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Approximation Algorithms:

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Approximation Algorithms: va Tardos. Cornell University. problems, techniques, and ... Blue or green. Red or green. Red or blue. s2. s1. s3. FOCS 2002. 16 ... – PowerPoint PPT presentation

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Title: Approximation Algorithms:


1
Approximation Algorithms
problems, techniques, and their use in game
theory
  • Éva Tardos
  • Cornell University

2
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.

3
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

4
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

5
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
6
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.
7
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
8
Disjoint paths open 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 …

9
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

10
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
11
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
12
Theorem Greedy is a 2-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.
13
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
14
Example
s1
cheap
medium
expensive
s2
s3
  • Does greedy work?
  • For each terminal in turn
  • Find min cut separating si from other terminals

15
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
16
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.)

17
A possible -move
Thm Boykov, Vekler, Zabih The best -move
can be found via an (s,t) min-cut
18
Idea of the flow network for finding a -move
s all other terminals retain current color
G
sc change color to c
19
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
20
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
21
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
22
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

23
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
24
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
25
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

26
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 ? ?
27
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

28
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..

29
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

30
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.

31
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).
32
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

33
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

34
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.
35
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

36
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
37
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

38
Primal-Dual Algorithm (2)
a 2
Starts contributing towards facility cost
  • Each client raises evenly what it is willing to
    pay

39
Primal-Dual Algorithm (3)
a 3
Three clients contributing
  • Each client raises evenly what it is willing to
    pay

40
Primal-Dual Algorithm (4)
4
a 3
Open facility
clients connected to open facility
  • Open facility, when cost is covered by
    contributions

41
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

42
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.

43
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

44
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 ??

45
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

46
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
47
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 ??

48
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

49
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

50
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

51
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

52
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.

53
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

54
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

55
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

56
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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