Page 1

arXiv:1112.1136v1 [cs.DS] 6 Dec 2011

Secretary Problems with Convex Costs

Siddharth BarmanSeeun Umboh

David Malec

Shuchi Chawla

University of Wisconsin–Madison

{sid, seeun, shuchi, dmalec}@cs.wisc.edu

Abstract

We consider online resource allocation problems where given a set of requests our goal is to select

a subset that maximizes a value minus cost type of objective function. Requests are presented online

in random order, and each request possesses an adversarial value and an adversarial size. The online

algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The “profit”

of a set of accepted requests is its total value minus a convex cost function of its total size. This problem

falls within the framework of secretary problems. Unlike previous work in that area, one of the main

challenges we face is that the objective function can be positive or negative and we must guard against

accepting requests that look good early on but cause the solution to have an arbitrarily large cost as more

requests are accepted. This requires designing new techniques.

We study this problem under various feasibility constraints and present online algorithms with com-

petitive ratios only a constant factor worse than those known in the absence of costs for the same feasi-

bility constraints. We also consider a multi-dimensional version of the problem that generalizes multi-

dimensional knapsack within a secretary framework. In the absence of any feasibility constraints, we

presentanO(ℓ) competitivealgorithmwhereℓis thenumberofdimensions; thismatcheswithinconstant

factors the best known ratio for multi-dimensional knapsack secretary.

1 Introduction

We study online resource allocation problems under a natural profit objective: a single server accepts or

rejects requests for service so as to maximize the total value of the accepted requests minus the cost imposed

by them on the system. This model captures, for example, the optimization problem faced by a cloud

computing service accepting jobs, a wireless access point accepting connections from mobile nodes, or an

advertiser in a sponsored search auction deciding which keywords to bid on. In many of these settings, the

server must make accept or reject decisions in an online fashion as soon as requests are received without

knowledge of the quality future requests. We design online algorithms with the goal of achieving a small

competitive ratio—ratio of the algorithm’s performance to that of the best possible (offline optimal) solution.

A classical example of online decision making is the secretary problem. Here a company is interested in

hiring a candidate for a single position; candidates arrive for interview in random order, and the company

must accept or reject each candidate following the interview. The goal is to select the best candidate as

often as possible. What makes the problem challenging is that each interview merely reveals the rank of the

candidate relative to the ones seen previously, but not the ones following. Nevertheless, Dynkin [11] showed

that it is possible to succeed with constant probability using the following algorithm: unconditionally reject

the first 1/e fraction of the candidates; then hire the next candidate that is better than all of the ones seen

1

Page 2

previously. Dynkin showed that as the number of candidates goes to infinity, this algorithm hires the best

candidate with probability approaching 1/e and in fact this is the best possible.

More general resource allocation settings may allow picking multiple candidates subject to a certain

feasibility constraint. We call such a problem a generalized secretary problem (GSP) and use (Φ,F) to

denote an instance of the problem. Here F denotes a feasibility constraint that the set of accepted requests

must satisfy (e.g. the size of the set cannot exceed a given bound), and Φ denotes an objective function

that we wish to maximize. As in the classical setting, we assume that requests arrive in random order; the

feasibility constraint F is known in advance but the quality of each request, in particular its contribution

to Φ, is only revealed when the request arrives. Recent work has explored variants of the GSP where Φ

is the sum over the accepted requests of the “value” of each request. For such a sum-of-values objective,

constant factor competitive ratios are known for various kinds of feasibility constraints including cardinality

constraints [17, 19], knapsack constraints [4], and certain matroid constraints [5].

In many settings, the linear sum-of-values objective does not adequately capture the tradeoffs that the

server faces in accepting or rejecting a request, and feasibility constraints provide only a rough approxima-

tion. Consider, for example, a wireless access point accepting connections. Each accepted request improves

resource utilization and brings value to the access point. However as the number of accepted requests grows

the access point performs greater multiplexing of the spectrum, and must use more and more transmitting

power in order to maintain a reasonable connection bandwidth for each request. The power consumption

and its associated cost are non-linear functions of the total load on the access point. This directly translates

into a value minus cost type of objective function where the cost is an increasing function of the load or total

size of all the requests accepted.

