Rounding algorithms for covering problems - Massachusetts Institute ...

Mathematical Programming 80 (1998) 63 89

Rounding algorithms for covering problems Dimitris Bertsimas a,,,1, Rakesh Vohra

b,2

a Massachusetts Institute of Technology, Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142-1347, USA b Department of Management Science, Ohio State University, Ohio, USA

Received 1 February 1994; received in revised form 1 January 1996

Abstract In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Recent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless P -- NP. As a result, as far as negative results are concerned a unifying picture is emerging. On the other hand, as far as particular approximation algorithms for different problems are concerned, the picture is not very clear. Different algorithms work for different problems and the insights gained from a successful analysis of a particular problem rarely transfer to another. Our goal in this paper is to present a framework for the approximation of a class of integer programming problems (covering problems) through generic heuristics all based on rounding (deterministic using primal and dual information or randomized but with nonlinear rounding functions) of the optimal solution of a linear programming (LP) relaxation. We apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering, facility location, general covering, network design and cut covering problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

1. Introduction G i v e n o u r inability to efficiently solve several discrete o p t i m i z a t i o n p r o b l e m s (in p a r t i c u l a r N P - h a r d p r o b l e m s ) exactly, it is n a t u r a l to a s k w h e t h e r it is possible to a p p r o x i m a t e them. W e will focus on m i n i m i z a t i o n p r o b l e m s , b u t a p a r a l l e l t h e o r y can be d e v e l o p e d for m a x i m i z a t i o n p r o b l e m s (see for e x a m p l e [1]). A l g o r i t h m A constitutes an a p p r o x i m a t i o n a l g o r i t h m for m i n i m i z a t i o n p r o b l e m I I with g u a r a n t e e f ( n ) , if for each instance I o f size n o f 17, A l g o r i t h m A runs in p o l y n o m i a l time in

* Corresponding author. E-mail: [email protected]. 1 Research partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory. 2 Research partially supported by a Deans Summer Fellowship of the College of Business of the Ohio State University. S1383-7621/98/$19.00 © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. PIIS1383-7621(98)00068-3

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n and returns a value ZA (I) such that ZA (I) 0 f o r j E N and a family ofm sets Si _c N, find a set S C N such that IS fq Si[ ~> 1, such that ~ j c s @ is minimized. In order to formulate the problem we let aij = 1 if j E Si. Then the set covering problem can be formulated as follows: (IP2)

IZ2 = minimize

~-'~_1 CjXj

subject to

j=laijxJ>~ 1,

i= 1,...,m,

(0, 1}. Lund and Yannakakis [3] proved that the set covering problem cannot be approximated within a factor smaller than O(logm) unless P = NP, i.e., if there exists a polynomial time algorithm within ¼1ogre from the optimal value, then P = NP. Johnson [11] and Lovfisz [13] propose a greedy heuristic with value Z~ for the problem with cj = 1, such that g~ P{V/}P{Vj}.

jcSiNSj

Therefore, P{U~ N Uj} = 1 - P{V~} - P{Vj} +P{V/A V/} >~ 1 - P{Vi} - P{V/} +P{V,.}P{V/} = P{U,.}P{Uj), which is intuitively obvious, since P{U,.IU/} >~ P{U,}. In general,

I7, e -t(1-a)e[xl} ~
~ 1. m U~ be the event that the solution XH is feasible. Then Let F = N~=I

since P{U~[Uj} > / P { G } . Thus

P{F} >t (1

(~-l)e h m _ e-2-~mx) .

Moreover,

jER

jERe

,/

2kamax)(k 1)2x~m ~ k/(1 -- e-k)

=.

jERc

jeR

Since E[Zn] >t E[ZHIF]P{F}, E[GHIF]-- ~ k / ( 1

-- e

Z3 For k = 2ama~ log m + 2, we obtain Z3

~

(2amaxlogm + 2)

1-

---- O(amaxlogm).

By derandomizing as in the proof of Theorem 1 we can find deterministically a solution xn that satisfies ZH = O(amax log m). Z3

[]

Dobson [21] achieves a bound O(logmaxj ~ i a i j ) by analyzing a greedy heuristic. The bound in this case is O(logm-Flogamax) as opposed to our weaker bound O(amax log m). For amax constant both bounds are O(log m).

