Priority Models Sashka Davis University of California, San Diego June 1, 2003

Priority Models

Sashka Davis

University of California, San Diego

June 1, 2003

2

Goal

Define priority models, which are a formal framework of greedy algorithms

Develop a technique for proving lower bounds for priority algorithms

3

The big picture

Divide and Conquer

GreedyDynamic Programming

Hill-climbing

Polynomial Time Algorithms

Can we build a formal model for the different algorithmic design paradigms?Evaluate the limitations of each technique?Classify the kinds of problems on which the different heuristics perform well

Are the known algorithms optimal or can they be improved?

4

Greedy heuristics

Priority algorithms are a formal model for greedy algorithms.

ADAPTIVE

FIXED

ShortPathPRIORITY ALGORITHMS

5

Common structure of greedy algorithms

They sort items (edges, intervals, etc.)

Consider each item once and either add it to the solution or throw it away

6

Interval scheduling on a single machine

Instance:

Set of intervals I=(i1, i2,…,in), j ij=[rj, dj] Problem: schedule intervals on a single

machine Solution: S I Objective function: maxiS (dj - rj)

7

A simple solution (LPT)

Longest Processing Time algorithm input I=(i1, i2,…,in)

1. Initialize S ← 2. Sort the intervals in decreasing order (dj – rj)

3. while (I is not empty) Let ik be the next in the sorted order

If ik can be scheduled then S ← S U {ik};

I ← I \ {ik}

4. Output S

8

LPT is a 3-approximation

LPT sorts the intervals in decreasing order according to their length

3 LPT ≥ OPT

OPT OPT OPT

LPT

ri di

9

The minimum cost spanning tree problem

Instance:

Edge weighted graph

Problem:

Find a tree of edges that spans V

Objective function

Minimize the cost of the tree

( , ); :G V E E R

T E

min ( )e T

w e

10

A solution for MST problem

Kruskal’s algorithmInput (G=(V,E), w: E →R)1. Initialize empty solution T2. L = Sorted list of edges in increasing order

according to their weight3. while (L is not empty)

e = next edge in L Add the edge to T, as long as T remains a forest

and remove e from L4. Output T

11

Another solution to the MST problem

Prim’s algorithm

Input G=(V,E), w: E →R

1. Initialize an empty tree T ← ; S ← 2. Pick a vertex u; S={u};

3. for (i=1 to |V|-1) (u,v) = min(u,v)cut(S, V-S)w(u,v) S←S {v}; T←T{(u,v)}

Output T

12

Classification of the example algorithms

ADAPTIVE

FIXED

Prim

PRIORITY ALGORITHMS

KruskalLPT

13

Talk outline

1. History of priority algorithms

2. Priority algorithm framework for scheduling problems

3. Priority algorithms for facility location

4. General framework of priority algorithms

5. Future research

14

Results [BNR02]

Defined fixed and adaptive priority algorithms

Proved that fixed priority algorithms are less powerful than adaptive priority algorithms

Considered variety of scheduling problems and proved many non-trivial upper and lower bounds

15

Results [AB02]

Proved tight bounds on performance of

Adaptive priority algorithms for facility location in arbitrary spaces

Fixed priority algorithms for uniform metric facility location

Adaptive priority algorithms for set cover

16

Results [DI02]

Defined a general model of priority algorithms

Proved a strong separation between the classes of fixed and adaptive priority algorithms

Proved a separation between the class of memoryless adaptive priority algorithms and adaptive priority algorithms with memory

Proved tight bound for performance of adaptivepriority algorithms for weighted vertex cover problem

17

Talk outline


2. Priority algorithm framework for scheduling problems [BNR02]



5. Future research

18

The defining characteristics of greedy algorithms [BNR02]

The order in which data items are considered is determined by a priority function, which orders all possible data items

The algorithm sees one input at a time

Decision made for each data item is irrevocable

19

Priority models

Difference: How the next data item is chosen

Fixed PriorityAlgorithms

Adaptive PriorityAlgorithms

Ø

20

Fixed priority algorithms [BNR02]

Input: a set of jobs S

Ordering: Determine without looking at S a total ordering of all possible jobs

while (S is not empty) Jnext - next job in S according to the order above

Decision: Make irrevocable decision for Jnext

S ← S \ {Jnext }

21

Adaptive priority algorithms [BNR02]

Input: a set of jobs S

while (not empty S)

Ordering: Determine without looking at S a total ordering of all possible jobsJnext - next job in S according to the order above

Decision: Make irrevocable decision for Jnext

S ← S \ {Jnext }

22

Separation between fixed and adaptive priority algorithms [BNR02]

