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Worst-case analysis of the greedy algorithm for a generalization of
the maximum p-facility location problem
M.I. Sviridenko
a;b;∗aSobolev Institute of Mathematics, Novosibirsk, Russia
bBasic Research Institute in Computer Science, University of Aarhus, Denmark
Received 1 February 1999; received in revised form 1 December 1999
Abstract
In this work we consider the maximum p-facility location problem withkadditional resource constraints. We prove that, the simple greedy algorithm has performance guarantee (1−e−(k+1)
)=(k+ 1). In the casek= 0 our performance guarantee coincides with bound due to [4]. c 2000 Elsevier Science B.V. All rights reserved.
Keywords:Approximation algorithm; Greedy algorithm; Worst-case analysis
1. Introduction
LetI={1; : : : ; n}andJ={1; : : : ; m}. In this work we consider the following optimization problem:
max
S⊆I{f(S):|S|=P}; (1)
wheref(S) is a polynomially computable set function. Nemhauser et al. [4] consider this problem with nonnegative nondecreasing submodular functions f(S) (function is submodular if f(S) + f(T)¿ f(S∪T) +f(S∩T) for allS; T⊆I). They prove that the simple greedy algorithm has performance guaran-tee 1−e−1. The main property used in their proof is that nondecreasing set functionf(S) is submodular if
∗Correspondence address: Basic Research Institute in Computer
Science, University of Aarhus, Denmark.
E-mail address: sviri@brics.dk, svir@math.nsc.ru (M.I. Sviridenko)
and only if
f(T)6f(S) + X
p∈T\S
(f(S∪ {p})−f(S))
for allS; T⊆I. Wolsey [5] studies problem (1) with the following objective function:
f(S) = max
xij¿0
X
i∈S X
j∈J
fijxij; (2)
subject to
X
i∈S
xij6Aj; j∈J; (3)
X
j∈J
xij6Bi; i∈S: (4)
Wolsey proves that function (2) – (4) is a submodular set function and, consequently, the greedy algorithm has performance guarantee 1−e−1.
In this work we study more general class of ob-jective functions. We prove that the greedy algorithm gives a (1−e−(k+1))=(k+ 1)-approximation solution for problem (1) with nonnegative nondecreasing ob-jective function satisfying the following inequality:
f(T)6(k+ 1)f(S) + X
p∈T\S
(f(S∪ {p})−f(S))
(5)
for all S; T⊆I. We introduce the natural members of this class of functions and prove that function (2) – (4) withkadditional resource constraints:
f(S) = max
satises inequality (5). Let us denote function (6) – (9) by f(S;A;C) (or f(S) for short) where A = (A1; : : : ; Am) and C = (C1; : : : ; Ck). We
as-sume that all input numbers of the problem, i.e. (fij; cijl; Aj; Bi; Cl), are nonnegative. Notice
that we may only consider the feasible solu-tions S of problem (6) – (9) satisfying the equal-ity P
i∈Sxij =Aj. This can be done by
introduc-ing a dummy facility q such that fqj = 0; Bq =m;
cqjl= 0 for allj∈J; l= 1; : : : ; k. We also assume that
q ∈ S for any feasible solution S and |S|=P+ 1. The value of the functionf(S) can be computed in polynomial time by any known polynomial algorithm for linear programming. Ifk= 1 and allBi¿mthen
problem (6) – (9) is the continuous multiple-choice knapsack problem andf(S) can be computed by the algorithm with running time O(m|S|) [1,6].
2. Algorithm and its analysis
We now describe the greedy algorithm for solving problem (1).
Let the maximum in (10) is attained on the indexit.
SetSt=St−1∪ {i
t}. If|St|¡ P+ 1, sett=t+ 1 and
do the next step, otherwise, stop.
In the proof of the performance guarantee we will use the inequality due to Wolsey [5]: ifP andDare arbitrary positive integers,i; i= 1; : : : ; Pare arbitrary
nonnegative reals and1¿0 (notice that Wolsey uses slightly more general conditions), then
PP
Theorem 1. The worst-case ratio of the greedy algo-rithm for solving problem(1)with objective function satisfying condition(5)is equal(1−e−(k+1))=(k+1).
Proof. The proof is a straightforward modication of the Nemhauser’s, Wolsey’s and Fisher’s one. Let St; t= 0; : : : ; Pbe the sets dened in the description
of the algorithm and letS∗ be an optimal solution of the problem (1). For allt= 0; : : : ; P−1 we have
f(S∗)6(k+ 1)f(St) +X
i∈S∗\St
(f(St∪ {i})−f(St))
6(k+ 1)f(St) +Pt+1: (12)
The last inequality follows from the facts that f(St∪{i})−f(St)6
The last inequality is equivalent to the following one:
ek¿k(1−e−1) + 1
and therefore holds for allk¿0.
