Improved Competitive Ratios for Submodular
Secretary Problems
(Extended Abstract)
Moran Feldman ⋆ , Joseph (Seffi) Naor ⋆⋆ , and Roy Schwartz
Computer Science Dept., Technion, Haifa, Israel
{moranfe,naor,schwartz}@cs.technion.ac.il
Abstract. The Classical Secretary Problem was introduced during the
60’s of the 20 th century, nobody is sure exactly when. Since its introduction,
many variants of the problem have been proposed and researched.
In the classical secretary problem, and many of its variant, the input
(which is a set of secretaries, or elements) arrives in a random order. In
this paper we apply to the secretary problem a simple observation which
states that the random order of the input can be generated by independently
choosing a random continuous arrival time for each secretary.
Surprisingly, this simple observation enables us to improve the competitive
ratio of several known and studied variants of the secretary problem.
In addition, in some cases the proofs we provide assuming random arrival
times are shorter and simpler in comparison to existing proofs. In this
work we consider three variants of the secretary problem, all of which
have the same objective of maximizing the value of the chosen set of
secretaries given a monotone submodular function f. In the first variant
we are allowed to hire a set of secretaries only if it is an independent set
of a given partition matroid. The second variant allows us to choose any
set of up to k secretaries. In the last and third variant, we can hire any
set of secretaries satisfying a given knapsack constraint.
1 Introduction
In the (classical) secretary problem (CS), a set of n secretaries arrives in a random
order for an interview. Each secretary is associated with a distinct non-negative
value which is revealed upon arrival, and the objective of the interviewer is to
choose the best secretary (the one having maximum value). The interviewer must
decide after the interview whether to choose the candidate or not. This decision
is irrevocable and cannot be altered later. The goal is to maximize the probability
of choosing the best secretary. 1 It is known that the optimal algorithm for CS
⋆
Moran Feldman is a recipient of the Google Europe Fellowship in Market Algorithms,
and this research is supported in part by this Google Fellowship.
⋆⋆
This work was partly supported by ISF grant 1366/07.
1
The above definition of CS is slightly different from the standard one, though both
definitions are equivalent. In the standard definition, the secretaries have ranks instead
of values, and the relative rank of each secretary is revealed upon arrival.is to reject the first n/e secretaries, and then choose the first secretary that is
better than any of the first n/e secretaries. This algorithm succeeds in finding
the best secretary with probability of e −1 [10, 24].
Throughout the years, many variants of CS have been considered. In this
work we focus on variants where a subset of the secretaries can be chosen and
the goal is to maximize some function of this chosen subset. Such variants of
CS have attracted much attention (see, e.g., [3–5, 17, 19]). Some examples of
these variants include: requiring the subset of chosen secretaries to form an
independent set in a given matroid, limiting the size of the subset to at most
k, and requiring the chosen secretaries to satisfy some knapsack constraints. All
these variants have been studied with both linear and submodular objectives.
More details on previous results can be found in Section 1.2.
In this work we use a simple observation which has been employed in problems
other than CS. The observation states that the random order in which the
secretaries arrive can be modeled by assigning each secretary an independent
uniform random variable in the range [0, 1). This continuous random variable
determines the time in which the secretary arrives. Obviously, this modeling is
equivalent to a random arrival order. 2 Though this modeling of the arrival times
as continuous random variables is very simple, it has several advantages when
applied to variants of the secretary problem, on which we elaborate now. First, it
enables us to achieve better competitive ratios for several variants of CS. Second,
the proofs of the performance guarantees of the algorithms are much simpler.
The latter can be exemplified by the following simple problem.