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D. Bertsimas, R. Vohra / Mathematical Programming 80 (1998) 63-89

2.3. Facility location We are given an undirected graph G = (V,E) (IVI = n) with costs cij ~> 0 for (i,j) E E and di ~> 0 for each i E V. The goal is to find a set S _c V of facilities that minimizes ~ i c s d + ~i~vminjcsCij • The problem is NP-hard [1]. H o c h b a u m [24] presents a greedy algorithm for the problem with cost within O(log n) of the optimum. Since the problem can be formulated as a set covering problem (this is a nonstandard formulation) involving O(n 2) rows, this is not a surprise. Here we show that the O(log n) can be obtained directly from the standard formulation of this problem. In this section we show that the generic randomized heuristic returns a solution with cost within O(logn) of the optimum cost, i.e., it performs as well as any heuristic can (unless P = NP). We start with the classical strong integer programming formulation of the problem. Let y~ = 1 if we locate a facility at node i 6 V, 0 otherwise. Let x~2 = 1 if customer i 6 V is assigned to facility j. Then, the problem can be formulated as follows.

(IP4)

I Z 4 = minimize

~-~(i,j)6E CijXij ~- ~ j 6 v djyj

xij ~ 1, i = 1 , . . . , n , xij, yj C {0, 1}.

subject to

Let Z4 be the value of the LP relaxation. The randomized heuristic is as follows: 1. Solve the LP relaxation and find the solution xu, yj. 2. R o u n d as follows:

P{yj = 1} = 1 - (1 - y;)~, P{x~j = 1 [yj = 1}

1 -

k = logn,

(1 - x;j)

-

1 -

P{xij = l lyj = O} = O. In this way the constraints xij ~< yj are always satisfied. Let xn be the solution generated by this algorithm and Zn its cost. Notice that

P{xij=l}=l-(1-x~j)

~

and

P{~j

}=Hs (1-(1-x~)k).

We bound the performance of the heuristic as follows. Theorem 4.

E[ZHIxH is feasible]

n

HP{ } = H ( 1 - P{Ai}) >~ ( 1 j=l

e ~)n.

j=l

Moreover, cij(1 - (1 _ xij),k ) + ~-~4(1 - (1 _ g)k) ~ kZ4.

E[Zu] = ~ (i,j)cE

jcV

Therefore,

E[ZH] >>.E[ZH [F]P{F} >~E[Zn [F](1 - e-k) ", which implies that

E[ZH IF] ~
~ b, x~O,

and its dual Z5 = maximize y'b subject to yA 2, otherwise (IPs) is trivial. We apply the generic dual heuristic to (IP5) starting with an optimal solution x*,y*. Let ZH be the value of the heuristic. Theorem 5. The dual heuristic produces a solution with

ZH Z5

--

4/+1.

Proof. Let x* and y* be an optimal primaLdual pair for (LP5) and (D5). The heuristic solution xj = Fx)] is feasible, since n

n

n

ai,x, = j--1

a,,Fx;1

a,,x;

j=l

b,

j--I

Let K = {j : x) > 0}. Then,

z.=

,,y, r ;1 , a..'

,Fx;l= Z ,GK

Z jcK m

,

,

since cj = ~i=l auYi for j E K by complementary slackness. Since [xjl ~< x) + 1 and ~jcX aUx) > b~ implies that y,.* = 0 by complementarity we obtain

z. ~

b, + m

1.

au y; = jEK

i=1

]

b,y; + i=1

auY; ,=1

jGK

m

~

aijxj =

jCK(y,z)

aij > b~.

jCK(y,z)

Ify~ = 0 for all i E A(y,z) then (7) of the proposition holds with y* = y,z* = z. Suppose then, there exists a p E A(y,z) such that Yv > 0. Let

D. Bertsimas, R. Vohra / Mathematical Programming 80 (1998) 63~89 •

0

.

79

Z"

mm

jc/@a):~pj>0ap9)

t.