3. Properties of the functionf(S;A;C)
We will prove the inequality f(S;A;C1−; C2; : : : ; Ck)
Without loss of generality, we assume that g1=c1¿
g2=c2¿· · ·¿gn=cn and Pi∈Icibi¿ B. If the input
data violate the second assumption then, trivially,xi=
bi for alli ∈I. The optimal solution of this problem
has the following property:xi=bifori= 1; : : : ; s−1,
xi= 0 fori=s+ 1; : : : ; nandxs=B−Psi=1−1 cibifor
some index s (See the proof of this property in the case when all bi= 1 in [3], Theorem 2:1. The
gen-eral case can be treated in the same way). Using this property we obtain that for any ∈ [0; B] the fol-lowing inequality holds:G(B−)=(B−)¿G(B)=B and thereforeG(B−)=G(B)¿1−=B. Moreover, we may assume that the optimal solution of the knap-sack problem with budgetB−is not bigger in each coordinate then the optimal solution of the problem with budgetB. Using the same argument we now can prove inequality (13). Let (yij); i∈S; j∈J be an
op-timal solution of problem (6) – (9) corresponding the functionf(S;A;C). Consider the following knapsack problem
The optimal solution of this problem with B = C1 − is a feasible solution of problem (6) – (9)
The next theorem is the main statement of this paper. It generalizes the property of the capacitated maximum p-facility location problem proved by Wolsey [5].
Theorem 2. The functionf(S;A;C)satises the fol-lowing inequality for allS; T⊆I
f(T)6(k+ 1)f(S) + X
p∈T\S
(f(S∪ {p})−f(S));
wherek is a number of constraints(9).
Proof. We will prove an equivalent inequality
(k+ 1)f(S;A;C) +X
p∈T
f(S∪ {p};A;C)
¿f(T;A;C) +|T|f(S;A;C):
Let xpj(T), p ∈ T; j ∈ J be an optimal solution
of problem (6) – (9) corresponding the function f(T;A;C). Let pl = Pj∈J cpjlxpj(T), p =
Inequality (14) holds since from the optimal solution (yij),i∈S; j∈J of problem (6) – (9) corresponding
the functionf(S;A−Xp;C−p) one can obtain the
feasible solution (xij),i∈S∪ {p}; j∈J of problem
(6) – (9) corresponding the functionf(S∪ {p};A;C) with objective value equal the value of the right-hand side of (14). This solution is dened as follows:xij=
Therefore, it is sucient to prove the inequality
(k+ 1)f(S) +X
p∈T
f(S;A−Xp;C−p)¿|T|f(S)
for proving the statement of the theorem. Using in-equality (13) we obtain
f(S;A−Xp;C−p)
Summing this inequalities over p∈T and using the facts that P
Applying the same argumentktimes we obtain
kf(S) +X
Hence, it is sucient to prove the following inequality
f(S;A;C) +X
Proof. We partition the vector z into k vectors y1; : : : ; yk with nonnegative components such that
kyik=
iandPki=1yi=z. Then we denezi=z−yi. It is easy to see that all properties of the claim are held for the vectorszi.
By using Proposition 1, we will nd a feasible solu-tions of the problems in the left hand side of (15) such that the inequality will hold. Therefore the inequality will hold for the optimal solutions too. Let (xij(S))
be an optimal solution of problem (6) – (9) corre-sponding the function f(S;A;C). Recall that Xp =
(xp1(T); : : : ; xpm(T)). Dene vectors
By assumptions made in introduction, kxj(S)k = kxj(T)k=Aj¿0. We construct (by Proposition 1) a feasible solution of problem (6) – (9) corresponding the functionf(S;A−Xp;C) and
We have given the performance guarantee of the greedy algorithm for generalization of the maximum capacitated p-facility location problem.
There are a number of interesting questions raised by this work. Clearly, the analysis of Section 2 can be applied to the set functions satisfying the following condition: for allS; T⊆I
f(T)6(k+ 1)f(S) +L X
p∈T\S
(f(S∪ {p})−f(S));
(16)
where L ¿0 is a some constant. Are there natural members of this class of functions? Does there exist a characterization of set functions satisfying conditions (5) or (16)? Are other results from the theory of sub-modular set functions generalizable to these classes of set functions?
proves that 1−e−1 is the best possible performance guarantee for the Max p-Cover unlessP=NP. Max p-Cover is a special case of the problem (1) with func-tions (2) – (4), i.e. the casek= 0. We conjecture that the result of Feige can be extended to the case of gen-eral integerk¿0.
References
[1] M.E. Dyer, An O(n) algorithm for the multiple choice knapsack problem, Math. Programming 29 (1984) 57–63.
[2] U. Feige, A Threshold of lnnfor Approximating Set Cover, J. Assoc. Comput. Math. 45 (1998) 634–652.
[3] S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990. [4] G.L. Nemhauser, L.A. Wolsey, M.L. Fisher, An analysis of
approximations for maximizing submodular set functions-1, Math. Programming 14 (1978) 265–294.
[5] L.A. Wolsey, Maximising real-valued submodular functions: primal and dual heuristics for location problems, Math. Oper. Res. 7 (1982) 410–425.