Assume one wishes to partition the arriving secretaries into two sets where
each secretary independently and uniformly chooses one of the sets, and all
secretaries of one set arrive before all secretaries of the other set. An obvious
difficulty is that the position of a secretary in the random arrival order depends
on the positions of all other secretaries. For example, if a set S contains many
secretaries that have arrived early, then a secretary outside of S is likely to
arrive late, since many of the early positions are already taken by members of
S. This difficulty complicates both the algorithms and their analysis. To get
around this dependence [3, 19], for example, partition the secretaries into two
sets: one containing the first m secretaries and the other containing the last
n−m secretaries. The value of m is binomially distributed Bin(n, 1/2). It can be
shown that this partition, together with the randomness of the input, guarantees
that every secretary is uniformly and independently assigned to one of the two
sets. Such an elaborate argument is needed to create the desired partitioning
because of the dependencies between positions of secretaries in the arrival order.
In contrast, using the modeling of the arrival times as continuous random
variables, creating a subset of secretaries where each secretary independently
belongs to it with probability 1/2 is simple. One just has to choose all secretaries
that arrive before time t = 1/2. This simplifies the above argument, designed for
a random arrival order, considerably. This simple example shows how continuous
2
Given a random arrival order, we can sample n independent uniformly random arrival
times, sort them and assign them sequentially to the secretaries upon arrival.arrival times can be used to simplify both the algorithm and its analysis. In the
variants of CS considered in this paper, this simplification enables us to obtain
improved competitive ratios.
For some variants, the algorithms presented in this paper can be viewed as
the continuous “counterparts” of known algorithms (see the uniform matroid and
knapsack variants), but this is not always the case. For the partition matroid
variant, the algorithm we define in this work using continuous arrival times uses
different techniques than previous known algorithms for this case.
It is important to note that the continuous time “counterparts” of algorithms
designed using a random arrival order are not equivalent to the original algorithms.
For example, the optimal algorithm for CS assuming continuous random
arrival times is to inspect secretaries up to time e−1 , and hire the first secretary
after this time which is better than any previously seen secretary (see
Appendix A for an analysis of this algorithm using continuous arrival times).
Observe that this algorithm inspects n/e secretaries in expectation, while the
classical algorithm inspects that number of secretaries exactly. This subtle difference
is what enables us to improve and simplify previous results.
Formally, all problems considered in this paper are online problems in which
the input consists of a set of secretaries (elements) arriving in a random order.
Consider an algorithm A for such a problem, and denote by opt an optimal offline
algorithm for the problem. Let I be an instance of the problem, and let A(I)
and opt(I) be the values of the outputs of A and opt, respectively, given I. We
≥ α,
where the expectation is over the random arrival order of the secretaries of I
and the randomness of A (unless A is deterministic). The competitive ratio is a
standard measure for the quality of an online algorithm.
say that A is α-competitive (or has an α-competitive ratio) if infI E[A(I)]
opt(I)
1.1 Our Results
In this paper we consider variants of CS where the objective function is normalized,
monotone and submodular. 3 There are three variants for which we provide
improved competitive ratios. The first is the submodular partition matroid secretary
problem (SPMS) in which the secretaries are partitioned into subsets, and
at most one secretary from each subset can be chosen. The second is the submodular
cardinality secretary problem (SCS) in which up to k secretaries can be
chosen. The third and last variant is the submodular knapsack secretary problem
(SKS), in which each secretary also has a cost (which is revealed upon arrival),
and any subset of secretaries is feasible as long as the total cost of the subset
does not exceed a given budget.
For SPMS we present a competitive ratio of (1 − ln 2)/2 ≈ 0.153, which
improves on the current best result of Ω(1) by Gupta et al. [17]. We note that
the exact competitive ratio given by [17] is not stated explicitly, however, by
3
Given a groundset S, a function f : 2
S
→ R is called submodular if for every
A, B ⊆ S, f(A) + f(B) ≥ f(A ∪ B) + f(A ∩ B). Additionally, f is called normalized
if f(∅) = 0 and monotone if for every A ⊆ B ⊆ S, f(A) ≤ f(B).inspecting their result carefully it seems that the competitive ratio they achieve
is at most 71/1280000. We note that for SPMS the algorithm we provide is not
a continuous time “counterpart” of the algorithm of [17]. This demonstrates the
fact that modeling the arrival times as continuous random variables helps in
designing and analyzing algorithms for submodular variants of CS.
