Schwiegelshohns Proof of the Kawaguchi-Kyan Bound Martin Skutella - PowerPoint PPT Presentation

Schwiegelshohn’s Proof of the Kawaguchi-Kyan Bound Martin Skutella TU Berlin

Single Machine Scheduling to Minimize � w j C j Given: n jobs j = 1 , . . . , n , processing times p j > 0, weights w j > 0 Task: schedule jobs on a single machine; minimize � j w j C j 2 3 1 2 1 3 time 0 C 3 C 1 C 2 Weighted Shortest Processing Time (WSPT) rule: Theorem (Smith 1956). Sequencing jobs in order of non-increasing ratios w j / p j is optimal. “Photographer’s Rule”

Two-Dimensional Gantt Charts Eastman, Even & Isaacs 1964; Goemans & Williamson 2000 1 2 3 3 1 2 time 0 C 1 C 2 C 3 weight w 1 w 2 w 3 p 1 p 2 p 3 time w j / p j = diagonal slope of rectangle representing job j

Swap Weights and Processing Times 1 2 3 time weight time weight

Parallel Machine Scheduling to Minimize � w j C j Given: n jobs j = 1 , . . . , n , processing times p j > 0, weights w j > 0 Task: schedule jobs on m parallel machines; minimize � j w j C j 3 2 5 1 5 8 7 1 2 4 6 4 3 7 8 6 time ◮ weakly NP-hard for two machines (Bruno, Coffman & Sethi 1974) ◮ strongly NP-hard if m part of input (Garey & Johnson, problem SS13) ◮ PTAS (Sk. & Woeginger 2000)

List Scheduling in Order of Non-Increasing w j / p j w 1 / p 1 ≥ w 2 / p 2 ≥ · · · ≥ w n / p n 1 1 5 8 2 2 4 6 3 3 7 Theorem (Conway, Maxwell & Miller 1967). Optimal if w j = 1 for all j (or: p j = 1 for all j ). Theorem (Kawaguchi & Kyan 1986). time √ Tight performance ratio: 1 + 2 ≈ 1 , 207 . . . 2

Outline 1 WSPT has performance ratio ≤ 3 / 2 √ 2 WSPT has performance ratio exactly 1 2 ( 1 + 2 ) ≈ 1 , 207 . . . 3 WSEPT for stochastic scheduling 4 Open problem

Fast Single Machine Lower Bound Lemma (Eastman, Even & Isaacs 1964). � � � � 1 OPT 1 − 1 ≤ OPT m − 1 j w j p j j w j p j m 2 2 weight weight + 1 m time time

WSPT has Performance Ratio ≤ 3 / 2 Lemma (Eastman, Even & Isaacs 1964). � � � � 1 OPT 1 − 1 ≤ OPT m − 1 j w j p j j w j p j m 2 2 WSPT 1 5 8 2 4 6 WSPT start times ≤ single machine start times 3 7 Thus: OPT 1 / m � � � + � WSPT m ≤ 1 OPT 1 − 1 j w j p j j w j p j m 2 � ≤ OPT m + 1 3 j w j p j ≤ 2 OPT m 2

Schwiegelshohn’s Proof of the Kawaguchi-Kyan Bound Theorem (Kawaguchi & Kyan 1986). √ WSPT has performance ratio exactly 1 + 2 ≈ 1 , 207 . . . 2 Proof idea: explicit construction of worst-case instance (for m → ∞ ) Refined: exact performance ratio for each fixed m (Jäger & Sk. 2018) Sequence of reductions to worst-case instances with: 1 w j = p j for all j 2 at most m − 1 large jobs and many tiny jobs 3 all but one large job are extra-large 4 all XL jobs have same size

First Reduction: w j = p j ∀ j w j w j for j = 1 , . . . , k for j = k + 1 , . . . , n p j ≥ R p j ≤ r 1 5 8 11 R > r 2 4 6 9 3 7 10 n k n k r � 1 − r � � � � � w j C j = w j C j + + w j C j w j C j R R j = 1 j = 1 j = k + 1 j = 1 � �� � � �� � =: A =: B � A WSPT � WSPT = A WSPT + B WSPT , B WSPT = ≤ max ⇒ OPT A OPT + B OPT A OPT B OPT

