.2 60 3 12 5 10.9 12 22.0 70 3 14 3 11.0 12 22.3

~--~--- ---~

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References

[1] Y.P. Aneja, V. Aggarwal and K.P.K. Nair, "Shortest chain subject to side con- straints", Networks 13, 295-302 (1983).

M.R. Garey and D.S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness, Freeman, San 1979.

[3] S.E. Gvozdev, "On two discrete optimization problems" , In Upravliaemye temYJ Novosibirsk, 19 22-30 (1979) (in Russian).

[4] G. Handler and 1. Zang, "A dual algorithm for the constrained shortest path problem" ^j Networks 10, 293-310 (1980).

[5] R. Hassin, "Approximation schemes for the restricted shortest Math. Oper. Res. 17, 36-42 (1992).

problem" ,

[6] M. "The shortest path problem with two objective functions", European J. Oper. Res. 25, 281-291 (1985).

[7] H.C. Joksch, The shortest route problem with constraints, J. A1ath. Anal. Appl.

14, 191-197 (1966).

[8] E.L. Lawler, Combinatorial Optimization: Networks and Mathroids, Hold, Rine- hart and Winston, New York, 1976.

[9J 1.1. Melamed, S.L Sergeev, and 1.Kh. Sigal, "The Traveling Salesman Prob- lem: Part 1, Theoretical Issues" Automation and Remote (}ontroI50 1147-1173 (1989).

[10J B.M.E. Moret and H.D. Shapiro, "How to find a minimum spanning tree in practice", Lecture Notes in CS 555, 192-203 (1991).

[11] S.A. Morozov, "On the minimum spanning tree problem with restrictions" In Upravliaemye systemYJ Novosibirsk, 15, 40-47 (1976) (in Russian).

[12] C.A. Phillips, "The network inhibition problem", in Proc. 25th Ann. ACM Symp. Theory Computing, 776-785 (1993).

M.J. Rosenblatt, Z. Sinuani-Stern, "Generating the discrete efficient frontier to the capital budgeting problem" ¹Oper. Res., 37, 384-394 (1987).

[14J H.A. Taha, Operation Research: An Introduction, New York, 1992.

[15] A. Warburton, "Approximation of Pareto optima in multi-objective, shortest path problems", Oper. Res. 35, 70-79 (1987).

(Received 23/11/95; revised 16/5/96)

approximate algorithm for combinatorial optimization with two parameters