Theorem: No fixed priority algorithm can achieve an approximation ratio better than 3 for the interval scheduling problem on multiple machine configuration

Theorem: CHAIN-2 is an adaptive priority algorithm achieving an approximation ratio 2, for interval scheduling on a two machine configuration

23

Online algorithms

1. Must service each request before the next request is received

2. Several alternatives in servicing each request

3. Online cost is determined by the options selected

24

Connection between online and priority algorithms

Similarities The instance is viewed one input at a time Decision is irrevocable

Difference The order of the data items

25

Competitive analysis of online algorithms

t=1; I=∅ Round t

Adversary picks a data item t; I ← I U {t}

Algorithm makes a decision σt for t: A ← A U { (t, σt )} Adversary chooses whether to end the game.

If not, the next round begins: t ←t+1

Adversary picks a solution B for I offline Algorithm is awarded payoff value(A) / value(B)

26

Fixed priority game

Adversary selects a finite set of data items S0; I←;t ←1 Algorithm picks a total order on S0

Adversary restricts the remaining data items: S1 S0

Round t Let t St be next data item in the order Algorithm makes a decision σt for t : A ← A U { (t, σt )} Adversary restricts the set St+1 St – t; I ← I U {t}; Adversary chooses whether to end the game. If

not, the next round begins: t ←t+1 Adversary picks a solution B for I Algorithm is awarded payoff value (A) / value (B)

27

Example lower bound [BNR02]

Theorem1: No priority algorithm can achieve an approximation ratio better than 3 for the interval scheduling problem with proportional profit for a single machine configuration

28

Proof of Theorem 1

Adversary’s move

Algorithm’s move: Algorithm selects an ordering

Let i be the interval with highest priority

3

2

1

q

q-1q-11

2

3

e

29

Adversary’s strategy

If Algorithm decides not to schedule i During next round Adversary removes all

remaining intervals and schedules interval i

3

2

1

i

jk1

2

3

i

30


If i = and Algorithm schedules i During next round the Adversary restricts the

sequence:

3

2

1

i

jk1

2

3

i

i

kj

31


If i = and Algorithm schedules i During next round Adversary restricts the

sequence:

3

2

1

m

jk1

2

3

i

i

m

32

Conclusion

Adversary can pick (q, e) so that the advantage gained is arbitrarily close to 3

No priority algorithm (fixed or adaptive) can achieve an approximation ratio better than 3

LPT achieves an approximation ratio 3

LPT is optimal within the class of priority algorithms

33

Talk outline



3. Priority algorithms for facility location [AB02]


5. Future research

34

[AB02] work on priority algorithms

[AB02] proved lower bounds on performance of adaptive and fixed priority algorithms for the facility location problem in metric and arbitrary spaces, and the set cover problems

35

[AB02] result

Theorem: No adaptive priority algorithm can achieve an approximation ratio better than log(n) for facility location in arbitrary spaces

36

Adaptive priority game

Adversary selects a finite set of data items S0; I←;t ←1

Round t Algorithm picks a data item t and a decision σt for t :

A ← A U { (t, σt )}

Adversary restricts the set St+1 St – t ; I ← I U {t}; Adversary chooses whether to end the game, if not next

round begins t ← t+1

Adversary picks a solution B for I Algorithm is awarded payoff value(A)/value(B)

37

Facility location problem

Instance is a set of cities and set of facilities The set of cities is C={1,2,…,n} Each facility fi has an opening cost cost(fi) and connection

costs for each city: {ci1, ci2,…, cin}

Problem: open a collection of facilities such that each city is connected to at least one facility

Objective function: minimize the opening and connection costs min(ΣfScost(fi) + ΣjCmin fiScij )

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Adversary presents the instance:

Cities: C={1,2,…,n}, where n=2k

Facilities: Each facility has opening cost n City connection costs are 1 or ∞ Each facility covers exactly n/2 cities cover(fj) = {i | i C,cji=1}

Cu denotes the set of cities not yet covered by the solution of the Algorithm

39


At the beginning of each round t The Adversary chooses St to consist of

facilities f such that fSt iff |cover(f) ∩ Cu| = n/(2t)

The number of uncovered cities Cu is n/(2t-1)

Two facilities are complementary if together they cover all cities in C. For any round t St consists of complementary facilities

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The game

Uncovered cities Cu

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End of the game

Either Algorithm opened log(n) facilities or failed to produce a valid solution

Cost of Algorithm’s solution is n.log(n)+n

Adversary opens two facilities incurs total cost 2n+n

42

Conlusion

The Adversary has a winning strategy

No adaptive priority algorithm can achieve an approximation ratio better than log(n)

