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Whom to marry, how to cook andwhere to buy gas: solving

dilemmas of daily life, onealgorithm at a time

SAMIR KHULLERDept. of Computer Science

University of Maryland

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y y

A typical conversation

Person: What do you do?

Me: I am a computer science professor.

Person: I have a problem with my PC, can you fix it?

Me: No, I dont think I can do that.

Person: You will not fix my PC?

Me: I cannot fix my PC, let alone yours.

Person: Then what exactly do you do?

Me: I study algorithms.

Person: Oh, I know that.

Me: Really?

Person: Yes! I learnt logarithms in high school.

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Algorithms not Logarithms!

AL GO R I T H M

Al-Khowarizmi

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Algorithms Introduction

Recipe for baking a cake.

2 sticks butter

2 cups flour

1 cup sugar

4 eggs

1 cup milk

1 tsp baking powder

Cocoa powder (1/2 pound)

Mix the sugar, baking powder and flour, mix in beateneggs, melted butter and bake at 325F for 40 mins.

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ALGORITHMS

Set ofinstructions for solving a problem, tofind a solution.

What is a problem?

What is an instruction?

What is a solution?

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Computer Science

What is the computeractually doing?

Its running a program

(a set of instructions),but what is theprogram doing?

Typically, analgorithm is what theprogram implements.

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Outline of talk

Algorithms and their Applications

Whom to Marry?

How to Cook? Where to buy gas?

A few favorite projects of mine..

Acknowledgements

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Why are algorithms central tocomputing?

An airport shuttle company needs to schedulepickups delivering everyone to the airport ontime. Who goes in which shuttle, and in what

order do the pickups occur? A delivery company has several customers and

trucks that can carry objects. How should theyschedule deliveries to the customers to minimize

their cost?

This leads to interesting algorithmic problems

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There are lots of feasible solutions!

How should we pick amongst thesesolutions?

Some solutions are cheap, and others

may be expensive or undesirable. The number of potential solutions is so

large that even a fast computer cannot

evaluate all these solutions. Algorithms tell us how to find good

solutions!

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A disclaimer

I have chosen a set of problems whosealgorithms are quite simple.

Towards the end of the talk I will alsomention some recent projects that are alittle more involved, and its hard to reallydescribe the algorithms and methods used

since they are quite complex.

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The Marriage Problem

N men, N women

Each person provides a ranking of themembers of the opposite sex

Can we find a good marriage?

First studied by Gale and Shapley (1962)

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An application: resident matchingprogram

Each resident rank orders the hospitals,and each hospital rank orders theresidents.

How do we choose an assignment ofresidents to hospitals?

We do not want a situation that a resident

prefers another hospital, and that hospitalpreferred this resident to the personassigned to them.

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Mens Preference Lists

Brad (B)

George(G)

Vince(V)

Jennifer(J) Laura(L) Angelina(A)1 2 3

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Womens Preference List

Jennifer(J)

Angelina(A)

Laura(L)

1 2 3

Brad(B) George(G) Vince(V)

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Stable Marriage Problem

A marriage is unstable if there is a pair ofpeople, not married to each other, suchthat both prefer each other to their currentpartners. In other words, they have anincentive to elope.

Can we find a stable marriage?

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Stable marriage?

(Brad, Jen)

(Vince, Angelina)

(George, Laura)

Unstable since

Jen and Vinceboth prefereach other totheir currentpartners.

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Running the Algorithm

FIRST ROUND:

Brad proposes to Jen

Vince proposes to Laura

George proposes to Jen

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Brad proposes!

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FIRST ROUND:

Brad proposes to Jen

Vince proposes to Laura

George proposes to Jen

Jen rejects George, engaged to Brad

Laura engaged to Vince

Angelina gets no proposals.

(Brad,Jen) and (Vince, Laura)

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SECOND ROUND:

George proposes to Laura

Laura breaks engagement with

Vince, and gets engaged toGeorge

(Brad,Jen) and (George,Laura)

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THIRD ROUND:

Vince proposes to Jen

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Jen dumps Brad!

