Multi-parameter complexity analysis of scheduling problems

Let any instance p of problem P be characterized by n integer-valued parameters: )) ( , ), ( ( ) ( 1 p x p x p x n ! = ; } {∞ = " Z Z , | { ) ( p y P = } ) ( y p x ≤ be the subproblem defined for an arbitrary vector n Z y∈ . To determine the complexity of all subproblems ) ( y P , n Z y∈ , a notion of the basis system of subproblems is introduced. It is shown that if such a basis system exists, then it is unique and (under natural assumptions) finite. As an illustration, we perform a 4-parameter complexity analysis of the Open Shop problem and the Connected List Coloring problem. (The latter has a scheduling interpretation.) It is shown that the latter problem possesses a basis system consisting of 8 items, whereas the Open Shop problem has no basis system. At the same time, when we are restricted to the subset of nontrivial subproblems (containing instances with arbitrarily large number of operations), such a basis system exists and consists of either 10 or 14 items. 1. Basis system of subproblems Suppose that, given a discrete problem P , n integer-valued parameters n x x , , 1 ! are chosen that partly characterize any instance p of P ; )) ( , ), ( ( ) ( 1 p x p x p x n ! = is called a characteristic vector of instance p . Let Z be the set of integers, } {∞ = " Z Z . For an arbitrary vector n Z y∈ we define a subproblem } ) ( | { ) ( y p x p y P ≤ = ; | { n P Z y Y ∈ = Ø} ) ( ≠ y P . The objective is to determine the complexity of all subproblems ) ( y P , P Y y∈ . For any P Y y ∈ ' we define domains } ' | { ) ' ( y y Y y y D P ≤ ∈ = − and = + ) ' ( y D } ' | { y y Y y P ≥ ∈ . Evidently, if a subproblem ) ' ( y P is polynomially solvable, then ) ( y P is polynomially solvable for every ) ' (y D y − ∈ . Alternatively, if ) ' ( y P is either hard to solve or NP-hard, then ) ( y P is hard to solve or NP-hard for every ) ' (y D y + ∈ . We show that a set of subproblems } ' | ) ( { Y y y P ∈ , P Y Y ⊆ ' , admits an exhaustive n parameter complexity analysis if it contains a so called basis system of subproblems } ~ | ) ( { Y y y P ∈ defined for a subset ' ~ Y Y ⊆ and satisfying the following 3 properties. 1) Definiteness: a partition P̂ P ~ ~ ~ Y Y Y " = is given such that all subproblems ) ( y P ) ~ ( P Y y∈ are polynomially solvable, and all subproblems ) ( y P ) ~ ( P̂ Y y∈ are hard to solve or NP-hard. 2) Completeness: )) ( ( ' P ~ y D Y Y y − ∈ ⊆ " " )) ( ( P̂ ~ y D Y y + ∈ " . 3) Independence: the relation 2 1 y y < between two vectors Y y y ~ , 2 1 ∈ may hold only when P 1 ~ Y y ∈ , P̂ 2 ~ Y y ∈ . In the case that a basis system of subproblems } ~ | ) ( { Y y y P ∈ is known for a subset P Y Y ⊆ ' , to determine the complexity of a subproblem ' ), ( Y y y P ∈ , it suffices to compare y with every vector Y y ~ '∈ . Natural questions arise: whether such a basis system always exists? And when it exists, is it finite? Is there a unique basis system for a given subset P Y Y ⊆ ' , or there may be many of them? Two theorems below give some general answers to these questions. Theorem 1 [2]. If, given a problem P and a subset of vectors P Y Y ⊆ ' , there exists a basis system for the set of subproblems } ' | ) ( { Y y y P ∈ , then such a system is unique (under assumption that P ≠ NP). Theorem 2 [2]. If all feasible values of the key parameters are bounded from below and there exists a basis system of subproblems for a given subset P Y Y ⊆ ' , then it is finite. As we show, for some problems P the whole set of subproblems } | ) ( { P Y y y P ∈ may have no basis system. We illustrate our approach on two discrete problems: one from graph theory and another from scheduling theory. In both cases a 4-parameter complexity analysis is performed. 2. Four-parameter complexity analysis of the CLC-problem In the Connected List Coloring problem (shortly, CLC-problem) posed by Vising [4] we are given a simple graph ) , ( E V G = , a finite set of colors C , and lists of admissible colors C v A ⊆ ) ( ) ( V v∈ . A function C V → : ρ is called a feasible vertex coloring of graph G , if V v v A v ∈ ∀ ∈ ), ( ) ( ρ , and each subgraph induced by all vertices received the same color is connected. The question is: does there exist a feasible coloring? Vising conjectured that the above decision problem is NP-complete even if graph G is a chain. As follows from our 4-parameter complexity analysis of this problem, it is true. The following 4 parameters are chosen to characterize an instance p : ) ( 1 p x is the type of graph G ; 0 ) ( 1 = p x if G is a forest, and 1 ) ( 1 = p x otherwise (so, 1 ) ( 1 ≤ p x means that G is a graph of an arbitrary type); = ) ( 2 p x ) (G ∆ is the maximum vertex degree of graph G ; = = # max 3 ) ( A p