## epsilon-Transformation: Exploiting Phase Transitions to Solve Combinatorial Optimization Problems (1994)

Venue: | Artificial Intelligence |

Citations: | 14 - 5 self |

### BibTeX

@ARTICLE{Zhang94epsilon-transformation:exploiting,

author = {Weixiong Zhang and Joseph C. Pemberton},

title = {epsilon-Transformation: Exploiting Phase Transitions to Solve Combinatorial Optimization Problems},

journal = {Artificial Intelligence},

year = {1994},

volume = {81},

pages = {297--325}

}

### Years of Citing Articles

### OpenURL

### Abstract

It has been shown that there exists a transition in the averagecase complexity of tree search problems, from exponential to polynomial in the search depth. We develop a new method, called ffl- transformation, which makes use of this complexity transition, to find a suboptimal solution. With a random tree model, we show that the expected number of nodes expanded by branch-and-bound (BnB) using ffl-transformation is at most cubic in the search depth, and that the error of the solution cost found relative to the optimal solution cost is a small constant. We also present an iterative version of ffl-transformation that can be used to find both optimal and suboptimal goal nodes. Depth-first BnB (DFBnB) using iterative ffl-transformation significantly improves upon truncated DFBnB on random trees with large branching factors and deep goal nodes, finding better solutions sooner on average. Our experiments on the asymmetric traveling salesman problem show that DFBnB using ffl- transformati...

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Citation Context ...vement over truncated DFBnB is less, as seen by comparing Figure 5(b) to Figure 4(b). 6.2 The Asymmetric Traveling Salesman Problem The asymmetric traveling salesman problem (ATSP) [19] is an NP-hard =-=[7]-=- combinatorial problem. Given n cities and an asymmetric matrix (c i;j ) that 12 defines a cost between each pair of cities, the ATSP is to find a minimumcost tour that visits each city exactly once a... |

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Citation Context ... as the traveling salesman problem [33, 34]. Best-first search (BFS) and depth-first branch-and-bound (DFBnB) can be used to search these random trees. Both are special cases of branch-andbound (BnB) =-=[17, 20, 26]-=-, which is a general technique for problem solving. BFS maintains a partially expanded state space, and at each cycle expands a minimum-cost node among all those generated but not yet expanded, until ... |

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Citation Context ... a UCLA Chancellor's Dissertation Year Fellowship (1993-94). 1 1 Introduction It has been observed that phase transitions exist in many intelligent systems [9] and combinatorial optimization problems =-=[3, 4, 15, 18, 22, 23, 24, 31, 33, 34]-=-. A phase transition is a dramatic change to some problem property as some order parameter changes across a critical point. The simplest phase transition example is that water changes from a solid pha... |

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Citation Context ...s is that the territory explored by iterative ffl-DFBnB is generally smaller than the territory explored by truncated DFBnB, although iterative ffl-DFBnB may re-expand a node many times. Local search =-=[12, 13, 21]-=- is a well-known approximation method for many difficult combinatorial problems. Starting at an initial solution, such as one generated by a polynomial-time approximation algorithm, local search conti... |

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Citation Context ...nsformation for finding both suboptimal and optimal goal nodes (Section 5). We identify the conditions when both methods are effective on random trees and on the asymmetric traveling salesman problem =-=[19]-=- (Section 6). Related work is discussed in Section 7, and our conclusions appear in Section 8. 2 ffl-Transformation ffl-transformation is based on the following very simple observation of Figure 1(b).... |

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Citation Context ... a UCLA Chancellor's Dissertation Year Fellowship (1993-94). 1 1 Introduction It has been observed that phase transitions exist in many intelligent systems [9] and combinatorial optimization problems =-=[3, 4, 15, 18, 22, 23, 24, 31, 33, 34]-=-. A phase transition is a dramatic change to some problem property as some order parameter changes across a critical point. The simplest phase transition example is that water changes from a solid pha... |

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Citation Context ... a UCLA Chancellor's Dissertation Year Fellowship (1993-94). 1 1 Introduction It has been observed that phase transitions exist in many intelligent systems [9] and combinatorial optimization problems =-=[3, 4, 15, 18, 22, 23, 24, 31, 33, 34]-=-. A phase transition is a dramatic change to some problem property as some order parameter changes across a critical point. The simplest phase transition example is that water changes from a solid pha... |

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Citation Context ...of the edge costs along the path from the root to that node. An optimal goal node is a node of minimum cost at depth d. Compared to the conventional tree model that assumes independence of node costs =-=[8, 27, 28]-=-, this random tree is more realistic because it naturally introduces dependences among node costs. The costs of two nodes are dependent on each other if they share common edges on their paths to the r... |