43

Talk outline




4. General framework of priority algorithms [DI02]

5. Future research

44

Fixed priority algorithms [DI02]

Input: instance ={1,2,…, n},

Output: solution

1. Determine an ordering function

2. Order I according to

3. Repeat Let the next data item in the ordering be Make a decision Update the partial solution S

until (decisions are made for all data items)

4. Output

1 2{ , ,..., }dI I {( , ) | 1,..., }i iS i d

+ { } R

i

{( , ) |1 }S i d

( )

45

Adaptive priority algorithms [DI02]

Input: instance ={1,2,…, n},

Output: solution

1. Initialization

2. Repeat Determine an ordering function Pick the highest priority data item according to Make an irrevocable decision Update:

until (decisions are made for all data items; )

3. Output

1 2{ , ,..., }dI I {( , ) | 1,..., }i iS i d

* + { }S It

R

{( , ) |1 }S i d

*; ; ; 1U I S I t

(t t U

* *\{ }; {( , )}; { }; 1t t t tU U S S I I t t U

46

Strong separation between fixed and adaptive priority algorithms

Theorem: No fixed priority algorithm can achieve any constant approximation ratio for the ShortPath problem

Dijkstra algorithm for the Single source shortest path problem solves the ShortPath problem exactly.

47

The ShortPath problem

Instance: Given an edge-weighted directed graph G=(V,E) and two nodes s and t

Problem: Find a directed tree of edges, rooted at s

Objective function: Minimize the combined weight of the edges on the path from s to t

48


t

b

s

a u(k)

w(k)

x(1)

v(1)

y(1)

z(1)

0 , , , , ,S u v w x y z

49

Algorithm selects an order on S0

If then the Adversary presents: ( ) ( )y z

t

b

s

a u(k)

w(k)

x(1)

v(1)

y(1)

z(1)

1 , , ,S u x y z

51


Wait until Algorithm considers edge Y(1).

Y(1) will be considered before Z(1)

Adversary can remove data items not yet considered

52

Case 1: y(1) is taken

t

u(k)

x(1)

y(1)

z(1)

b

a

s

The Algorithm constructs a path {u,y}

The Adversary outputs solution {x,z}

1

2

Alg k

Adv

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Case 2: y(1) is rejected

The Algorithm has failed to construct a path. The Adversary outputs a solution {u,y} and wins the game.

t

u(k)

x(1)

y(1)

z(1)

b

a

s

54

The outcome of the game:

Algorithm fails to output a solution or Algorithm achieves an approximation ratio (k+1)/2

The Adversary can set k arbitrarily large and thus can force the Algorithm to claim arbitrarily large approximation ratio

55

Conclusion

No fixed priority algorithm can achieve any constant approximation ratio

Dijkstra’s algorithm for the SSSP can be classified as an adaptive priority algorithm problem solves the ShortPath problem exactly

56

Future work

Improve upper and lower bounds for Priority algorithms

Define extended models of priority algorithms

Beyond greedy algorithms

57

Close the gaps

Metric Steiner Tree problem: The known 2-approximation belongs to the class

of fixed priority algorithms Current lower bound for adaptive priority algorithm

is 1.18, for space with distances {1,2}

Can we close the gap? Lower bound for metric space with arbitrary

distances?

58

Priority algorithms for other problems

What kind of lower bounds can we prove for weighted independent set problem?

What kind of lower bounds can we prove for graph coloring?

59

Extended priority models

Global information: Suppose the algorithm knows the length of the instance ( |V| or |E| or number of jobs, etc.)

There are ‘greedy algorithms’ that use this information

What kinds of lower bounds can we prove for this model?

60

More extensions of the model

Local information - information encoded in a single data item

What if the algorithm is allowed to see the neighborhood of the current vertex?

Or the two highest priority jobs, in case of job scheduling problems?

What kinds of lower bounds can we prove for this model?

61

Beyond greedy

Define similar framework for backtracking and dynamic programming algorithms

What are the limits of these techniques?

62

Defining characteristics of backtracking algorithms

Backtracking algorithms build a depth-first search pruning tree

Leaves of the search tree are solutions

Children of internal node represent the choices for a given data item

63

n n nnn

Fixed backtracking algorithms

22

1 2 n 1

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Fixed backtracking algorithms

Algorithm orders the universe of data items

Decision is irrevocable, algorithm commits to a set of options for each data item

Want to relate the quality of the solution with the fraction of leaves inspected

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Priority Models Sashka Davis University of California, San Diego June 1, 2003