Our goal then is to accept a set A out of a universe U of requests such that the “profit” π(A) = v(A) −

C(s(A)) is maximized; here v(A) is the total value of all requests in A, s(A) is the total size, and C is a

known increasing convex cost function1.

Note that when the cost function takes on only the values 0 and ∞ it captures a knapsack constraint,

and therefore the problem (π,2U) (i.e. where the feasibility constraint is trivial) is a generalization of the

knapsack secretary problem [4]. We further consider objectives that generalize the ℓ-dimensional knapsack

secretary problem. Here, we are given ℓ different (known) convex cost functions Cifor 1 ≤ i ≤ ℓ, and

each request is endowed with ℓ sizes, one for each dimension. The profit of a set is given by π(A) =

v(A) −?ℓ

costs, we obtain online algorithms with competitive ratios within a constant factor of those achievable for a

sum-of-values objective with the same feasibility constraints. For ℓ-dimensional costs, in the absence of any

constraints, we obtain an O(ℓ) competitive ratio. We remark that this is essentially the best approximation

achievable even in the offline setting: Dean et al. [9] show an Ω(ℓ1−ǫ)hardness for the simpler ℓ-dimensional

knapsack problem under a standard complexity-theoretic assumption. For the multi-dimensional problem

with general feasibility constraints, our competitive ratios are worse by a factor of O(ℓ5) over the corre-

sponding versions without costs. Improving this factor is a possible avenue for future research.

We remark that the profit function π is a submodular function. Recently several works [13, 6, 16] have

looked at secretary problems with submodular objective functions and developed constant competitive algo-

rithms. However, all of these works make the crucial assumption that the objective is always nonnegative;

it therefore does not capture π as a special case. In particular, if Φ is a monotone increasing submodular

function (that is, if adding more elements to the solution cannot decrease its objective value), then to obtain

i=1Ci(si(A)) where si(A) is the total size of the set in dimension i.

Weconsider theprofitmaximization problem under various feasibility constraints. Forsingle-dimensional

1Convexity is crucial in obtaining any non-trivial competitive ratio—if the cost function were concave, the only solutions with

a nonnegative objective function value may be to accept everything or nothing.

2

Page 3

a good competitive ratio it suffices to show that the online solution captures a good fraction of the optimal

solution. In the case of [6] and [16], the objective function is not necessarily monotone. Nevertheless,

nonnegativity implies that the universe of elements can be divided into two parts, over each of which the

objective essentially behaves like a monotone submodular function in the sense that adding extra elements

to a good subset of the optimal solution does not decrease its objective function value. In our setting, in

contrast, adding elements with too large a size to the solution can cause the cost of the solution to become

too large and therefore imply a negative profit, even if the rest of the elements are good in terms of their

value-size tradeoff. As a consequence we can only guarantee good profit when no “bad” elements are added

to the solution, and must ensure that this holds with constant probability. This necessitates designing new

techniques.

Our techniques.

classify elements as “good” or“bad” based onathreshold on their value tosize ratio (a.k.a. density) such that

any large enough subset of the good elements provides a good approximation to profit; the optimal threshold

is defined according to the offline optimal fractional solution. Our algorithm learns an estimate of this

threshold from the first few elements (that we call the sample) and accepts all the elements in the remaining

stream that cross the threshold. Learning the threshold from the sample is challenging. First, following the

intuition about avoiding all bad elements, our estimate must be conservative, i.e. exceed the true threshold,

with constant probability. Second, the optimal threshold for the sample can differ significantly from the

optimal threshold for the entire stream and is therefore not a good candidate for our estimate. Our key

observation is that the optimal profit over the sample is a much better behaved random variable and is, in

particular, sufficiently concentrated; we use this observation to carefully pick an estimate for the density

threshold.

