274
个编辑
更改
跳到导航
跳到搜索
第168行：
第168行： − + − + − + 第179行：
第179行： − 这意味着相应的决策问题是 NP 完全问题。在计算机科学中，有趣的优化问题通常具有上述性质，因此是 NPO 问题。如果存在一个在多项式时间内找到最优解的算法，那么这个问题又称为 p 优化问题。通常，在处理类 NPO 时，人们对决策版本为 NP-complete 的优化问题感兴趣。请注意，硬度关系总是与某种程度的还原有关。由于近似算法和计算优化问题之间的联系，在某些方面保留近似值的约化比通常的图灵约化和卡普约化更受青睐。这种削减的一个例子是 l- 削减。由于这个原因，NP-complete 决策版本的优化问题不一定称为 npo 完成。+ − − 第192行：
第190行： − + − + − + − + − + 第206行：
第204行： − + − 一个 NPO 问题被称为多项式有界(PB) ，如果对于每个实例 < math > x </math > 和 f (x) </math > 中的每个解 < math > y，度量值 < math > 第219行：
第216行： − +
→NP optimization problem
* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
在<math>f(x)</math>中，对给定实例的大小<math>x</math>中，每个可行解的大小都是多项式的[[有界集|有界的]]，
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
语言<math>\{\,x\,\mid\,x\ in I\,\}</math>和<math>f(x)\,\}</math>中的<math>\{\,(x,y)\,\mid\,y\</math>在[[多项式时间]]内可以[[可判定语言|识别]]，并且，
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
<math>m</math>是[[多项式时间|多项式时间可计算]]。
This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.
This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.
这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是非营利组织问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''<font color="#FF8000">P-优化（PO）问题 P-optimization problem </font>'''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些还原有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的规约比通常的'''<font color="#FF8000">图灵和Karp规约 Turing and Karp Reductions </font>'''更为可取。这种减少的一个例子就是'''<font color="#FF8000">L-规约 L-reduction </font>'''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。
NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />
NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />
* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
''NPO(I)''：等价于[[FPTAS]]。包含[[背包问题]]。
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
''NPO(II)''：等价于[[多项式时间近似方案| PTAS]]。包含[[Makespan]]调度问题。
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的代价最多为''c''乘以最优成本（对于最小化问题）或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在[[Juraj Hromkovi|Hromkovič]]的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。不排除，等于APX。包含[[MAX-SAT]]和度量[[旅行商问题| TSP]]。
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
''NPO(IV)''：用多项式时间算法近似最优解的一类NPO问题，其比率为输入大小的对数的多项式。在赫罗姆科维奇的书中，除非P=NP，否则所有的NPO(III)-问题都被排除在这个类之外。包含[[set cover]]问题。
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''NPO(V)''：用多项式时间算法逼近最优解的一类NPO问题。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都被排除在这类问题之外。包含[[旅行推销员问题| TSP]]和[[集团问题|最大集团问题]]。
An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
如果对于每个实例<math>x</math>和对于<mat>f(x)</math>中的每个解<math>y\in f(x)<math>测量，则NPO问题称为多项式有界（PB）
m(x, y)
m(x, y)
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
被一个 < math > x </math > 大小的多项式函数所限制。NPOPB 类是一类多项式有界的 NPO 问题。
被一个 <math>x</math> 大小的多项式函数所限制。NPOPB 类是一类多项式有界的 NPO 问题。