St ackelberg Scheduling St rat egies Tim Roughgarden Cornell - PDF document

St ackelberg Scheduling St rat egies Tim Roughgarden Cornell Universit y 1

The Model • m machines 1,2,...,m • A quant it y r of j obs – j obs are small (model in a ct s way) • For each machine i, a load- dependent lat ency f unct ion l i (•) – assume cont inuous, nondecreasing Example: (r=1) j obs have j obs have lat ency ¼ lat ency 1 ½ ½ l 2 (x)=x 2 l 1 (x)=1 2

Equilibria • which j ob assignment s ar e “st able”? – j obs cont r olled by self ish, noncooper at ive agent s – no j ob want s t o swit ch machines (no j ob should be envious) vs. ½ ½ 0 1 l 1 (x)=1 l 2 (x)=x 2 1 x 2 3

More on Equilibrium Assignment s vs. ½ ½ 0 1 l 1 (x)=1 l 2 (x)=x 2 1 x 2 Def : an assignment is at Nash equilibrium (is a Nash assignment ) if : •all used machines have equal lat ency •unused machines have great er lat ency Fact : always have exist ence, uniqueness 4

How Good is an Assignment ? The Cost of an Assignment : • C(x) = cost or t ot al lat ency exper ienced by assignment x: Σ i x i • l i (x i ) • our not ion of syst em perf ormance • can opt imize in poly-t ime Example: cost = ½ ½ ½ •1 +½ •¼= ? 1 x 2 5

How Good are Nash Assignment s? Goal : prove t hat Nash assignment s ar e near-opt imal • want a laissez f aire approach t o regulat ing users Problem : f alse in general! Example: (r=1, k large) vs. ? 0 1 1- ? x k x k 1 1 OPT cost s ≈ 0 Nash cost s 1 6

Near-Opt imal Nash Assignment s Old Approach: weaken model, compar e Nash vs. OPT (due t o [Roughgarden/ Tardos 00]) • gener al lat ency f ns, weaker OPT Thm 1: cost of Nash =cost of OPT at rat e 2r • st r onger OPT, linear lat ency f ns (l i (x)=a i x+b i ) Thm 2: cost of Nash = 4/ 3 cost of OP T (at rat e r) 7

Taming Self ishness t hr ough a Manager New Appr oach: • not all j obs need be cont r olled by self ish users – “cent rally cont rolled” vs. “self ishly cont rolled” j obs – behavior of self ish users depends on assignment of managed j obs Goal: • assign cent rally cont rolled j obs t o induce “good” self ish behavior – see also [Kor lis, Lazar, Or da 97] 8

St ackelberg St rat egies • St ackelberg st r at egy = assignment of cent rally cont r olled j obs ⇒ yields an induced equilibrium • Basic Quest ions: • what ’s t he best st r at egy? • can we comput e/ charact erize it ? • how inef f icient is t he best induced equilibr ium? • are we provably near-opt imal? 9

Our Result s - General Lat ency Funct ions Theorem 1: Can ef f icient ly comput e a st r at egy inducing an equilibrium wit h cost =(1/ ß) × cost of opt assignment (ß = f ract ion of cent rally assigned j obs) Fact : (1/ ß) × OPT is best ß 1-ß possible [x/ (1-ß)] k 1 10

Our Result s - Linear Lat ency Funct ions Theorem 2: Can ef f icient ly comput e a st r at egy inducing an equilibrium wit h cost = [4/ (3+ß)] × cost of OPT (ß = f ract ion of cent rally assigned j obs) Fact : [4/ (3+ß)] × OPT is best ß 1-ß possible 1 x/ (1-ß) 11

What makes a st rat egy (in)ef f ect ive? The Scale St rat egy • comput e opt imal assignment x of all j obs, assign cent r ally cont r olled j obs via ß • x 2/ 3 2/ 3 1/ 3 1/ 3 1 (3/ 2)x 1 (3/ 2)x OPT I nduced Eq Moral: avoid machines t hat self ish users will (over)use anyways 12

The LLF St rat egy Largest Lat ency First (LLF): • comput e opt , x, f or all j obs • assign x i j obs t o i in or der of decreasing l i (x i )’s (unt il no managed j obs remain) l 1 =1 l 2 =½ l 3 =½ ¼ 1/ 12 5/ 12 5/ 12 0 1/ 3 (3/ 2)x (3/ 2)x 1 2x 1 2x OPT (r=1) LLF (ß=½ ) 13

LLF wit h General Lat ency Funct ions Theor em 1: The LLF st r at egy induces an equilibr ium wit h cost = (1/ ß) × cost of opt assignment . Proof idea: Exploit it er at ive st r uct ur e of LLF t o pr oceed by induct ion on # of machines. Base common case: lat ency = L ß •LLF ⇒ Machine 1’s lat ency ≥ L •OPT has ≥ ß j obs on machine 1 ⇒ OPT pays ≥ ßL, we pay L 14

LLF wit h Linear Lat ency Funct ions Theorem 2: The LLF st rat egy induces an equilibr ium wit h cost = [4/ (3+ß)] × cost of OPT. Main dif f icult y: • previous argument t oo weak f or: • need det ailed st udy of Nash, OPT when all lat encies are linear 15

Comput ing Opt imal St rat egies We' ve seen: LLF has t he best possible worst -case guanant ee Quest ion: is t he LLF st rat egy opt imal on all inst ances? Bad news: no, in f act : Theorem 3: Comput ing t he opt imal st r at egy is NP-hard (even f or linear lat ency f ns). • Compare t o: Opt , Nash assignment s 16

Open Quest ions Approximat ing t he opt imal st rat egy: • LLF is best possible using OPT as a lower bound • bet t er guar ant ees f or LLF via a bet t er lower bound? • mor e sophist icat ed algor it hms? – Thm 3 is reduct ion f rom Part it ion – exist ence of a (F)P TAS? 17

General Graphs Open: • f or gener al lat ency f ns, f ixed ß: a st rat egy inducing an equilibrium w/ cost = f (ß) × opt – 1/ ß not achievable in general graphs (!) – maybe 2/ ß? (or O (n)) • f or linear lat ency f ns, a st r at egy w/ cost < 4/ 3 × opt – e.g., is 8/ 7 achievable f or ß=½ ? 18