With general feasibility constraints, it is no longer sufficient to merely classify elements as good and

bad: an arbitrary feasible subset of the good elements is not necessarily a good approximation. Instead, we

decompose the profit function into two parts, each of which can be optimized by maximizing a certain sum-

of-values function (see Section 4). This suggests a reduction from our problem to two different instances

of the GSP with sum-of-values objectives. The catch is that the new objectives are not necessarily non-

negative and so previous approaches for the GSP don’t work directly. We show that if the decomposition of

the profit function is done with respect to a good density threshold and an extra filtering step is applied to

weed out bad elements, then the two new objectives on the remaining elements are always non-negative and

admit good solutions. At this point we can employ previous work on GSP with a sum-of-values objective

to obtain a good approximation to one or the other component of profit. We note that while the exposition

in Section 4 focuses on a matroid feasibility constraint, the results of that section extend to any downwards-

closed feasibility constraint that admits good offline and online algorithms with a sum-of-values objective2.

In the multi-dimensional setting (discussed in Section 5), elements have different sizes along different

dimensions. Therefore, a single density does not capture the value-size tradeoff that an element offers.

Instead we can decompose the value of an element into ℓ different values, one for each dimension, and

define densities in each dimension accordingly. This decomposes the profit across dimensions as well.

Then, at a loss of a factor of ℓ, we can approximate the profit objective along the “best” dimension. The

problem with this approach is that a solution that is good (or even best) in one dimension may in fact be

terrible with respect to the overall profit, if its profit along other dimensions is negative. Surprisingly we

show that it is possible to partition values across dimensions in such a way that there is a single ordering over

In the absence of feasibility constraints (see Section 3), we note that it is possible to

2We obtain an O(α4β) competitive algorithm where α is the best offline approximation and β is the best online competitive

ratio for the sum-of-values objective.

3

Page 4

elements in terms of their value-size tradeoff that is respected in each dimension; this allows us to prove that

a solution that is good in one dimension is also good in other dimensions. We present an O(ℓ) competitive

algorithm for the unconstrained setting based on this approach in Section 5, and defer a discussion of the

constrained setting to Section 6.

Related work.

survey. Recently a number of papers have explored variants of the GSP with a sum-of-values objective.

Hajiaghayi et al. [17] considered the variant where up to k secretaries can be selected (a.k.a. the k-secretary

problem) in a game-theoretic setting and gave a strategyproof constant-competitive mechanism. Klein-

berg [19] later showed an improved 1 − O(1/√k) competitive algorithm for the classical setting. Babaioff

et al. [4] generalized this to a setting where different candidates have different sizes and the total size of the

selected set must be bounded by a given amount, and gave a constant factor approximation. In [5] Babaioff

et al. considered another generalization of the k-secretary problem to matroid feasibility constraints. A

matroid is a set system over U that is downwards closed (that is, subsets of feasible sets are feasible), and

satisfies a certain exchange property (see [21] for a comprehensive treatment). They presented an O(logr)

competitive algorithm, where r is the rank of the matroid, or the size of a maximal feasible set. This was

subsequently improved to a O(√logr)-competitive algorithm by Chakraborty and Lachish [7]. Several pa-

pers have improved upon the competitive ratio for special classes of matroids [1, 10, 20]. Bateni et al. [6]

and Gupta et al. [16] were the first to (independently) consider non-linear objectives in this context. They

gave online algorithms for non-monotone nonnegative submodular objective functions with competitive ra-

tios within constant factors of the ratios known for the sum-of-values objective under the same feasibility

constraint. Other versions of the problem that have been studied recently include: settings where elements

are drawn from known or unknown distributions but arrive in an adversarial order [8, 18, 22], versions where

values are permuted randomly across elements of a non-symmetric set system [24], and settings where the

algorithm is allowed to reverse some of its decisions at a cost [2, 3].

The classical secretary problem has been studied extensively; see [14, 15] and [23] for a

2 Notation and Preliminaries

We consider instances of the generalized secretary problem represented by the pair (π,F), and an implicit

number n of requests or elements that arrive in an online fashion. U denotes the universe of elements.

F ⊂ 2Uis a known downwards-closed feasibility constraint. Our goal is to accept a subset of elements

A ⊂ U with A ∈ F such that the objective function π(A) is maximized. For a given set T ⊂ U, we use

O∗(T) = argmaxA∈F∩2T π(A) to denote the optimal solution over T; O∗is used as shorthand for O∗(U).