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THIRD ROUND:


Jen breaks engagement with

Brad, and gets engaged toVince

(Vince,Jen) and (George,Laura)

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FOURTH ROUND:

Brad proposes to Angelina

Angelina accepts and gets

engaged to Brad

(Vince,Jen), (George,Laura) and(Brad,Angelina)

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The couples

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Stable marriage?

(Brad, Angelina)

(Vince, Jen)

(George, Laura)

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This solution is stable!

(Vince, Jen) (George, Laura) (Brad, Angelina)

Vince prefers Laura to his partner Jen, butLaura would rather be with George.

Brad prefers Jen to Angelina, but Jenwould rather be with Vince.

George prefers Jen to Laura, but Jenwould rather be with Vince.

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Optimal from the mens point of

view

Each man gets the best possible partner

in ANY stable solution.

Unintuitive: looks like the marriage is a

good one for the women as well, or is it?

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Consider a different instance

Brad proposes to Angelina


George proposes to Laura

All women accept since they only

get one offer.

NOTE: Each woman is paired withthe worst possible partner.Now run the algorithm with the womenproposing..

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Online stable marriages

Assume that womens preferences are

known in advance. The men arrive one ata time and pick their most preferred

available partner. This does not give a stable solution, and in

fact may have MANY unstable pairs.

Paper by Khuller, Mitchell and Vazirani (1991).

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What went wrong? Peoples preferences change.(?)

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Scheduling Problems

Arise in many industrial applications.

Computers schedule multiple tasks,people multi-task, complex projects haveseveral interacting sub-parts.

With large companies manufacturing manyproducts, many interesting scheduling

problems arise.

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Cooking example

Salad:

25m prep, 0m cooking

Chicken noodle:10m prep, 40 min cooking

Rice pudding:15 mins prep, 20m cooking

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In what order should Martha makethe dishes?

Martha can work on preparing one dish ata time, however once something iscooking, she can prepare another dish.

How quickly can she get all the dishesready?

She starts at 5pm, and her guests will

arrive at 6pm.

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First try

5:00pm 5:25pm 5:35pm5:50pm

6:15pm 6:10pm

(25,0) (10,40) (15,20)

Prep time Cook time

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Second try

5:00pm 5:10pm 5:25pm 5:50pm

5:50pm 5:45pm

(10,40) (15,20)(25,0)

First work on dishes with shortest preparation time?

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This rule may not work all the time

Suppose the required times are:

Bulgur (5,10) Lentils (10, 60) Lamb (15, 75)

Shortest prep time order: start at 5pm, andfinish lamb at 6:45pm

Longest cooking time first: food ready at6:30pm.

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What if she had to make severaldishes?

For 3 dishes, there are only 6 possibleorders. SCR,SRC,RSC,RCS,CSR,CRS.

The number of possible orderings of 10dishes is 3,628,800.

For 15 dishes the number of possibleorderings is 1,307,674,368,000!

This leads to a combinatorial explosion.

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Key Idea

Order dishes in longest cooking time order.

Chicken noodle soup goes first (40 mins of cooktime), next is the Rice pudding (20 mins of cooktime), followed by the Salad (0 mins of cooktime).

This is the best ordering. In other words, noother order can take less time.

This does not work if there are very fewstovetops (now the problem becomes reallydifficult).

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What if we had a small number ofburners?

Problem becomesvery difficult if wehave 2, 3, 4 burners..

Problem can besolved optimally if weonly have one burner(Johnson, 1954)

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Where to fill gas?

Suppose you want to go on a road trip across theUS. You start from New York City and would like todrive to San Francisco.

You have :

roadmap

gas station locations and their gas prices

Want to: minimize travel cost

The Gas Station Problem (Khuller, Malekian,Mestre), Eur. Symp. of Algorithms

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Finding gas prices online

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Two vertices s & t

A fixed path

Optimal solution involves stops at everystation!

Thus we permit at most stops.

