View by Category

Loading...

PPT – Approximation Algorithms: PowerPoint presentation | free to download - id: 2081e-MzQxO

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Approximation Algorithms

problems, techniques, and their use in game

theory

- Éva Tardos
- Cornell University

What is approximation?

- Find solution for an optimization problem

guaranteed to have value close to the best

possible. - How close?
- additive error (rare)
- E.g., 3-coloring planar graphs is NP-complete,

but 4-coloring always possible - multiplicative error
- ?-approximation finds solution for an

optimization problem within an ? factor to the

best possible.

Why approximate?

- NP-hard to find the true optimum
- Just too slow to do it exactly
- Decisions made on-line
- Decisions made by selfish players

Outline of talk

- Techniques
- Greedy
- Local search
- LP techniques
- rounding
- Primal-dual

- Problems
- Disjoint paths
- Multi-way cut and labeling
- network design, facility location

- Relation to Games
- local search ? price of anarchy
- primal dual ? cost sharing

Max disjoint paths problem

- Given graph G, n nodes, m edges, and source-sink

pairs. - Connect as many as possible via edge-disjoint

path.

t

s

t

s

t

s

s

t

Greedy Algorithm

Greedily connect s-t pairs via disjoint paths, if

there is a free path using at most m½ edges

If there is no short path at all, take a single

long one.

Greedy Algorithm

Theorem m½ approximation. Kleinberg96 Proo

f One path used can block m½ better paths

Essentially best possible m½-? lower bound

unless PNP by Guruswami, Khanna, Rajaraman,

Shepherd, Yannakakis99

Disjoint pathsopen problem

- Connect as many as pairs possible via paths where

2 paths may share any edge

- Same practical motivation
- Best greedy algorithm n½ - (and also m1/3 -)

approximation Awerbuch, Azar, Plotkin93. - No lower bound

Outline of talk

- Techniques
- Greedy
- Local search
- LP techniques
- rounding
- Primal-dual

- Problems
- Disjoint paths
- Multi-way cut and labeling
- network design, facility location

- Relation to Games
- local search ? Price of anarchy
- primal dual ? Cost sharing

Multi-way Cut Problem

- Given
- a graph G (V,E)
- k terminals s1, , sk
- cost we for each edge e
- Goal Find a partition that separates terminals,

and minimizes the cost - ?e separated we

Separated edges

s3

s1

s4

s2

Greedy Algorithm

- For each terminal in turn
- Find min cut separating si from other terminals

s1

s3

s4

s2

s1

s3

The next cut

s4

s2

Theorem Greedy is a2-approximation

- Proof Each cut costs at most the optimums cut

Dahlhaus, Johnson, Papadimitriou, Seymour, and

Yannakakis94 - Cuts found by algorithm

s3

s1

Optimum partition

s4

s2

Selected cuts, cheaper than optimums cut,

but each edge in optimum is counted twice.

Multi-way cuts extension

- Given
- graph G (V,E), we?0 for e ?E
- Labels L1,,k
- Lv ? L for each node v
- Objective Find a labeling of nodes such that

each node v assigned to a label in Lv and it

minimizes cost ?e separated we

part 3

part 1

Separated edges

part 2

part 4

Example

s1

cheap

medium

expensive

s2

s3

- Does greedy work?
- For each terminal in turn
- Find min cut separating si from other terminals

Greedy doesnt work

- Greedy
- For each terminal in turn
- Find min cut separating si from other terminals
- The first two cuts

s1

Remaining part not valid!

s2

s3

Local search

- Boykov Veksler Zabih CVPR98 2-approximation
- Start with any valid labeling.
- 2. Repeat (until we are tired)
- Choose a color c.
- b. Find the optimal move where a subset of the

vertices can be recolored, but only with the

color c. - (We will call this a c-move.)