For SCS we present a competitive ratio of (e − 1)/(e 2 + e) ≈ 0.170, and the
current best result for this problem is due to Bateni et al. [5] who provided a
(1 − e −1 )/7 ≈ 0.0903 competitive ratio. There are two points to notice when
comparing our result and that of [5]. First, [5] did not optimize the competitive
ratio analysis of their algorithm. In fact, their algorithm provides better ratios
then stated, however, it does not seem that their competitive ratio reaches 0.17.
Hence, in this paper we still obtain improved competitive ratios, though the
improvement is smaller than stated above. Second, the algorithm presented in
this paper for SCS can be seen as a continuous time “counterpart” of [5]’s algorithm.
However, our analysis is simpler than the analysis presented in [5], and
also enables us to provide improved competitive ratios.
For SKS we provide a competitive ratio of (20e) −1 ≈ 0.0184. The current best
result is due to Bateni et al. [5] which provide a ratio of Ω(1). The exact competitive
ratio is not stated in [5], but careful inspection of their algorithm shows
that it is at most 96 −1 ≈ 0.0104. Notice that the best known competitive ratio
for the linear version of SKS is only 10e −1 ≈ 0.0368 [3]. As before, the algorithm
presented in this paper for SKS can be seen as a continuous time “counterpart”
of [5]’s algorithm. However, our analysis is simpler than the analysis presented
in [5], enabling us to provide improved competitive ratios. Table 1 summarizes
the above results.
1.2 Related Work
Many variants of CS have been considered throughout the years and we shall
mention here only those most relevant to this work. Babaioff et al. [4] considered
the case where the chosen subset of secretaries needs to be an independent
set of a given matroid, and the objective function f is linear. They provided
a competitive ratio of Ω(log −1 r) for this problem, where r is the rank of the
matroid. For several specific matroids, better constant competitive ratios are
Table 1. Comparison of our results with the known results for the monotone
submodular and linear variants of the problems we consider.
Problem Our Result Previous Result Best Result for Linear Variant
SPMS 0.153 0.0000555 [17] 0.368 1
SCS 0.170 0.0903 [5] 0.368 [3]
SKS 0.0184 0.0104 [5] 0.0368 [3]
1
For linear objective functions one can apply the algorithm for the classical
secretary problem to each subset of secretaries independently.known [4, 9, 18, 20]. The special variant of SCS where the objective function is
linear has also been studied. Two incomparable competitive ratios were obtained
by Babaioff et al. [3] and Kleinberg [19], achieving competitive ratios of e −1 and
1 − O(1/ √ k), respectively. An interesting variant of SCS with a linear objective
function gives each of the k possible slots of secretaries a different weight. The
value of the objective function in this case is the sum of the products of a slots’
weights with the values of the secretaries assigned to them. Babaioff et al. [2]
provide a competitive ratio of 1/4 for this special variant. Additional variants of
CS can found in [1, 6, 13, 14, 16].
Another rich field of study is that of submodular optimization, namely optimization
problems in which the objective function is submodular. In recent years
many new results in this field have been achieved. The most basic problem, in
this field, is that of unconstrained maximization of a nonmonotone submodular
function [12, 15]. Other works consider the maximization of a nonmonotone
submodular function under various combinatorial constraints [17, 22, 25]. The
maximization of a monotone submodular function under various combinatorial
constraints (such as a matroid, the intersection of several matroids and knapsack
constraints) has also been widely studied [7, 8, 21, 23, 25].