We now describe the function π.

In the single-dimensional cost setting, each element e ∈ U is endowed with a value v(e) and a size

s(e). Values and sizes are integral and are a priori unknown. The size and value functions extend to sets

of elements as s(A) =?

following quantities will be useful in our analysis:

e∈As(e) and v(A) =?

e∈Av(e). Then the “profit” of a subset is given by

π(A) = v(A) − C(s(A)) where C is a non-decreasing convex function on size: C : Z+→ Z+. The

• The density of an element, ρ(e) := v(e)/s(e). We assume without loss of generality that densities of

elements are unique and denote the unique element with density γ by eγ.

• The marginal cost function, c(s) := C(s) − C(s − 1). Note that this is an increasing function.

4

Page 5

• The inverse marginal cost function, ¯ s(ρ) which is defined to be the maximum size for which an

element of density ρ will have a non-negative profit increment, that is, the maximum s for which

ρ ≥ c(s).

• The density prefix for a given density γ and a set T, PT

density prefix,¯PT

γ := {e ∈ T : ρ(e) ≥ γ}, and the partial

γand¯PU

γ:= PT

γ\ {eγ}. We use Pγand¯Pγas shorthand for PU

γrespectively.

We will sometimes find it useful to discuss fractional relaxations of the offline problem of maximizing

π subject to F. To this end, we extend the definition of subsets of U to allow for fractional membership.

We use αe to denote an α-fraction of element e; this has value v(αe) = αv(e) and size s(αe) = αs(e). We

say that a fractional subset A is feasible if its support supp(A) is feasible. Note that when the feasibility

constraint can be expressed as a set of linear constraints, this relaxation is more restrictive than the natural

linear relaxation.

Note that since costs are a convex non-decreasing function of size, it may at times be more profitable

to accept a fraction of an element rather than the whole. That is, argmaxαπ(αe) may be strictly less than

1. For such elements, ρ(e) < c(s(e)). We use F to denote the set of all such elements: F = {e ∈ U :

argmaxαπ(αe) < 1}, and I = U \ F to denote the remaining elements. Our solutions will generally

approximate the optimal profit from F by running Dynkin’s algorithm for the classical secretary problem;

most of our analysis will focus on I. Let F∗(T) denote the optimal (feasible) fractional subset of T ∩ I for

a given set T. Then π(F∗(T)) ≥ π(O∗(T ∩ I)). We use F∗as shorthand for F∗(U), and let s∗be the size

of this solution.

In the multi-dimensional setting each element has an ℓ-dimensional size s(e) = (s1(e),...,sℓ(e)). The

cost function is composed of ℓ different non-decreasing convex functions, Ci: Z+→ Z+. The cost of a set

of elements is defined to be C(A) =?

iCi(si(A)) and the profit of A, as before, is its value minus its cost:

π(A) = v(A) − C(A).

2.1Balanced Sampling

Our algorithms learn the distribution of element values and sizes by observing the first few elements. Be-

cause of the random order of arrival, these elements form a random subset of the universe U. The following

concentration result is useful in formalizing the representativeness of the sample.

Lemma 2.1. Given constant c ≥ 3 and a set of elements I with associated non-negative weights, wifor

i ∈ I, say we construct a random subset J by including each element of I uniformly at random with

probability 1/2. If for all k ∈ I, wk≤1

least 0.76:

c

?

?

i∈Iwithen the following inequality holds with probability at

j∈J

wj≥ β(c)

?

i∈I

wi,

where β(c) is a non-decreasing function of c (and furthermore is independent of I).

We begin the proof of Lemma 2.1 with a restatement of Lemma 1 from [12] since it plays a crucial role

in our argument. Note that we choose a different parameterization than they do, since in our setting the

balance between approximation ratio and probability of success is different.

Lemma 2.2. Let Xi, for i ≥ 1, be indicator random variables for a sequence of independent, fair coin flips.

Then, for Si=?i

5

k=1Xk, we have Pr[∀i, Si≥ ⌊i/3⌋] ≥ 0.76.