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 I t of items I t = { i 1 , . . . i t } and a function s : I t → [ 0 , 1 ] . Find for each t ∈ N a function B t : { i 1 , . . . , i t } → N + , such that � i : B t ( i )= j s ( i ) ≤ 1 for all j . Minimize max i { B t ( 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 max i B t ( 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 B t and B t + 1 : 1 � s ( i j ) s ( i t + 1 ) j ≤ t : B t ( i j ) � = B t + 1 ( i j ) 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 ) 2 2 O ( 1 /ǫ ) and migration factor 2 O ( 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 s 1 , . . . , s m . Suppose that for each item i k ∈ I there is a size s j with s ( i k ) = s j . A configuration C i is a multiset of sizes { a ( C i , 1 ) : s 1 , a ( C i , 2 ) : s 2 , . . . a ( C i , m ) : s m } with � 1 ≤ j ≤ m a ( C i , j ) s j ≤ 1, where a ( C i , j ) denotes how often size s j appears in configuration C i . min � x � 1 � x i a ( C i , j ) ≥ b j ∀ 1 ≤ j ≤ m C i ∈ C x i ≥ 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 {� x � 1 | Ax ≥ b ′ , x ≥ 0 } . Find a solution x ′′ of min {� x � 1 | 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 ǫ 9 ) and migration factor O ( 1 /ǫ 4 ) K. Klein A Robust AFPTAS for Online Bin Packing with Polynomial Migration
Theorem Consider the LP min {� x � 1 | 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 B t be a packing of instance I t with max i B t ( i ) ≤ ( 1 + ǫ ) OPT . Find a packing B ′ t with max i 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 OPT � x ′ � 1 x ≥ x ′ − α ( 1 /δ + 1 ) x ′ � x ′ � 1 � x i ≤ ( 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 x var := α ( 1 /δ + 1 ) x ′ , x fix := x ′ − x var and � x ′ � b var := b − A ( x fix ) x = min {� x � 1 | Ax ≥ b var , x ≥ 0 } ◮ Solve the LP ˆ ◮ Generate a new solution x ′′ = x fix + 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
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