Thus, it comes as no surprise that recent works have combined the secretary
problem with submodular optimization. Gupta et al. [17] were the first to
consider this combination. For the variant where the goal is to maximize a submodular
function of the chosen subset of secretaries under the constraint that
this subset is independent in a given matroid, Gupta et al. [17] provide a competitive
ratio of Ω(log −1 r) (where r is the rank of the matroid). If the constraint
is that the chosen subset of secretaries belongs to the intersection of ℓ matroids,
Bateni et al. [5] provide a competitive ratio of Ω(ℓ −1 log −2 r). If the objective
function is submodular and monotone, the special case of a partition matroid is
exactly SPMS, and the special case of a uniform matroid is exactly SCS. As mentioned
before, for SPMS, Gupta et al. [17] provide a competitive ratio of Ω(1) for
SPMS which is at most 71/1280000 (the exact constant is not explicitly stated
in their work). They also get a similar competitive ratio for a variant of SPMS
with a non-monotone submodular objective function. For SCS, Gupta et al. [17]
provide a competitive ratio of Ω(1) which is at most 1/1417 (again, the exact
constant is not explicitly stated in their work), and a bit more complex Ω(1)
ratio for a variant of SCS with a non-monotone submodular objective function.
Both ratios were improved by Bateni et al. [5] who provided a competitive ratio
of (1 − e −1 )/7 ≈ 0.0903 for SCS, and a e −2 /8 ≈ 0.0169-competitive algorithm
for its non-monotone variant. For SKS the current best result is due to Bateni et
al. [5] who provide a competitive ratio of at most 96 −1 ≈ 0.0104 (as before, the
exact constant is not explicitly stated in their work). Another extension considered
by [5] is a generalization of SKS where every secretary has a ℓ dimensional
cost, and the total cost of the secretaries in each dimension should not exceed
the budget of this dimension (i.e., the hired secretaries should obey ℓ knapsack
constraints). For this problem [5] gives an Ω(ℓ −1 ) competitive algorithm.Organization. Section 2 contains formal definitions and several technical lemmata.
Sections 3, 4 and 5 provide the improved competitive ratio for SPMS, SCS
and SKS, respectively.
2 Preliminaries
An instance of a constrained secretary problem consists of three components
S, I and f.
– S is a set of n secretaries arriving in a random order. 4
– I ⊆ 2 S is a collection of independent sets of secretaries. The sets in I
are known in advance in some settings (e.g., SCS), and are revealed over
time in other settings (e.g., SKS). However, at any given time, we know all
independent sets containing only secretaries that already arrived.
– f : 2 S → R is a function over the set of secretaries accessed using an oracle
which given a set S of secretaries that have already arrived, returns f(S).
The goal is to maintain an independent set R of secretaries, and maximize the
final value of f(R) (i.e., its value after all secretaries have arrived). Upon arrival
of a secretary s, the algorithm has to either add it to R (assuming R ∪ {s} ∈ I),
or reject it. Either way, the decision is irrevocable. Given a submodular function
f : S → R + , the discrete derivative of f with respect to s is fs(R) = f(R ∪
{s}) − f(R). We use this shorthand throughout the paper.
Most algorithms for secretary problems with linear objective functions require
every secretary to have a distinct value. This requirement does not make
sense for submodular objective functions, and therefore, we work around it by
introducing a total order over the secretaries, which is a standard practice (see,
e.g., [9]). Formally, we assume the existence of an arbitrary fixed order Z over
the secretaries. If such an order does not exist, it can be mimicked by starting
with an empty ordering, and placing every secretary at a random place in this
ordering upon arrival. The resulting order is independent of the arrival order of
the secretaries, and therefore, can be used instead of a fixed order. Let s1, s2
be two secretaries, and let S be a set of secretaries. Using order Z we define
s1 ≻S s2 to denote “fs1 (S) > fs2 (S), or fs1 (S) = fs2 (S) and s1 precedes s2 in
Z”. Notice that ≻S is defined using f and Z. Whenever we use ≻S, we assume
f is understood from context and Z is the order defined above.
