Improved Competitive Ratios for Submodular Secretary Problems

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Moran Feldman Roy SchwartzJoseph (Seffi) Naor

Technion – Israel Institute of Technology

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Outline• The classical secretary problem– Algorithm– Alternative arrival process

• Prelimineries– Set functions– Competitive ratio

• Submodular Secretary Problems– State of the art and our results– Partition matroid– Uniform matroid

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The Secretary Problem• n secretaries, each with a unique

internal value vi.• The Secretaries are interviewed one

after the other, in a random order.• In the interview:– The value of the secretary is revealed.– We can hire the secretary, or dismiss her.– In both cases, the decision is final.

We can succeed with probability 1/e! [Bruss84]

v1v2v3v4v5v6v7v8v9

Goal: Hire the best secretary with high probability.Question: How good can we do?

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The AlgorithmAlgorithm1. Inspect the first n/e secretaries; let v be the value of the best

secretary among these.2. Interview the rest of the secretaries; hire the first one with

value over v.

It can be shown that this algorithm succeeds with probability at least 1/e. [Bruss84]

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The Arrival ProcessThe Arrival ProcessSecretaries arrive in a random order (random permutation).

Alternative Arrival ProcessSecretaries arrive at random times within the range [0, 1].

Equivalence of the Arrival Processes Random arrival times induce a random permutation. Choose n random arrival times, sort them, and assign them to the secretaries

upon arrival.

1/3 1/2 4/5ClaimPresenting and analyzing algorithms for secretary problems under this alternative arrival process is easier.

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Alternative AlgorithmAlternative Algorithm for the Secretary Problem1. Inspect all secretaries till time t = 1/e; let v be the best

secretary among these.2. Interview the rest of the secretaries; hire the first one with

value over v.

Remarks• This algorithm is not identical to the previous one.– For example, this algorithm might inspect no secretaries in step 1.

• Since this is a very simple case, the algorithms and analysis for both arrival processes are very similar.

• For more complex cases, the alternative arrival process yields cleaner proofs.

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Analysis of the Alternative AlgorithmTheoremThe alternative algorithm succeeds with probability at least 1/e.Proof• Let x be the arrival time of the best secretary.• If x ≤ t, the algorithm fails.• If x > t, the algorithm succeeds if:

– The best secretary in the range [0,x) arrives before time t.– Success probability: t/x

• x ~ U(0, 1), thus, the success probability is:e

ttdxxt

t

1ln1

t x

FailSuccess

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Set FunctionsDefinitionGiven a ground set E, a set function f : 2E assigns a number to every subset of the ground set.

PropertiesProperty Definition

Normalization f() = 0

Monotonicity For every two sets A B E:f(A) f(B)

Submodularity For all sets A, B E:f(A) + f(B) f(A B) + f(A B)

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The Improtance of SubmodularityAlternative (more intuitive) Definition• A function f is submodular if for sets A B E and e B:

fe(A) fe(B).• The “economy of scale” feeling of this definition made submodular

functions common in economics and game theory.

Submodular Function in Combinatorics• Submodular functions appear frequently in combinatorial settings.• Here are two simple examples:

Ground Set Submodular FunctionNodes of a graph The number of edges leaving a

set of nodes.Collection of sets The number of elements in the

union of a sub-collection.

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Variants of the Secretary Problem

• Many variants of the secretary problem were considered.• For example:

– Maximize the probability of every secretary to get hired, given that this probability must be independent of the arrival time of the secretary. [BJS10]

– Hire up to k secretaries with maximum total value. [Kleinberg05]– Hire a set of secretaries with maximum total value which is independent in a

given matroid. [BIK07]

• Two kinds of objectives:– Complete the job with maximum probability.– Maximize a given objective functions.

• For the second kind of objectives, algorithms are usually evaluated by their competitive ratio.