A possible -move

Thm Boykov, Vekler, Zabih The best -move

can be found via an (s,t) min-cut

Idea of the flow networkfor finding a -move

s all other terminals retain current color

G

sc change color to c

Theorem local optimum is a 2-approximation

Partition found by algorithm

Cuts used by optimum

The parts in optimum each give a possible local

move

Theorem local optimum is a 2-approximation

Partition found by algorithm

Possible move using the optimum

Changing partition does not help ? current cut

cheaper Sum over all colors Each edge in optimum

counted twice

Metric labeling ? classification open problem

- Given
- graph G (V,E) we?0 for e ?E
- k labels L
- subsets of allowed labels Lv
- a metric d(.,.) on the labels.
- Objective Find labeling f(v)?Lv for each node v

to minimize - ?e(v,w) we d(f(v),f(w))

Best approximation known O(ln k ln ln k)

Kleinberg-T99

Outline of talk

- Techniques
- Greedy
- Local search
- LP techniques
- rounding
- Primal-dual

- Problems
- Disjoint paths
- Multi-way cut and labeling
- network design, facility location

- Relation to Games
- local search ? Price of anarchy
- primal dual ? Cost sharing

Using Linear Programs for multi-way cuts

- Using a linear program
- fractional cut
- ? probabilistic assignment of nodes to parts

Idea Find optimal fractional labeling via

linear programming

Fractional Labeling

- Variables
- 0 ? xva ? 1 pnode, alabel in Lv
- xva ? fraction of label a
- used on node v
- Constraints

? xva 1

for all nodes v ? V

a?Lv

- each node is assigned to a label
- cost as a linear function of x
- ? we ½ ? xua - xva

e(u,v)

a?L

From Fractional x to multi-way cut

- The Algorithm (Calinescu, Karloff, Rabani, 98,

Kleinberg-T,99) - While there are unassigned nodes
- select a label a at random

The Algorithm (Cont.)

- While there are unassigned nodes
- select a label a at random

select 0 ? ? ? 1 at random assign all unassigned

nodes v to selected label a if xva ? ?

Why Is This Choice Good?

- select 0 ? ? ? 1 at random
- assign all unassigned nodes v to selected label a

if xva ? ? - Note
- Probability of assigning node v to label a is

? xva - Probability of separating nodes u and v in this

iteration is ?xua xva

From Fractional x to Multi-way cut (Cont.)

- Theorem Given a fractional x, we find multi-way

cut with expected - separation cost ? 2 (LP cost of x)
- Corollary if x is LP optimum . ?

2-approximation - Calinescu, Karloff, Rabani, 98
- 1.5 approximation for multi-way cut (does not

work for labeling) - Karger, Klein, Stein, Thorup, Young99 improved

bound ? 1.3438..

Outline of talk

- Techniques
- Greedy
- Local search
- LP techniques
- rounding
- Primal-dual

- Problems
- Disjoint paths
- Multi-way cut and labeling
- network design, facility location

- Relation to Games
- local search ? Price of anarchy
- primal dual ? Cost sharing

Metric Facility Location

- F is a set of facilities (servers).
- D is a set of clients.
- cij is the distance between any i and j in D ? F.
- Facility i in F has cost fi.

Problem Statement

We need to 1) Pick a set S of facilities to

open. 2) Assign every client to an open

facility (a facility in S). Goal Minimize

cost of S ?p dist(p,S).

What is known?

- All techniques can be used
- Clever greedy Jain, Mahdian, Saberi 02
- Local search starting with Korupolu, Plaxton,

and Rajaraman 98, can handle capacities - LP and rounding starting with Shmoys, T, Aardal

97 - Here primal-dual starting with Jain-Vazirani99

What is the primal-dual method?

- Uses economic intuition from cost sharing
- For each requirement, like
- ?a?Lv xva 1, someone has to pay to make it

true - Uses ideas from linear programming
- dual LP and weak duality
- But does not solve linear programs

Dual Problem Collect Fees

- Client p has a fee ap (cost-share)
- Goal collect as much as possible max ?p ap
- Fairness Do no overcharge for any subset A of

clients and any possible facility i we must have - ?p ?A ap dist(p,i) ? fi

amount client p would contribute to building

facility i.

Exact cost-sharing

- All clients connected to a facility
- Cost share ap covers connection costs for each

client p - Costs are fair
- Cost fi of selecting a facility i is covered by

clients using it - ?p ap f(S) ?p dist(p,S) , and
- both facilities are fees are optimal

Approximate cost-sharing

- Idea 1 each client starts unconnected, and with

fee ap0 - Then it starts raising what it is willing to pay

to get connected - Raise all shares evenly a
- Example

client

possible facility with its cost

Primal-Dual Algorithm (1)

Its a 1 share could be used towards building a

connection to either facility

a 1

- Each client raises his fee a evenly what it is

willing to pay

Primal-Dual Algorithm (2)

a 2

Starts contributing towards facility cost

- Each client raises evenly what it is willing to

pay

Primal-Dual Algorithm (3)

a 3

Three clients contributing

- Each client raises evenly what it is willing to

pay

Primal-Dual Algorithm (4)

4

a 3

Open facility

clients connected to open facility

- Open facility, when cost is covered by

contributions

Primal-Dual Algorithm Trouble

4

i

j

a 3

p

Open facility

- Trouble
- one client p connected to facility i, but

contributes to also to facility j

Primal-Dual Algorithm (5)

ghost

4

i

j

a 3

p

Open facility

- Close facility j will not open this facility.
- Will this cause trouble?
- Client p is close to both i and j ? facilities i

and j are at most 2a from each other.

