The Complexity of Dynamic Languages and ... - Semantic Scholar
STOC(Mi lwaukee 1981), 218-227
THE COMPLEXITY OF DYNAMIC LANGUAGES AND DYNAMIC OPTIMIZATION PROBLEMS James B. Orlin Sloan School of Management M.I.T.
1.
In Section 4, we define "dynamic languages" and prove some elementary properties of these languages. In Sections 5 and 6, we prove the P-space completeness of certain dynamic languages derived from NP-complete languages. These results contrast with those by Orlin [01], [02), [03], 04], and [05], who provides polynomial time recognition algorithms for various dynamic languages derived from polynomial time languages, including the following dynamic languages: dynamic linear programming, dynamic network flows, coloring periodic interval graphs, dynamic matching, and dynamic 2-satisfiabil ity. In Section 7, we discuss the real-time, complexity of dynamic optimization problems. Here, dynamic optimization problems are viewed as realtime scheduling problems with known periodic data. We show that there is a polynomial real-time scheduling algorithm for the dynamic optimization problems presented in Section 3 if nd only if P = P-space. In Section 8, we show that the dynamic knapsack problem may be solved in pseudo-polynomial time (i.e., in polynomial time if the data are unary encoded), and thus it is meaningful to describe other P-space problems as "strongly P-space complete", a counterpart to strong'NP-comp'eteness as described by Garey and Johnson GJ]. Finally, in Section 9, we present some variations of dynamic language problems.
Introduction
Dynamic/period optimization problems arise naturally in various quantitative disciplines including Computer Science, Economics, and Operations Research. These periodic models may be applied to long-range economic planning, workforce scheduling, vehicle routing, machine maintenance, and a host of industrial applications. In this paper we offer a unifying framework for dynamic problems in terms of "dynamic languages", and we discuss the complexity of these languages. In particular, many dynamic languages derived from NP-complete languages can be shown to be polynomial space (P-space) complete. Among these are the following: the dynamic 3-satisfiability problem, and dynamic 3-dimensional matching problem, the dynamic partition problem, the dynamic hamiltonian circuit problem, and the dynamic independent set problem. We provide a general technique for showing how to prove the P-space completeness of dynamic problems derived from NP-complete problems.
I1
The Outline of This Paper In Sections 2 and 3, we give examples of dynamic (periodic) optimization problems. In Section 2, we reference some of the research on various dynamic/periodic problems, whereas in Section 3 we illustraze how to derive dynamic problems from problems in the class NP.
I
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and htotice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise. or to republish, requires a fee and/or specific permission. °1981 ACM 0-89791-041-9 /80/0500/0218 $00.75
2. Some Dynamic Optimization Problems Dynamic/periodic optimization problems occur in a variety of settings. Below we reference some examples of such optimization problems. The list
218
:sas;;;;yc-i(f
-i4L'B·4PliP;ChJIEY;Bb,;Tf ·:tr-ikic Z_
-....
3. Dynamic
below is intended as a sampler and is far from comprehens i ve. Two types of periodic optimization problems arise typically in employment: shift scheduling and day-off scheduling. Both of these are surveyed by Baker [ B]. Many problems of this type arise in service industries, and they may be classified according to the following parameters: types of workshifts allowed, the demand structures, and the types of additional constraints (possibly imposed by union contracts).
the technique illustrated in Sections 5 and 6.
These problems also serve to motivate the definitions and theorems of Section 4 on dynamic languages. Below, let Z denote the set of integers. Let x denote x+...+x DYNAMIC BOUNDED INTEGER PROGRAMMING INSTANCE: mxn matrices A and B, m-vector d, scalar k. QUESTION: Is there an infinite sequence
consider the
problem of determining an infinite horizon tour for a tramp steamer so as to maximize its average
profit travel ceives port i
Languages Derived from
In this section we describe several dynamic languages that may be considered extensions of languages in the class NP. All of the languages below can be proved to be P-space complete using
Cyc1ic Staffinq
Vehicle Routing Dantzig, Blattner, and Rao [D]
Periodic
Languages in The Class NP
{x i:icZ
that
per day. The tramp steamer is required to among a fixed finite set of ports and rea profit ijeach time that it travels from to port j. The above problem (also called
Ixi
of non-negative integer n-vectors such
THE COMPLEXITY OF DYNAMIC LANGUAGES AND DYNAMIC OPTIMIZATION PROBLEMS James B. Orlin Sloan School of Management M.I.T.
