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A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Klaus Jansen Kim-Manuel Klein June 30, 2012

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Online Bin Packing

Given at time t ∈ N an instance It of items It = {i1, . . . it} and a function s : It → [0, 1]. Find for each t ∈ N a function Bt : {i1, . . . , it} → N+, such that

• i:Bt(i)=j s(i) ≤ 1 for all j. Minimize maxi {Bt(i)}.
• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Competitive Ratio of Online Bin Packing

Best known algorithm: Ratio of 1.58889 (Steven S. Seiden. On the online bin packing problem) Best known lower bound: Ratio of 1.5401 (Andre van Vliet. An improved lower bound for on-line bin packing algorithms)

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

We need a new model! Goal: Approximation guarantee maxi Bt(i) ≤ (1 + ǫ)OPT + f ( 1

ǫ)

and bounded migration.

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Migration

Migration Factor between Bt and Bt+1: 1 s(it+1)

• j≤t:Bt(ij)=Bt+1(ij)

s(ij) An algorithm is robust if the migration factor is bounded by a function f ( 1

ǫ).

Peter Sanders, Naveen Sivadasan and Martin Skutella. Online scheduling with bounded migration.

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Robust Bin Packing

Leah Epstein and Asaf Levin: "A robust APTAS for the classical bin packing problem" Running time: log(t)22O(1/ǫ) and migration factor 2O(1/ǫ)

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

LP-Formulation

Let I be an instance of bin packing with m different item sizes s1, . . . , sm. Suppose that for each item ik ∈ I there is a size sj with s(ik) = sj. A configuration Ci is a multiset of sizes {a(Ci, 1) : s1, a(Ci, 2) : s2, . . . a(Ci, m) : sm} with

• 1≤j≤m a(Ci, j)sj ≤ 1, where a(Ci, j) denotes how often size sj

appears in configuration Ci. min x1

• Ci∈C

xia(Ci, j) ≥ bj ∀1 ≤ j ≤ m xi ≥ 0 ∀1 ≤ j ≤ n

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Sensitivity Analysis

Problem: Let x′ be a solution of min {x1 |Ax ≥ b′, x ≥ 0}. Find a solution x′′ of min {x1 |Ax ≥ b′′, x ≥ 0} such that x′′ − x′1 is small. Theorem of Cook et al.: There exists a x′′ satisfying the LP and x′′ − x′∞ ≤ n∆ b′′ − b′∞

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Our Results:

Running time O(log(t) 1

ǫ9 ) and migration factor O(1/ǫ4)

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Theorem

Consider the LP min {x1 |Ax ≥ b, x ≥ 0} and an approximate solution x′ with x′1 = (1 + δ)OPT for some δ > 0. There exists a solution x′′ of the LP having value of at most x′′1 ≤ (1 + δ)OPT − α and x′ − x′′1 ≤ (2/δ + 2)α.

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Improve packing:

Let Bt be a packing of instance It with maxi Bt(i) ≤ (1 + ǫ)OPT. Find a packing B′

t with maxi B′ t(i) ≤ (1 + ǫ)OPT − α.

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

We prove feasibility of the following LP 1. Ax ≥ b (LP 1) x ≥ 0 x ≤ x′ + α(1/δ + 1) x′1 xOPT x ≥ x′ − α(1/δ + 1) x′1 x′

• xi ≤ (1 + δ)OPT − α
• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Algorithm

Let x′ be a LP solution with x′ ≤ (1 + δ)OPT

◮ Set xvar := α(1/δ+1) x′

x′, xfix := x′ − xvar and bvar := b − A(xfix)

◮ Solve the LP ˆ

x = min {x1 |Ax ≥ bvar, x ≥ 0}

◮ Generate a new solution x′′ = xfix + ˆ

x

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Problems:

◮ Given integral solution y′ with y′1 ≤ (1 + δ)OPT.

Compute integral solution y′′ with y′1 ≤ (1 + δ)OPT − α such that y′′ − y′′1 is small.

◮ Keep the number of non-zero components small ◮ Dynamic rounding

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Open questions:

◮ Smaller migration factor and running time ◮ Lower bounds for migration? ◮ Dynamic bin packing (allow departing of items) ◮ Use LP-techniques for other online problems (i.e. scheduling)

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Thank you!

• K. Klein

A Robust AFPTAS for Online Bin Packing with Polynomial Migration