Structure of the Optimal Solution

sv1 v2 v3 vn t

2.99$ 2.97$2.98$ 1.00$

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The Problem we want to solve

Input: Road map G=(V,E), source s, destination t

U: tank capacity

d: ER+ c: VR+ : No. of stops allowed : The initial amount of gas at s

Goal: Minimize the cost to go from s to t.

Output: The chosen path

The stops

The amount of gas filled at each stop

Gas cost is per mile and U is the range of the car in

miles.

s

U -

s

c(s)= 0

t

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Dynamic Programming

Assuming all values are integral, we can find anoptimal solution in O(n2 U2) time.

Not strongly polynomial time.The problem could be weakly NP-hard!

OPT[x,q,g] =Minimum cost to go from x to t in q stops,starting with g units of gas.

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Key Property

ui ui+1

c(ui) c(ui+1)

Suppose the optimal sequence of stops is u1,u2,,u.

If c(ui) < c(ui+1)

Fill up the whole tank If c(ui) > c(ui+1)Just fill enough to reach ui+1.

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Tour Gas Station problem

Would like to visit a set of cities T.

We have access to set of gas stations S.

Assume gas prices are uniform.

The problem is extremely hard even with thisrestriction.

We may have a deal with a particular gas

company.

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The research process

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Problems, Graphs and Algorithms

Is there a way to walk on every bridge exactlyonce and return to the starting point?

L. Euler (17071783)

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New England Kidney Exchange

A donors kidneymay not match theperson they wish todonate to.

In this case, perhapsanother pair has thesame problem and

the kidneys can beswapped.

A C

D

B

Use an algorithm for Maximum matchings in graphs (Edmonds 1965).

Each node here

is a COUPLE

Re-assigning Starbucks employees

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Re assigning Starbucks employeesto reduce commute times

Article from the Washington PostA

BC

DD

B

C

A

S S

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Energy Minimization

Consider fire monitoring.Sensors have: Fixed locations Limited battery power

If sensors are always on: Full coverage over the regions Short system life time

Better solution: Activating sensors in multiple time

slots

Benefits: Make use of overlap Turning sensors on and off

increase their life time

Sensors A

B

C

D

S1

S2

S3S4

Work with A. Deshpande, A.Malekian

and M. Toossi

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Data Placement & Migration

Data Layout: load balancing Disks have constraints on space, load, etc.

User data access pattern which changes with time

Primarily joint work with L. Golubchik, S. Kashyap, Y-A. Kim, S. Shargorodskaya,J. Wan, and A. Zhu

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Approximation Algorithms

For many problems, no simple (or complex!) rules seemto work.

Such problems arise very frequently the famousTraveling Salesperson Problem is an example.

How should we cope with this? Our attempt is to develop a set of tools that would give

rise to methods for approaching such problems. Even ifwe cannot find the optimal solutions quickly, perhaps wecan find almost optimal solutions quickly?

Greedy Methods, LP rounding methods, Primal-Dualmethods.

In general, these methods also provide lower boundingmethods

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Acknowledgements

Lecture dedicated to the memory of my grandfather, Prof. Ish Kumar (19021999).

Academic InfluenceProf. R. Karp, Prof. V. Vazirani, Prof. J. Mitchell, Prof. E. Arkin, Prof. U. Vishkin

My wonderful co-authors, especially, B. Raghavachari, N. Young, A. Srinivasan, L.Golubchik, B.Bhattacharjee, D. Mount, S. Mitchell, B. Schieber, A. Rosenfeld, J.

Naor, R. Hassin, S. Guha, M. Charikar, R. Thurimella, R. Pless, M Shayman, G.Kortsarz.

My Ph.D. students R. Bhatia, Y.Sussmann, R. Gandhi, Y-A. Kim, J. Wan, J. Mestre,S. Kashyap, A. Malekian.

Undergrads K. Matherly, D. Hakim, J. Pierce, B. Wulfe, A. Zhu, S. Shargorodskaya,C. Dixon, J. Chang, M. McCutchen.

Above all, a BIG thanks to all members of my family, friends and relatives. I cannotexpress my thanks deeply enough.

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