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Competitive RatioNotation• I – An instance of a secretary problem.• ALG(I) – The value of an algorithm ALG on I. Notice that ALG(I) is a

random variable of:– The arrival times.– The randomness of ALG itself.

• OPT(I) – The value of the optimal offline algorithm on I.– An offline algorithm know all secretaries from the beginning.

Competitive RatioDefinition:

Remark: This definition of the competitive ratio is adjusted for secretary problems.

IOPTIALGE

Iinf

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Submodular Secretary ProblemsWhat is it?A secretary problem where the objective function is to maximize a given submodular objective function.

What problems were considered?

•NMS – Normalized Monotone Submodular•NS – Nonnegative Submodular

Constraint Objective Function Previous Result Our Result

Partition matroid constraint(given k predefined sets, hire at most one secretary from each set)

NS 5.55 10∙ -5 [GRST10]

-

NMS 5.55 10∙ -5 [GRST10] 0.153

Uniform matroid constraint(hire up to k secretaries)

NS 0.0169 [BHZ10] -

NMS 0.0903 [BHZ10] 0.170

Knapsack constraint NMS 0.0104 [BHZ10] 0.0184

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The Secretary Problem over a Partition Matroid

Definition• Each secretary belongs to one of k sets G1,G2,…,Gk.• At most one secretary of each set can be hired.• Objective: maximize a given objective function over the set of hired secretaries.

Algorithm for linear objectivesApply the algorithm for the standard secretary problem to each set Gi, independently.

TheoremThe above algorithm is e-1-competitive.ProofFollows from the linearity of the expectation since each set is a completely independent instance of the secretary problem.

RemarkSuch a “black box” reduction works only for the alternative arrival process.

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The Secretary Problem over a Partition Matroid (cont.)

Algorithm for submodular objectives1. Inspect all secretaries till time t = ½.2. Interview the rest of the secretaries. Hire secretary s from set Gi, if:

No secretary of Gi was hired before. Among the secretaries of Gi seen so far, s maximizes fs(R), where

R is the set of secretaries hired so far.

RemarkFor linear functions the above algorithm degenerates to the previous one, with a different threshold time.

TheoremThe above algorithm is (1 – ln 2) / 2 ≈ 0.153-competitive.

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The Secretary Problem over a Uniform Matroid

Definition• Hire at most k secretaries.• Objective: maximize a given objective function over the set of hired

secretaries.

Algorithm for linear objectivesDivide time into k intervals. Apply the algorithm for the classical secretary problem to each interval separately.

TheoremThe above algorithm is (e – 1)∙e-2-competitive.

Remarks• The above algorithm is the analog for the alternative arrival process of the

algorithm of [BHZ10].• Better algorithms are known for the linear case.

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The Secretary Problem over a Uniform Matroid (cont.)

LemmaLet si be the best secretary in the ith interval, then:

Proof• The probability that an interval will get no secretary of OPT

is:

• For every interval i, let us select a random representative ri from the secretaries of OPT in interval i.

OPTeeswE

k

ii

11

111

e

k

k

Interval i

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The Secretary Problem over a Uniform Matroid (cont.)

• There are k intervals and in expectation at least (1-e-1)k representatives.

• By symmetry, every secretary is a representative with probability at least 1-e-1.

• By linearity of the expectation:

Proof of the Theorem [the algorithm’s compeitive ratio]• Fix the distribution of secretaries among intervals.• In expectation, the value collected in the ith interval is at least e-1 ∙

w(si).• The expected value of the algorithm is at least:

OPTeeOPTerwE

k

ii

1)1( 1

1

OPTeeswEe

k

ii

2

1

1 1

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The Secretary Problem over a Uniform Matroid (cont.)

Algorithm for submodular objectives• Same as the algorithm for linear objectives.• Inside every intervals, consider the marginal

contributions of the secretaries, instead of their weights.

RemarkFor linear functions the above algorithm degenerates to the previous one.

TheoremThe above algorithm is (e – 1) / (e2 + e) ≈ 0.170-competitive.

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