Remark: The probability that two secretaries arrive at the same time is 0, thus
we ignore this event.
The following theorem is used occasionally in our proofs. Similar theorems
appear in [12, 11].
4
If the input is given as a random permutation, the size n of S is assumed to be
known. Note that in order to generate the random arrival times of the secretaries
and assign them upon arrival, n has to be known in advance. On the other hand, if
the arrival times are part of the input, the algorithms in this work need not know n.Theorem 1. Given a normalized monotone submodular function f : 2 S → R + ,
a set A and a random set A ′ containing every element of A with probability at
least p (not necessarily independently). Then, E[f(A ′ )] ≥ p · f(A).
Proof. Order the elements of A in an arbitrary order {a1, a2, . . . , a |A|}. Let Xi
be an indicator for the event {ai ∈ A ′}, and let Ai = {a1, a2, . . . , ai}. Then,
E[f(A ′ ⎡
|A|
∑
)] = E ⎣ Xi · fai
(A ′ ⎤
∩ Ai−1) ⎦ =
|A|
i=1
∑
Pr[Xi = 1] · E [fai (A′ ∑
∩ Ai−1) |Xi = 1] ≥ p · fai (Ai−1) = p · f(A) ,
i=1
|A|
i=1
where the inequality follows from the submodularity of f.
3 (1 − ln 2)/2 ≈ 0.153-Competitive Algorithm for SPMS
The Submodular Partition Matroid Secretary Problem (SPMS) is a secretary
problem with a normalized monotone and submodular objective function f. The
collection I of independent sets is determined by G1 × . . . × Gk (where the Gi’s
are a partition of S). This variant corresponds to the scenario where the goal is
to hire secretaries of different types, one of each type. For every secretary s, the
index of the set Gi containing s is revealed when s arrives.
When designing an algorithm for SPMS, we want to select the best secretary
from every set Gi. If f was linear, we could apply the algorithm for the classical
secretary problem to each Gi separately. The following algorithm is based on a
similar idea.
SPMS Algorithm(f, k):
1. Initialize R ← ∅.
2. Observe the secretaries arriving till time t. a
3. After time t, for every secretary s arriving, let Gi be the set of s. Accept
s into R if:
(a) no previous secretary of Gi was accepted,
(b) and for every previously seen s ′ ∈ Gi, s ′ ≺R s.
4. Return R.
a
t is a constant to be determined later.
In this subsection we prove the following theorem.
Theorem 2. The above algorithm is a (1−ln 2)/2 ≈ 0.153-competitive algorithm
for SPMS.
The algorithm clearly maintains R as a feasible set of secretaries, hence, we only
need to show that, in expectation, it finds a good set of secretaries.
Observation 3 We can assume there is at least one secretary in every set Gi.
Proof. The behavior of the algorithm is not effected by empty sets Gi.3.1 Analysis of a single Gi
In this subsection we focus on a single Gi. Let Ei be an event consisting of the
arrival times of all secretaries in S − Gi, we assume throughout this subsection
that some fixed event Ei occurred. Let Rx be the set of secretaries from S − Gi
collected by the algorithm up to time x assuming no secretary of Gi arrives (observe
that Rx is not random because we fixed Ei). We define ˆsx as the maximum
secretary in Gi with respect to ≻Rx
. The analysis requires two additional event
types: Ax is the event that ˆsx arrives at time x, and Bx is the same event with
the additional requirement that the algorithm collected ˆsx.
Lemma 1. For every x > t, Pr[Bx|Ax] ≥ 1 − ln x + ln t.
Proof. Event Ax states that ˆsx arrived at time x. If no other secretary of Gi is
collected till time x, ˆsx is collected by the definition of the algorithm. Hence, it
is enough to bound the probability that no secretary of Gi is collected till time
x, given Ax.