x | ) ( | max v A V v∈ is the maximum list cardinality; and ) ( ) ( 4 A f p x = | )} ( | { | max v A i V v C i ∈ ∈ = ∈ # is the maximum frequency of a color in the lists. Theorem 3 [3]. The whole set of subproblems } | ) ( { CLC Y y y CLC ∈ of the CLC-problem has a unique and finite (due to theorems 1 and 2) basis system } ~ | ) ( { Y y y CLC ∈ which consists of 8 basis subproblems presented in Tab. 1. Table 1. Basis system of subproblems of the CLC-problem ), ( i y CLC ! = i Type of graph G i y1 ) (G ∆ i y2 max A i y3 ) (A f i y4 Complexity 1 1 0 ∞ ∞ P 2 1 ∞ ∞ 2 P 3 1 ∞ 1 ∞ P 4 1 2 2 ∞ P 5 1 3 2 3 P 6 0 1 3 3 NPC 7 0 4 2 3 NPC 8 0 3 2 4 NPC In the case when graph G is a family of chains, the CLC-problem admits the following natural interpretation as a scheduling problem on parallel machines. Suppose, we are given m parallel machines and one job consisting of n consecutive operations of unit length. Each operation can be performed on any machine. For each machine, a family of time intervals is specified when the machine is available. The question is: does there exist a schedule for processing the job on machines in the time interval ] , 0 [ n such that each machine works without an idle time during some connected time interval? To reduce this scheduling problem to the CLC-problem, it suffices to imagine that the machines are the colors, while the i th unit time interval ) , 1 [ i i − stands for the i th vertex of a chain consisting of n vertices. 3. Four-parameter complexity analysis of the Open Shop problem In the Open Shop problem (shortly, OS-problem), we are given sets J } , , { 1 n J J ! = ! M } , , { 1 m M M ! = ! and O } , , { 1 K O O ! = ! whose elements are called jobs, machines, and operations, respectively. Operation ∈ i O O belongs to a job ) ( i O J , and its processing requires a connected time interval of length ) ( i O p on machine ∈ ) ( i O M M. It is required to assign a starting time 0 ) ( ≥ i O s to each operation ∈ i O O so that for any two operations ∈ j i O O , O belonging to the same job or requiring the same machine, the time intervals )) ( ) ( ), ( ( i i i O p O s O s + and )) ( ) ( ), ( ( j j j O p O s O s + would not overlap, and the maximum completion time over all operations )) ( ) ( ( max ) ( } { max i i O O p O s S C i + =# would be minimal. A 4-parameter complexity analysis of the OS-problem is performed for the following key parameters: the number of jobs n , the maximum number ν of operations per job, the number of machines m , and the maximum number μ of operations on a machine. Theorem 4 [2]. Under assumption P ≠ NP, there is no basis system for the whole set of subproblems } | ) ( { OS Y y y OS ∈ of the Open Shop problem. Let )} ( ) ( | { 4 3 2 1 ∞ = ∨ ∞ = ∧ ∞ = ∨ ∞ = ∈ = y y y y Y y Y OS OS stand for the set of vectors defining nontrivial subproblems of the OS-problem (each contains instances with arbitrarily large number of operations). Theorem 5 [2]. For the set of nontrivial subproblems } | ) ( { OS Y y y OS ∈ of the OSproblem, there exists a basis system consisting of either 10 or 14 subproblems, presented in Tab. 2. The first 10 items in the list are undeniable items of the basis system, whereas the remaining four enter into the basis system if and only if ) ( 6 y OS and ) ( 7 y OS are proved to be polynomially solvable. Table 2. Complexity of basis subproblems of the OS-problem ), ( i y OS ! = i n ν m μ Complexity 1 ∞ 1 ∞ ∞ P 2 ∞ ∞ ∞ 1 P 3 2 ∞ ∞ ∞ P 4 ∞ ∞ 2 ∞ P 5 ∞ 2 ∞ 2 P 6 ∞ 2 3 ∞ ?? 7 3 ∞ ∞ 2 ?? 8 ∞ 2 ∞ 3 NPH 9 ∞ 3 ∞ 2 NPH 10 3 ∞ 3 ∞ NPH 11 3 ∞ ∞ 3 NPH 12 ∞ 3 3 ∞ NPH 13 4 ∞ ∞ 2 NPH 14 ∞ 2 4 ∞ NPH Therefore, the ultimate answer to the question of cardinality of the basis system for the OSproblem depends on an answer to the question on the complexity of the OS-problem with 3 machines and at most 2 operations per job (in our notation, ) ( 6 y OS ; subproblem ) ( 7 y OS is symmetric). Such a question was raised by T. Gonzalez and S. Sahni [1] in 1976 and is still open. AcknowledgementsSupported by the Russian Foundation for Fundamental Research (Grant 99-01-00581). References[1] Gonzalez, T. and Sahni, S. (1976). Open shop scheduling to minimize finish time, Journal ofthe Association for Computing Machinery, 23, 665–679.[2] Kashyrskikh, K.N., Sevastianov, S.V. and Tchernykh I.D. (2000). 4-parameter complexityanalysis of the open shop problem, Diskret. Analiz i Issled. Oper., Ser. 1, 7 (4), 59–77, (inRussian).[3] Kononov, A.V. and Sevastianov, S.V. (2000). On the complexity of the connected list vertex-coloring problem, Diskret. Analiz i Issled. Oper., Ser. 1, 7 (2), 21–46, (in Russian).[4] Vising, V.G. (1999). On the connected list coloring of graphs, Diskret. Analiz i Issled. Oper.,Ser. 1, 6 (4), 36–43, (in Russian).