On the performance of Smiths rule in single-machine scheduling with - PowerPoint PPT Presentation

On the performance of Smith’s rule in single-machine scheduling with nonlinear cost Wiebke H¨ ohn Technische Universit¨ at Berlin Tobias Jacobs NEC Laboratories Europe 18th Combinatorial Optimization Workshop Aussois 2014

Generalized min-sum scheduling p j w j Given: jobs j = 1 , . . . , n with weight w j > 0 time 0 C j processing time p j > 0 W. H¨ ohn and T. Jacobs

Generalized min-sum scheduling p j w j Given: jobs j = 1 , . . . , n with p j w j weight w j > 0 time 0 C j processing time p j > 0 Task: compute sequence with minimum cost � w j f ( C j ) j C j completion time of job j non-decreasing, non-negative cost function f W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 time W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed 1 C ( s ) time j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) j � s ( t ) dt = p j 1 0 i ≤ j C ( s ) time j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) j � p j = C (1) s ( t ) dt = 1 j 0 i ≤ j C ( s ) time j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) � � j C ( s ) � p j = C (1) S := s ( t ) dt = 1 j j 0 i ≤ j C ( s ) time j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) � � j C ( s ) � p j = C (1) S := s ( t ) dt = 1 j j 0 i ≤ j C ( s ) time j = S − 1 � � ⇔ C ( s ) C (1) j j W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) � � j C ( s ) � p j = C (1) S := s ( t ) dt = 1 j j 0 i ≤ j C ( s ) time j = S − 1 � � ⇔ C ( s ) C (1) j j increasing speed s ↔ concave cost f decreasing speed s ↔ convex cost f W. H¨ ohn and T. Jacobs

Motivation priorities and fairness � L k -norms/monomials compromise on worst and average case w j C ( s ) linear cost � but non-uniform speed s j j speed � C ( s ) � � j C ( s ) � p j = C (1) S := s ( t ) dt = 1 j j 0 i ≤ j C ( s ) time j = S − 1 � � ⇔ C ( s ) C (1) j j increasing speed s ↔ concave cost f decreasing speed s ↔ convex cost f Our main focus: convex / concave cost functions W. H¨ ohn and T. Jacobs

Outline 1 Analysis of Smith’s rule for convex (and concave) cost 2 Exact algorithms for monomials W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear weakly NP-hard [Yuan ’92] convex W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear weakly NP-hard weakly NP-hard [Yuan ’92] [Yuan ’92] convex FPTAS ? strongly NP-hard ? strongly NP-hard ? W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear weakly NP-hard weakly NP-hard [Yuan ’92] [Yuan ’92] convex FPTAS ? strongly NP-hard ? strongly NP-hard ? in P / FPTAS ? (strongly) NP-hard ? concave W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear weakly NP-hard weakly NP-hard [Yuan ’92] [Yuan ’92] convex FPTAS ? strongly NP-hard ? strongly NP-hard ? in P / FPTAS ? (strongly) NP-hard ? concave in P / FPTAS ? (strongly) NP-hard ? monomials t k W. H¨ ohn and T. Jacobs

Related work & complexity status linear in P [Smith 1956] exponential in P [Rothkopf 1966] general PTAS strongly NP-hard [H., Jacobs 2012] [Megow, Verschae 2012] piece-wise linear weakly NP-hard weakly NP-hard [Yuan ’92] [Yuan ’92] convex FPTAS ? strongly NP-hard ? strongly NP-hard ? in P / FPTAS ? (strongly) NP-hard ? concave in P / FPTAS ? (strongly) NP-hard ? monomials t k piece-wise linear, weakly NP-hard [Yuan ’92] FPTAS [Megow, Verschae ’12] const. # pieces W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Smith’s rule Schedule jobs in non-increasing order of their density w j p j . W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Smith’s rule Schedule jobs in non-increasing order of their density w j p j . How good is this simple algorithm for a fixed convex/concave cost function? W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Smith’s rule Schedule jobs in non-increasing order of their density w j p j . How good is this simple algorithm for a fixed convex/concave cost function? √ 3+1 Smith’s rule is a -approximation for [Stiller & Wiese ’10] 2 any concave cost function f W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Smith’s rule Schedule jobs in non-increasing order of their density w j p j . How good is this simple algorithm for a fixed convex/concave cost function? √ 3+1 Smith’s rule is a -approximation for [Stiller & Wiese ’10] 2 any concave cost function f Theorem The tight approximation ratio of Smith’s rule for fixed convex f is � q 0 f ( t ) dt + p · f ( q + p ) sup � p + q f ( t ) dt . p · f ( p )+ 0 < q , p p W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Smith’s rule Schedule jobs in non-increasing order of their density w j p j . How good is this simple algorithm for a fixed convex/concave cost function? √ 3+1 Smith’s rule is a -approximation for [Stiller & Wiese ’10] 2 any concave cost function f Theorem The tight approximation ratio of Smith’s rule for fixed convex f is � q 0 f ( t ) dt + p · f ( q + p ) sup � p + q f ( t ) dt . p · f ( p )+ 0 < q , p p � holds with inverse ratio for concave cost function W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Narrow space of worst-case instances for convex cost: W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Narrow space of worst-case instances for convex cost: w j p j We can assume w.l.o.g. that: W. H¨ ohn and T. Jacobs

Analysis of Smith’s rule Narrow space of worst-case instances for convex cost: w j p j We can assume w.l.o.g. that: 1. w j = p j for all jobs j W. H¨ ohn and T. Jacobs