Observe that Rx takes at most k − 1 values for x ∈ [0, 1). Hence, the range
[0, 1) can be divided into k intervals I1, . . . , Ik such that the set Rx is identical
for all times within one interval. Divide the range [0, x) into small steps of size
∆y such that ∆y divides t and x, and every step is entirely included in an interval
(this is guaranteed to happen if ∆y also divides the start time and the length
of every interval Ii). Since each step is entirely included in a single interval, for
every time x in step j, Rx = R (j−1)·∆y.
A secretary cannot be collected in step j if j · ∆y ≤ t. If this is not the case,
a secretary is collected in step j if the maximum secretary of Gi in the range
[0, j · ∆y) with respect to ≻R (j−1)·∆y
arrives at time (j − 1) · ∆y or later. The
probability that this happens is ∆y/(j · ∆y) = j −1 . We can now use the union
bound to upper bound the probability that any secretary is accepted in any of
the steps before time x:
Pr[Bx|Ax] ≥ 1 −
x/∆y
∑
j=t/∆y+1
j −1 ∫ x/∆y
dj
≥ 1 −
t/∆y j
= 1 − [ln j]x/∆y
t/∆y
= 1 − ln x + ln t .
Let s∗ i denote the single secretary of Gi ∩ OP T , and let ai be the secretary
of Gi collected by the algorithm. If no secretary is collected from Gi, assume
ai is a dummy secretary of value 0 (i.e., f is oblivious to the existence of this
dummy secretary in a set). We also define Ri to be the set R immediately before
the algorithm collects ai (if the algorithms collects no secretary of Gi, Ri is an
arbitrary set).
Observation 4 If Bx occurs for some x, fai (Ri) ≥ fs∗(R), where R is the set
i
returned by the algorithm.
Proof. a
(1)
(2)
fai
(Ri) = fai
(Rx) ≥ fs∗ (3)
(Rx) ≥ fs i ∗(R) .
iWhere (1) and (2) follow from the fact that Bx occurred, and therefore, ai
was collected at time x and ai = ˆsx. Also, Bx implies Rx ⊆ R, hence, the
submodularity of f implies (3).
Let Bi = ∪ x∈(t,1)Bx, and let Pi be {s ∗ i } if Bi occurred, and ∅, otherwise.
Corollary 1. fai(Ri) ≥ f(R ∪ Pi) − f(R).
Proof. If Pi = ∅, the claim follows because fai (Ri) is nonnegative by the monotonicity
of f. If Pi = {s ∗ i }, we know that Bi occurred, and therefore, there must
be an x for which Bx occurred also. The corollary now follows immediately from
Observation 4.
Lemma 2. Pr[Bi] ≥ 2 + ln t − 2t.
Proof. Observe that the events Bx are disjoint, hence, Pr[Bi] is the sum of the
probabilities of the events Bx. Since Bx implies Ax, Pr[Bx] = Pr[Bx|Ax]·Pr[Ax].
The event Ax requires that the maximum secretary with respect to ≻Rx
arrives
in time x. The probability that this secretary arrives in an interval of size ∆x is
∆x. Hence, the probability that it arrives in an infinitesimal interval of size dx
is Pr[Ax] = dx. Therefore,
Pr[Bi] =
∫ 1
t
Pr[Bx|Ax]dx ≥
∫ 1
t
(1 − ln x + ln t)dx = 2 + ln t − 2t .
The last expression is maximized for t = 0.5, thus, we choose t = 0.5.
3.2 Analysis of the Entire Output
Throughout the previous subsection we assumed some fixed event Ei occurred.
Hence, Corollary 1 and Lemma 2 were proven given this assumption, however,
they are also true without it.
Lemma 3. Corollary 1 and Lemma 2 also hold without fixing an event Ei.
Proof. Corollary 1 states that for every fixed Ei, if Bi occurs then fai (Ri) ≥
f(R ∪ Pi) − f(R). Since some event Ei must occur (the secretaries of S − Gi
must arrive at some times), this is also true without fixing some Ei.