1.
In Section 4, we define "dynamic languages" and prove some elementary properties of these languages. In Sections 5 and 6, we prove the P-space completeness of certain dynamic languages derived from NP-complete languages. These results contrast with those by Orlin [01], [02), [03], 04], and [05], who provides polynomial time recognition algorithms for various dynamic languages derived from polynomial time languages, including the following dynamic languages: dynamic linear programming, dynamic network flows, coloring periodic interval graphs, dynamic matching, and dynamic 2-satisfiabil ity. In Section 7, we discuss the real-time, complexity of dynamic optimization problems. Here, dynamic optimization problems are viewed as realtime scheduling problems with known periodic data. We show that there is a polynomial real-time scheduling algorithm for the dynamic optimization problems presented in Section 3 if nd only if P = P-space. In Section 8, we show that the dynamic knapsack problem may be solved in pseudo-polynomial time (i.e., in polynomial time if the data are unary encoded), and thus it is meaningful to describe other P-space problems as "strongly P-space complete", a counterpart to strong'NP-comp'eteness as described by Garey and Johnson GJ]. Finally, in Section 9, we present some variations of dynamic language problems.
Introduction
Dynamic/period optimization problems arise naturally in various quantitative disciplines including Computer Science, Economics, and Operations Research. These periodic models may be applied to long-range economic planning, workforce scheduling, vehicle routing, machine maintenance, and a host of industrial applications. In this paper we offer a unifying framework for dynamic problems in terms of "dynamic languages", and we discuss the complexity of these languages. In particular, many dynamic languages derived from NP-complete languages can be shown to be polynomial space (P-space) complete. Among these are the following: the dynamic 3-satisfiability problem, and dynamic 3-dimensional matching problem, the dynamic partition problem, the dynamic hamiltonian circuit problem, and the dynamic independent set problem. We provide a general technique for showing how to prove the P-space completeness of dynamic problems derived from NP-complete problems.
I1
The Outline of This Paper In Sections 2 and 3, we give examples of dynamic (periodic) optimization problems. In Section 2, we reference some of the research on various dynamic/periodic problems, whereas in Section 3 we illustraze how to derive dynamic problems from problems in the class NP.
I
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and htotice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise. or to republish, requires a fee and/or specific permission. °1981 ACM 0-89791-041-9 /80/0500/0218 $00.75
2. Some Dynamic Optimization Problems Dynamic/periodic optimization problems occur in a variety of settings. Below we reference some examples of such optimization problems. The list
218
:sas;;;;yc-i(f
-i4L'B·4PliP;ChJIEY;Bb,;Tf ·:tr-ikic Z_
-....
3. Dynamic
below is intended as a sampler and is far from comprehens i ve. Two types of periodic optimization problems arise typically in employment: shift scheduling and day-off scheduling. Both of these are surveyed by Baker [ B]. Many problems of this type arise in service industries, and they may be classified according to the following parameters: types of workshifts allowed, the demand structures, and the types of additional constraints (possibly imposed by union contracts).
the technique illustrated in Sections 5 and 6.
These problems also serve to motivate the definitions and theorems of Section 4 on dynamic languages. Below, let Z denote the set of integers. Let x denote x+...+x DYNAMIC BOUNDED INTEGER PROGRAMMING INSTANCE: mxn matrices A and B, m-vector d, scalar k. QUESTION: Is there an infinite sequence
consider the
problem of determining an infinite horizon tour for a tramp steamer so as to maximize its average
profit travel ceives port i
Languages Derived from
In this section we describe several dynamic languages that may be considered extensions of languages in the class NP. All of the languages below can be proved to be P-space complete using
Cyc1ic Staffinq
Vehicle Routing Dantzig, Blattner, and Rao [D]
Periodic
Languages in The Class NP
{x i:icZ
that
per day. The tramp steamer is required to among a fixed finite set of ports and rea profit ijeach time that it travels from to port j. The above problem (also called
Ixi
of non-negative integer n-vectors such