Notice that 0 > 0. We define a new dual solution (y', z') as follows:

y~ (yi, =

{Yi

i¢p, i =p,

- - O,

zt={zj, J zj - -

jfgK(y,z), j E K(y,z).

apjO,

The process by which we obtain (y',z') from (y,z) we call reduction. We show first that the solution (y',z') is dual feasible: By the definition of 0, y~ /> 0, 4 ~> 0. Moreover, i f j C K(y,z) j

aijYi

i=1

apjO

zj + apjO

i=1

Z

aijYi

zj ~ cj.

i=1

If j ¢ K(y, z), then, m

Z ai,y; -

=

a,,yi - a ,O- z, 0. Therefore, by repeating the reduction operation either we find an optimal dual solution that has z) = 0 for a l l j ory~* = 0 for all i c A(y*,z*). In addition, since (y*, z*) is optimal, it should satisfy complementary slackness, i.e., ?,* = 0 for all i E C(y*,z*). Therefore, we have proved (6). We now proceed to prove (7) by showing

jcK(y* ,z*)

Notice that

z; 0 for some p c B(y*,z*). We apply a reduction operation again, i.e., construct a dual solution:

y, = { y~, i y*-O,

i ~ p, i=p,

~ = { z~, j ~ K(y*,z*), z~-apjO, jcK(y*,z*),

where 0 = min(yp, minjc/@, z.): ,,,j>0z)/apj) > 0. We first observe that, as before, the new solution is dual feasible (the only difference with the previous derivation, how-

D. Bertsimas, R. Vohra I Mathematical Programming 80 (1998) 63 89

80

ever, is that the new solution is not necessarily optimal, since p C B(y*, z*) rather than

p c A(y*,z*)). Given a dual feasible solution (y, z) we define

L(y,z)=

~

zj

and

R(y,z)=

jcK(y* ,z*)

~

~

aijyi.

iEB(v* ,z*) jGK(y* ,z*)

We then want to show that L(y*,z*) ~ 1 - - -

n b"

5. Concluding remarks

We presented two methods to construct approximation algorithms for covering problems: • Randomized rounding with nonlinear rounding functions and • Deterministic rounding using dual information. We saw that these two approximation methods match the best known bounds for several covering problems. Related to the question that motivated our research, whether there exists a systematic way to construct approximation algorithms, we believe that these two algo-

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D. Bertsimas, P~ Vohra / Mathematical Programming 80 (1998) 63~89

r i t h m i c ideas p r o v i d e a unification o f m e t h o d s for a p p r o x i m a t i n g discrete o p t i m i z a tion p r o b l e m s . W e close the p a p e r with a s u m m a r y o f research directions-conjectures that, we believe, will enhance o u r u n d e r s t a n d i n g o f a p p r o x i m a t i o n m e t h o d s : 1. R e l a t e d to the question o f which p r o b l e m s can be a p p r o x i m a t e d better t h a n others, we believe t h a t as far as m i m s u m (as o p p o s e d to r a i n - m a x ) p r o b l e m s , we believe t h a t o n l y covering p r o b l e m s can be a p p r o x i m a t e d within a O ( l o g n ) factor. A l t h o u g h this is n o t a f o r m a l m a t h e m a t i c a l statement we d o n o t k n o w o f a n y e x a m p l e o f a m i n - s u m p r o b l e m with a l o g a r i t h m i c o r a s u b l o g a r i t h m i c g u a r a n t e e t h a t c a n n o t be f o r m u l a t e d as a covering p r o b l e m . 2. T h e r a n d o m i z e d r o u n d i n g heuristic uses the n o n l i n e a r r o u n d i n g function f ( x ) . W h i l e we specified several r o u n d i n g functions, we did n o t p r o p o s e a systematic m e t h o d to c o n s t r u c t the r o u n d i n g function. T h e p r o b l e m o f finding the best r o u n d i n g function f ( x ) in o r d e r to minimize E[Zu (f)] reduces to a calculus o f variations p r o b l e m , which seems to be difficult to solve at its full generality. 3. I n v e s t i g a t i o n o f C o n j e c t u r e 1 seems interesting as an extension o f r a n d o m g r a p h theory; a first step in this direction was t a k e n b y A l o n [26]. A p p l i c a t i o n s to a p p r o x i m a b i l i t y c o u l d also be interesting to explore.

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