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Citation Context ...s is that the territory explored by iterative ffl-DFBnB is generally smaller than the territory explored by truncated DFBnB, although iterative ffl-DFBnB may re-expand a node many times. Local search =-=[12, 13, 21]-=- is a well-known approximation method for many difficult combinatorial problems. Starting at an initial solution, such as one generated by a polynomial-time approximation algorithm, local search conti... |

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Citation Context ... as the traveling salesman problem [33, 34]. Best-first search (BFS) and depth-first branch-and-bound (DFBnB) can be used to search these random trees. Both are special cases of branch-andbound (BnB) =-=[17, 20, 26]-=-, which is a general technique for problem solving. BFS maintains a partially expanded state space, and at each cycle expands a minimum-cost node among all those generated but not yet expanded, until ... |

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Citation Context ...ed ffl -DFBnB with a local search method [25] which was applied five different times for each problem instance in our experiments. The five initial tours were generated by the nearest-neighbor method =-=[6]-=-, nearest insertion farthest insertion, greedy algorithms, and the patching algorithm [6, 14]. We used random cost matrices and matrices converted from no-wait flowshop scheduling for 4 machines, whic... |

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Citation Context ...each problem instance in our experiments. The five initial tours were generated by the nearest-neighbor method [6], nearest insertion farthest insertion, greedy algorithms, and the patching algorithm =-=[6, 14]-=-. We used random cost matrices and matrices converted from no-wait flowshop scheduling for 4 machines, which is NP-hard [13]. No-wait flowshop scheduling involves determining a sequence for processing... |

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Citation Context ...nd returns to the starting city. The ATSP can be optimally solved by branch-and-bound (BnB), using the solution cost of the related assignment problem (AP) [26] as a monotonic node cost function. See =-=[2]-=- for a summary and detailed description of the method. Briefly, the state space of the ATSP under BnB is a tree without duplicate nodes. In our implementation of ffl-DFBnB and iterative fflDFBnB, we u... |

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Citation Context ...s is that the territory explored by iterative ffl-DFBnB is generally smaller than the territory explored by truncated DFBnB, although iterative ffl-DFBnB may re-expand a node many times. Local search =-=[12, 13, 21]-=- is a well-known approximation method for many difficult combinatorial problems. Starting at an initial solution, such as one generated by a polynomial-time approximation algorithm, local search conti... |

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Citation Context ...of the edge costs along the path from the root to that node. An optimal goal node is a node of minimum cost at depth d. Compared to the conventional tree model that assumes independence of node costs =-=[8, 27, 28]-=-, this random tree is more realistic because it naturally introduces dependences among node costs. The costs of two nodes are dependent on each other if they share common edges on their paths to the r... |

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Citation Context ...nt as an approximation. This method was also referred to as truncated DFBnB, which we adopted in this paper. The earliest study of this method that we found was made by Ashour [1, 16]. Ibaraki et al. =-=[10, 11]-=- systematically studied approximation methods based on BnB, which they called suboptimal BnB algorithms. Their experimental results indicate that a good cost function is crucial to realize a successfu... |

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Citation Context ...error of tour costs from these two algorithms relative to the AP lower bounds. This result is similar to the result on random trees in Figure 4. We also compared ffl -DFBnB with a local search method =-=[25]-=- which was applied five different times for each problem instance in our experiments. The five initial tours were generated by the nearest-neighbor method [6], nearest insertion farthest insertion, gr... |

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Citation Context ...tive ffl-transformation are effective. To this end, we compare fflDFBnB and iterative ffl-DFBnB with other approximation algorithms. Iterative ffl-DFBnB can be used in the same way as truncated DFBnB =-=[11, 32]-=- to find approximate and optimal solutions. Truncated DFBnB is a DFBnB that terminates prematurely when the total available computation has been exhausted. The best solution found up to that point can... |

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Citation Context ...tive ffl-transformation are effective. To this end, we compare fflDFBnB and iterative ffl-DFBnB with other approximation algorithms. Iterative ffl-DFBnB can be used in the same way as truncated DFBnB =-=[11, 32]-=- to find approximate and optimal solutions. Truncated DFBnB is a DFBnB that terminates prematurely when the total available computation has been exhausted. The best solution found up to that point can... |

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Citation Context ... as the traveling salesman problem [33, 34]. Best-first search (BFS) and depth-first branch-and-bound (DFBnB) can be used to search these random trees. Both are special cases of branch-andbound (BnB) =-=[17, 20, 26]-=-, which is a general technique for problem solving. BFS maintains a partially expanded state space, and at each cycle expands a minimum-cost node among all those generated but not yet expanded, until ... |

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Citation Context ...lution found to that point as an approximation. This method was also referred to as truncated DFBnB, which we adopted in this paper. The earliest study of this method that we found was made by Ashour =-=[1, 16]-=-. Ibaraki et al. [10, 11] systematically studied approximation methods based on BnB, which they called suboptimal BnB algorithms. Their experimental results indicate that a good cost function is cruci... |

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