An algorithm for ranking assignments using reoptimization

Pedersen, Christian Roed; Nielsen, Lars Relund; Andersen, Kim Allan

doi:10.1016/j.cor.2007.04.008

Cited by 27 publications

(17 citation statements)

References 18 publications

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Section: Finding the K-best Multimodal Assignmentsmentioning

confidence: 99%

“…An efficient algorithm for ranking classic assignments is given in Pedersen et al (2005a). The algorithm uses a reoptimization solution technique such that the minimum-cost assignments for the subsets can be found easily (see (Pedersen et al 2005a) for more details). Because the general branching technique described above does not create more subsets than the classic branching technique, the overall complexity for ranking the K-best multimodal assignments is the same.…”

Section: Finding the K-best Multimodal Assignmentsmentioning

confidence: 99%

“…That is, we just have to increase a local pointer by one to find the new minimal cost. For more details on the ranking implementation, see Pedersen et al (2005a), an earlier version of Pedersen et al (2007). All nondominated points found by the ranking procedure are stored in a single linked list available in both phase one and phase two.…”

Section: Implementational Detailsmentioning

confidence: 99%

“…The K-best multimodal assignment procedure was implemented using the reoptimization algorithm in Pedersen et al (2005a) for ranking assignments for the classical AP, with the slightly more general branching technique given in Section 4. In particular, note that, when considering a subset where we remove an entry in the ordered array of parametric costs, the new entry with minimal cost is the next entry in the array.…”

Section: Implementational Detailsmentioning

confidence: 99%

“…Acknowledging that ranking procedures have been applied with great success for other bicriterion optimization problems (see, e.g., (Nielsen et al 2003)), we employ ranking of multimodal assignments as a subroutine. The subroutine is an efficient extension of the algorithm for finding the K best assignments by Pedersen et al (2005a).…”

Section: Introductionmentioning

confidence: 99%

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The Bicriterion Multimodal Assignment Problem: Introduction, Analysis, and Experimental Results

Pedersen

Nielsen

Andersen

2008

INFORMS Journal on Computing

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Abstract:We consider the bicriterion multimodal assignment problem, which is a new generalization of the classical linear assignment problem. A two-phase solution method using an effective ranking scheme is presented. The algorithm is valid for generating all nondominated criterion points or an approximation. Extensive computational results are conducted on a large library of test instances to test the performance of the algorithm and to identify hard test instances. Also, test results of the algorithm applied to the bicriterion assignment problem are provided.

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Section: Finding the K-best Multimodal Assignmentsmentioning

confidence: 99%

Section: Finding the K-best Multimodal Assignmentsmentioning

confidence: 99%

Section: Implementational Detailsmentioning

confidence: 99%

Section: Implementational Detailsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Bicriterion Multimodal Assignment Problem: Introduction, Analysis, and Experimental Results

Pedersen

Nielsen

Andersen

2008

INFORMS Journal on Computing

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k-Best Enumeration

Eppstein¹

2016

Encyclopedia of Algorithms

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We survey k-best enumeration problems and the algorithms for solving them, including in particular the problems of finding the k shortest paths, k smallest spanning trees, and k best matchings in weighted graphs.in both the input size and k. There are two general techniques that can be used to achieve this, based on optimal substructures and solution space partitioning. In turn, these methods rely on fast algorithms for the selection problem in certain sets of structured values. SelectionUnderlying many k-best algorithms is the problem of selection, finding the k smallest values from a larger set of values [21]. If there are n values in the set, then selection may be solved in time O(n + k), for instance by quickselect, an algorithm that chooses a random pivot value from the set, partitions the rest of the set into subsets that are less than, equal to, or greater than the pivot, and then recurses into one of these subsets [131].However, in k-best problems, we do not wish to generate all solutions before finding the best of them: we wish to avoid the O(n) part of this O(n + k) time bound. Often, the solution values among which we are selecting the k best have some additional structure that allows finding the k best without examining all solutions. For instance, suppose that the space of solutions can be represented as the vertex set of an (implicitly defined) edge-weighted graph, that the degrees of the vertices in this graph are bounded, and that the quality of any particular solution equals the length of the shortest path in this graph from some designated starting vertex to the solution vertex. In such a case, Dijkstra's algorithm can be used to recover the k smallest of these values in time O(k log k), independent of the size of the graph, without requiring that the whole graph be explored. Other structures that allow the k best solutions to be found more efficiently than enumeration of the whole solution set include representations of the solution space as the set of sums of pairs of values drawn from two sorted sets [76,111], or as the elements of an (implicitly defined) matrix whose rows and columns are sorted [77]. Again, for such structures, the k smallest values can be found efficiently without examining the whole matrix or the whole set of pair sums.A general framework that can be used to describe many of these structured selection problems is the problem of selection in a d-ary heap. A min-heap is a rooted tree, with values associated with its nodes, such that each non-root node has a value that is at least as large as its parent. In order to avoid having to construct all solutions before performing a selection algorithm, and in order to handle situations in which this tree has infinitely many nodes, we will assume that our selection algorithm is given as input an implicit representation of such a heap. That is, its input consists of a binary representation of the root node of the heap together with pointers to two subroutines that take a single node as input: one subroutine to list the children of any...

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