Primal-Dual Algorithm (6)

ghost

a 3

4

a 6

a 3

a 3

Open facility

no not need to pay more than 3

- Not yet connected clients raise their fee evenly
- Until all clients get connected

Feasibility fairness ??

- ? All clients connected to a facility
- ? Cost share ap covers connection costs of

client p - ? Cost fi of opening a facility i is covered by

clients connected to it - ?? Are costs fair ??

Are costs fair??

- a set of clients A, and any possible facility i

we have ?p ?A ap dist(p,i)? fi - Why? we open facility i if there is enough

contribution, and do not raise fees any further - But closed facilities are ignored! and may

violate fairness

Are costs fair??

j

i

4

aq4

Closed facility, ignored

open facility

p

cause of closing

Fair till it reaches a ghost facility. Let aq

? aq be the fee till a ghost facility is reached

Feasibility fairness ??

- ? All clients connected to a facility
- ? Cost share ap covers connection costs for

client p - ? Cost ap also covers cost of selected a

facilities - ? Costs ap are fair
- How much smaller is a ? a ??

How much smaller is a ? a?

- q client met ghost facility j
- j became a ghost due to client p

j

i

4

q

p

- p stopped raising its share first
- ap ? aq ? aq
- Recall dist(i,j) ? 2 ap, so
- aq ? aq 2 ap ? 3aq

Primal-dual approximation

- The algorithm is a 3-approximation

algorithm for the facility location problem - Jain-Vazirani99, Mettu-Plaxton00
- Proof
- Fairness of the ap fees ?
- ?p ap ? min cost max ? min
- cost-recovery
- f(S) ?p dist(p,S) ?p ap
- a ? 3aq
- 3-approximation algorithm

Outline of talk

- Techniques
- Greedy
- Local search
- LP techniques
- rounding
- Primal-dual

- Problems
- Disjoint paths
- Multi-way cut and labeling
- network design, facility location

- Relation to Games
- primal dual ? Cost sharing
- local search ? Price of anarchy

primal dual ? Cost sharing

- Dual variables ap are natural cost-shares
- Recall
- fair no set is overcharged
- core allocation
- ?p ?Aap dist(p,i) ? fi for all A and i.
- Chardaire98 Goemans-Skutella00 strong

connection between core cost-allocation and

linear programming dual solutions - See also Shapley67, Bondareva63 for other games

Primal-Dual ? Cost-sharing

- Primal dual for each requirement someone

willing to pay to make it true - Cost-sharing only players can have shares.
- Not all requirements are naturally associated

with individual players. - Real players need to share the cost.

primal dual ? Cost sharing

- Fair ? no subset is overcharged
- Stronger desirable property population monotone

(cross-monotone) - Extra clients do not increase cost-shares.
- Spanning-tree game Kent and Skorin-Kapov96 and

Jain Vazirani01 - Facility location, single source rent-or-buy

Pal-T02

Local search (for facility location)

- Local search simple search steps to improve

objective - add(s) adds new facility s
- delete(t) closes open facility t
- swap(s,t) replaces open facility s by a new

facility t - Key to approximation bound
- How bad can be a local optima?
- 3-approximation Charikar, Guha00

Local search ? Price of anarchy in games

- Price of anarchy facilities are operated by

separate selfish agents - Agents open/close facilities when it benefits

their own objective. - Agents best response dynamic
- Simple local steps analogous to local search.
- Price of anarchy
- How bad can be a stable state?
- 2-approximation in a related maximization game

Vetta02

Conclusions for approximation

- Greedy, Local search
- clever greedy/local steps can lead to great

results - Primal-dual algorithms
- Elegant combinatorial methods
- Based on linear programming ideas, but fast,

avoids explicitly solving large linear programs - Linear programming
- very powerful tool, but slow to solve
- Interesting connections to issues in game theory

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Approximation Algorithms:" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Cornell University and other schools with their online training by sharing educational presentations for free