CSCI 3210: Computational Game Theory Approximation Algorithms Ref: - - PDF document

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CSCI 3210: Computational Game Theory

Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website: www.bowdoin.edu/~mirfan/CSCI-3210.html

Approximation Algorithms

Ref: Vazirani [Blackboard]

Task

u Take an NP-Hard optimization problem u Give an approximately optimal solution in

poly time

u Finding optimal is NP-Hard u Lower bound as a yardstick

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LP duality

Primal LP Maximize ... Dual LP Minimize ...

Increasing

• bjective

function

Gap?

Vertex cover problem

u Given a graph, select the minimum number

• f nodes such that at least one endpoint of

every edge is selected.

u Answer?

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Optimization

u xt is 1 if node t is picked and 0 otherwise u Is the following a usual linear program?

Minimize Σt xt Subject to xu + xv >= 1, for each edge (u, v) xt ∈ {0, 1}, for each node t

Integer linear program (ILP) In general NP-hard

LP relaxation

Minimize Σt xt Subject to xu + xv >= 1, for each edge (u, v) xt >= 0, for each node t

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Our first approximation algorithm

u LP relaxation u Rounding u 2-approximation algorithm for vertex cover

u Proof.

OPT_relaxed_LP <= OPT_ILP Rounded solution <= 2 * OPT_relaxed_LP <= 2 * OPT_ILP Factor 2 approximation!

Primal-dual approximation

u Primal (Relaxed LP – why?)

Minimize Σt xt Subject to xu + xv >= 1, for each edge (u, v) xt >= 0, for each node t

u Dual LP

Maximize Σe ye Subject to Σall edges e incident on t ye <= 1, for each node t ye >= 0, for each edge e

Maximum matching problem (relaxed)

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Maximum matching problem

u Select the maximum number of edges so that

the selected edges do not share any vertex.

u What's your solution?

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Paradox?

u Vertex cover is NP-Hard u Maximum matching is dual of vertex cover u Maximum matching is polynomial-time

solvable!

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Answer: “duality gap” for ILP

Maximum matching ILP Minimum vertex cover ILP

Increasing

• bjective

function

Gap?

Strong duality thm: Gap = 0 for relaxed primal and dual

Lower bounding technique for approx. algorithm

Maximum matching

Maximal matching

Minimum vertex cover

Increasing

• bjective

function

Gap?

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Maximum vs. maximal matching

u Find a matching that’s maximal but not

maximum

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Famous primal-dual

• approx. algorithm for

vertex cover & analysis

(class note)

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Set Cover Problem

Set Cover Problem

u Given a universe U of n elements and a

collection of subsets of U, S = {S1, ..., Sm}, find the minimum # of subsets to cover all n elements

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SET COVER

u Decision version is NP-complete

Greedy algorithm

• 1. C = { }
• 2. While C != Universe do

i.

Pick a subset S' which covers the max # of new elements

ii.

C ß C U S' 3.

Output the picked subsets

Cost: Cost of each picked subset is 1 Price: Each of the k new element covered by a picked subset pays a price of 1/k

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LP

Primal: ILP Primal Relaxed LP: Covering Dual LP: Packing

Lower bound

Increasing

• bj. func.

(Min) Primal Relaxed LP (Max) Dual of relaxed LP OPTf Primal ILP OPT

Integrality gap LB: Any feasible solution to dual

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Finding a dual feasible soln

u Claim. ye = price(e)/Hn is a feasible soln.

u Hn = 1 + ½ + ... + 1/n

u Proof. Show that no subset S is overpacked.

Dual: Packing LP

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Main result

u Theorem. Greedy set cover is an O(ln n) approximation. u Proof.

Size of set cover (Min) Relaxed LP (Max) Dual LP OPTf ILP OPT

LB: Any feasible solution to dual

Applications

u Geometric coverage problems

u Guarding problems u Facility location problems

u Crew scheduling problems

u [Yelbay et al., 2014]: American Airlines, bus driver, etc. u [Pezzella & Faggioli, 1997]: Italian Railways Company

u Testing programs: smartly choosing test cases u Clustering and outlier detection

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https://pdfs.semanticscholar.

• rg/199e/63ef433ba5340a59d6

b84e018229bea354e6.pdf

Summary

u Vertex cover: 2 approximation

u OPEN: Find a less than 2 approximation

u Set cover: log n approximation

u Inapproximable below log n

u Tight examples exist for both