Objective Function in Terms of Machine Loads (for w j = p j ) weight one machine i : p 1 � � � 2 � � 1 + 1 p j 2 p j C j = p j 2 2 p 2 j → i j → i j → i � �� � L i p 3 p 1 p 2 p 3 time L i m -machine schedule: n m n � � � L i 2 + 1 1 p j 2 p j C j = 1 2 2 j = 1 i = 1 j = 1 2 notice: 3 ◮ � i L i = � j p j (fixed) time i L i 2 minimal if L 1 = · · · = L m ◮ � L 3 L 2 L 1

Second Reduction: Large Jobs and Sand � � � L i 2 + 1 p j C j = 1 p j 2 2 2 j i j WSPT schedule WSPT: i L i 2 remains unchanged ◮ � ◮ � j p j 2 decreases by δ ≥ 0 OPT: i L i 2 unchanged or decreases ◮ � j p j 2 decreases by δ ◮ � WSPT time = unchanged or increases L min ⇒ OPT

Third Reduction: Make Large Jobs Extra-Large WSPT schedule OPT schedule x i x i x i x i old: L min = 1 L min y i y i new: Increase in objective: � � ( 1 + y i ) 2 + y i 2 − ( 1 + x i ) 2 − x i 2 � � � y i 2 − x i 2 � 1 1 ≥ 0 2 i 2 i � y i 2 − x i 2 � = � as � i x i = � i y i i

Fourth Reduction: All XL Jobs of Same Size WSPT schedule OPT schedule y i y i old: 1 z i z i new: Increase in objective: � � ( 1 + z i ) 2 + z i 2 − ( 1 + y i ) 2 − y i 2 � � � z i 2 − y i 2 � 1 ≤ 0 2 i i � z i 2 − y i 2 � = � as � i x i = � i y i i

Analyzing the Performance Ratio WSPT schedule OPT schedule 1 . x x . . k . y y . . m m + y 1 m − k OPT = k · x 2 + ( m + y ) 2 2 ( m − k ) + y 2 WSPT = m 2 + k · x ( 1 + x ) + y ( 1 + y ) 2 OPT = ( m − k )( 2 kx 2 + 2 kx + 2 y 2 + 2 y + m ) WSPT ( m − k )( 2 kx 2 + y 2 ) + ( y + m ) 2 m Observation: maximum at y = 0 and x = � k ( 2 m − k ) − k

Worst-Case Instance � � � k k k worst-case performance ratio: max k 1 − 2 m + 2 m ( 1 − 2 m ) √ �� � � 1 − 1 Observation: maximum at k = 2 m . 2 1.205 1.200 1.195 1.190 1.185 0 5 10 15 20 25

Stochastic Scheduling Given: distributions of independent random processing times p j ≥ 0 Pr [ p j ≥ t ] 1 1 1 1 t �� w j C j � Task: find scheduling policy minimizing E ◮ scheduling policy must be nonanticipatory, i.e., decision made at time t may only depend on the information known at time t time 0 t t

Weighted Shortest Expected Processing Time (WSEPT) WSEPT Rule List scheduling in order of non-increasing w j / E [ p j ] . ◮ WSEPT is optimal for single machine (Rothkopf 1966) 2 ( 1 + ∆) with ∆ ≥ Var [ p j ] ◮ WSEPT has performance ratio 1 + 1 E [ p j ] 2 for all j . (Möhring, Schulz & Uetz 1999) ◮ WSEPT has no constant performance ratio. (Cheung, Fischer, Matuschke & Megow 2014; Im, Moseley & Pruhs 2015) √ ◮ WSEPT has performance ratio 1 + 1 2 ( 2 − 1 )( 1 + ∆) . (Jäger & Sk. 2018)

Open Problem Online setting: ◮ jobs arrive one by one; must be immediately assigned to machines ◮ on each machine, assigned jobs are optimally sequenced (WSPT) Algorithm MinIncrease ◮ assign job to machine minimizing increase of current objective value Known results: ◮ MinIncrease has competitive ratio 3 1 2 m . 2 − ◮ If jobs arrive in order of non-increasing or non-decreasing w j / p j , then √ MinIncrease achieves competitive ratio 1 2 ( 1 + 2 ) . Conjecture (Stougie 2017). √ MinIncrease has competitive ratio 1 2 ( 1 + 2 ) .