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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

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