## Integer Programming: Optimization and Evaluation are Equivalent

### BibTeX

@MISC{Orlin_integerprogramming:,

author = {James B. Orlin and Abraham P. Punnen and Andreas S. Schulz},

title = {Integer Programming: Optimization and Evaluation are Equivalent},

year = {}

}

### OpenURL

### Abstract

We show that if one can find the optimal value of an integer programming problem min{cx: Ax ≥ b, x ∈ Z n +} in polynomial time, then one can find an optimal solution in polynomial time. We also present a proper generalization to general integer programs and to local search problems of the well-known result that optimization and augmentation are equivalent for 0/1-integer programs. Our results imply that, among other things, PLS-complete problems cannot have “near-exact” neighborhoods, unless PLS = P.

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Citation Context ...uch that X = {x ∈ Z n + : Ax ≥ b, x ≤ u}.) For input vector c ∈ Z n , 6 The complexity classes PLS and PLS-complete were introduced by Johnson et al. to capture the difficulty of finding local optima =-=[9]-=-. Prominent PLS-complete problems include the max-cut problem with the flip neighborhood and the graph partitioning problem with the swap neighborhood [14], the traveling salesman problem with the k-e... |

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Citation Context ... value [4]. Ausiello et al. and Johnson pointed out that evaluation is actually as hard as finding an optimal solution for all optimization problems whose associated decision problems are NP-complete =-=[2,8]-=-. Schulz reports on the relative complexity of 15 problems related to 0/1-integer programming, 3 including augmentation, optimization and evaluation, and shows that they are all oraclepolynomial-time ... |

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Citation Context ...rhood [14], the traveling salesman problem with the k-exchange neighborhood (for sufficiently large, but constant k) [10], and the problem of finding pure-strategy Nash equilibria in congestion games =-=[5]-=-. 7 We may assume, without loss of generality, that there exists a feasible solution, i.e., X ̸= ∅. Otherwise both problems, evaluation and optimization, would have to detect infeasibility.4 J.B. Orl... |

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Citation Context ...apture the difficulty of finding local optima [9]. Prominent PLS-complete problems include the max-cut problem with the flip neighborhood and the graph partitioning problem with the swap neighborhood =-=[14]-=-, the traveling salesman problem with the k-exchange neighborhood (for sufficiently large, but constant k) [10], and the problem of finding pure-strategy Nash equilibria in congestion games [5]. 7 We ... |

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Citation Context ...oximation scheme if and only of the corresponding augmentation problem has a (fully) polynomial-time approximation scheme. The same result was obtained by Orlin et al. using quite different arguments =-=[11]-=-.Optimization and evaluation are equivalent 9 5 The unit increment problem and local optimization In this section we consider the complexity of computing a locally optimal solution with respect to a ... |

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Citation Context ...with the flip neighborhood and the graph partitioning problem with the swap neighborhood [14], the traveling salesman problem with the k-exchange neighborhood (for sufficiently large, but constant k) =-=[10]-=-, and the problem of finding pure-strategy Nash equilibria in congestion games [5]. 7 We may assume, without loss of generality, that there exists a feasible solution, i.e., X ̸= ∅. Otherwise both pro... |

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Citation Context ...m. 5 Hence, we have a proper generalization of a result by Grötschel and Lovász and Schulz et al. who showed that optimization and augmentation are polynomial-time equivalent for 0/1-integer programs =-=[6,17]-=-. A relaxation to the augmentation problem, the ɛ-augmentation problem, can be defined as follows: Given an objective function vector c and a feasible solution x, find a feasible solution with better ... |

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Citation Context ...r exact” if every locally optimal solution is no worse than all but a polynomial number of feasible solutions. Near-exact neighborhoods are related to the domination number of local search heuristics =-=[7]-=-. We show that, for 0/1-integer programs, polynomial solvability of the local augmentation problem implies polynomial solvability of the local optimization problem whenever the corresponding neighborh... |

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Citation Context ...on available for the unit increment problem is not of much help for NP-hard problems. This provides additional evidence that postoptimality analysis is typically hard for NP-hard problems (see, e.g., =-=[3,18,13]-=- for related results). We now examine the relationship between the unit increment problem and the augmentation problem. Let x 0 be an optimal solution to min{cx : x ∈ X}, and let j be a given index, 1... |

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(Show Context)
Citation Context ... value [4]. Ausiello et al. and Johnson pointed out that evaluation is actually as hard as finding an optimal solution for all optimization problems whose associated decision problems are NP-complete =-=[2,8]-=-. Schulz reports on the relative complexity of 15 problems related to 0/1-integer programming, 3 including augmentation, optimization and evaluation, and shows that they are all oraclepolynomial-time ... |

7 |
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Citation Context ...on available for the unit increment problem is not of much help for NP-hard problems. This provides additional evidence that postoptimality analysis is typically hard for NP-hard problems (see, e.g., =-=[3,18,13]-=- for related results). We now examine the relationship between the unit increment problem and the augmentation problem. Let x 0 be an optimal solution to min{cx : x ∈ X}, and let j be a given index, 1... |

6 |
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(Show Context)
Citation Context ...on available for the unit increment problem is not of much help for NP-hard problems. This provides additional evidence that postoptimality analysis is typically hard for NP-hard problems (see, e.g., =-=[3,18,13]-=- for related results). We now examine the relationship between the unit increment problem and the augmentation problem. Let x 0 be an optimal solution to min{cx : x ∈ X}, and let j be a given index, 1... |

4 |
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(Show Context)
Citation Context ...ting an optimal solution and provided sufficient and necessary conditions for the existence of optimization problems for which obtaining an optimal solution is harder than computing the optimal value =-=[4]-=-. Ausiello et al. and Johnson pointed out that evaluation is actually as hard as finding an optimal solution for all optimization problems whose associated decision problems are NP-complete [2,8]. Sch... |

4 | On the relative complexity of 15 problems related to 0/1-integer programming
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- 2009
(Show Context)
Citation Context ...ports on the relative complexity of 15 problems related to 0/1-integer programming, 3 including augmentation, optimization and evaluation, and shows that they are all oraclepolynomial-time equivalent =-=[16]-=-. In this paper, we prove that evaluation and optimization are polynomial-time equivalent for all integer linear programming problems. That is, given a matrix A ∈ Z m×n and a vector b ∈ Z m , a polyno... |

3 |
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- Groetschel, Lovasz
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(Show Context)
Citation Context ...m. 5 Hence, we have a proper generalization of a result by Grötschel and Lovász and Schulz et al. who showed that optimization and augmentation are polynomial-time equivalent for 0/1-integer programs =-=[6,17]-=-. A relaxation to the augmentation problem, the ɛ-augmentation problem, can be defined as follows: Given an objective function vector c and a feasible solution x, find a feasible solution with better ... |