Let us rephrase Lemma 2 to explicitly present the assumption that some
fixed Ei occurred: Pr[Bi|Ei] ≥ 1 − ln 2 (recall that we chose t = 0.5). Therefore,
Pr[Bi] = ∑
Pr[Ei] · Pr[Bi|Ei] ≥ (1 − ln 2) · ∑
Pr[Ei] = 1 − ln 2 .
Ei
Ei
Let P = ∪ k i=1 Pi, and notice that P ⊆ OP T . The following lemma lower
bounds the expected value of f(P ).
Lemma 4. E[f(P )] ≥ (1 − ln 2)f(OP T ).Proof. Every element s∗ i ∈ OP T appears in P with probability Pr[Bi]. By
Lemma 2 and the value we chose for t, the last probability is at least 1 − ln 2.
Hence, the lemma follows from Theorem 1.
We are now ready to prove Theorem 2.
Proof (Proof of Theorem 2). Observe the following.
[
k∑
] [
k∑
]
E[f(R)] = E fai(Ri) ≥ E f(R ∪ Pi) − f(R) ≥ E[f(R∪P )]−E[f(R)].
i=1
i=1
Rearranging terms, we get: E[f(R)] ≥
f(R∪P ) f(P ) 1−ln 2
2
≥
2
≥
2
· f(OP T ).
4 The Submodular Cardinality Secretary Problem
The Submodular Cardinality Secretary Problem (SCS) is a secretary problem in
which the objective function f is a normalized monotone submodular function,
and we are allowed to hire up to k secretaries (the scollection I of independent
sets contains every set of up to k secretaries).
Theorem 5. There is a (e − 1)/(e 2 + e) ≈ 0.170-competitive algorithm for SCS.
Due to space limitations, the proof of Theorem 5 is omitted from this extended
abstract.
5 The Submodular Knapsack Secretary Problem
The Submodular Knapsack Secretary Problem (SKS) is a secretary problem in
which the objective function f is a normalized monotone submodular function
and every secretary s has a cost c(s) (revealed upon arrival). A budget B is also
given as part of the input, and the algorithm is allowed to hire secretaries as
long as it does not exceed the budget. In other words, the collection I of allowed
sets contains every set of secretaries whose total cost is at most B.
Theorem 6. There is a 1/(20e)-competitive algorithm for SKS.
Due to space limitations, the proof of Theorem 6 is omitted from this extended
abstract.
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A
Example - the Classical Secretary Problem
The Classical Secretary Problem (CS) is a secretary problem with the set I of
independent sets consisting of all singletons. We demonstrate the usefulness of
the continuous model by analyzing an algorithm for this problem.
CS Algorithm(f):
1. Observe the secretaries arriving till time t = e −1 , and let L be the set
of secretaries arriving until this time.
2. Let ˆs be the maximum secretary in L with respect to ≻∅. a
3. After time t, accept the first secretary s such that f(s) ≻∅ f(ˆs).
a
If no secretary arrives till time t, we assume s ≻∅ ˆs for every secretary s.
Theorem 7. The above algorithm for CS is e −1 -competitive.
Proof. Let s ∗ be the secretary of the optimal solution (breaking ties in favor
of the earlier secretary according to ≻∅). Given that s ∗ arrives at some time
x ∈ (t, 1), s ∗ is accepted if one of the following conditions hold:
– No secretary arrives before time x.
– The best secretary arriving in the time range [0, x) arrives before time t.
Since the secretaries are independent, with probability at least t/x, at least one
of these conditions holds. The probability that s∗ arrives in an interval of size
ℓ is ℓ. Hence, the probability it arrives in an infinitesimal interval of size dx is
dx. Therefore, by the law of total probability, the probability that the above
algorithm accept s∗ is at least
∫ 1
t
t
x dx = t[ln x]1 t = −t ln t = e −1 .