Cardinality constrained combinatorial optimization: Complexity and polyhedra

Stephan, Rüdiger

doi:10.1016/j.disopt.2010.03.002

Cited by 14 publications

(7 citation statements)

References 11 publications

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“…, 20 are summarized as tables in Appendix C. On each network, each setting (depicted by a rounded rectangle in Figure 3) shows the averages and standard deviations of the optimal results for 20 runs for p = 10. After obtaining the averages and standard deviations of the (population size, generation number) settings (20,1), and (20, 100), we can trace the evolution of the optimal result for p = 10. For example, "0.74, 0.12" indicates that the average of the optimal p = 10 results in (20, 100) is 0.74 times that of (20, 1), and the standard deviation is 0.12 times.…”

Section: Resultsmentioning

confidence: 99%

“…Upon close examination, if a CCOP possesses a non-increasing property and η is accidentally set to 0, the Mucard algorithm cannot push the optimal results of every scenario onto the Pareto front; further, it is ambiguous whether the missing results in a certain scenario are caused by Pareto dominance or by the algorithm itself. Thus, η in (6) cannot be zero when (20,1) to setting (20, 100); B: setting (20, 100) to setting (220, 100); C: setting (20, 100) to setting (20, 1100). MSCCOP possesses the non-increasing property and when the optimum is pursued in every scenario.…”

Section: Resultsmentioning

confidence: 99%

“…For instance, the cardinality of set {1, 2, 4} is three, which can be denoted as card({1, 2, 4}) = 3. The minimized single-scenario version 1) of a cardinality constrained optimization problem (CCOP), which searches for a cardinality-specified subset that optimally minimizes a function, can be expressed as follows [1]:…”

Section: Introductionmentioning

confidence: 99%

“…99A: setting(20,1) to setting(20, 100); B: setting (20, 100) to setting (220, 100); C: setting(20, 100) to setting(20, 1100). + sign is constant.…”

mentioning

confidence: 99%

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Solving multi-scenario cardinality constrained optimization problems via multi-objective evolutionary algorithms

Zhou

Wang

Peng

et al. 2019

Sci. China Inf. Sci.

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Cardinality constrained optimization problems (CCOPs) are fixed-size subset selection problems with applications in several fields. CCOPs comprising multiple scenarios, such as cardinality values that form an interval, can be defined as multi-scenario CCOPs (MSCCOPs). An MSCCOP is expected to optimize the objective value of each cardinality to support decision-making processes. When the computation is conducted using traditional optimization algorithms, an MSCCOP often requires several passes (algorithmic runs) to obtain all the (near-)optima, where each pass handles a specific cardinality. Such separate passes abandon most of the knowledge (including the potential superior solution structures) learned in one pass that can also be used to improve the results of other passes. By considering this situation, we propose a generic transformation strategy that can be referred to as the Mucard strategy, which converts an MSCCOP into a low-dimensional multi-objective optimization problem (MOP) to simultaneously obtain all the (near-)optima of the MSCCOP in a single algorithmic run. In essence, the Mucard strategy combines separate passes that deal with distinct variable spaces into a single pass, enabling knowledge reuse and knowledge interchange of each cardinality among genetic individuals. The performance of the Mucard strategy was demonstrated using two typical MSCCOPs. For a given number of evolved individuals, the Mucard strategy improved the accuracy of the obtained solutions because of the in-process knowledge than that obtained by untransformed evolutionary algorithms, while reducing the average runtime. Furthermore, the equivalence between the optimal solutions of the transformed MOP and the untransformed MSCCOP can be theoretically proved.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…99A: setting(20,1) to setting(20, 100); B: setting (20, 100) to setting (220, 100); C: setting(20, 100) to setting(20, 1100). + sign is constant.…”

mentioning

confidence: 99%

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Solving multi-scenario cardinality constrained optimization problems via multi-objective evolutionary algorithms

Zhou

Wang

Peng

et al. 2019

Sci. China Inf. Sci.

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“…into the ThSP. The cardinality constraint (17) represents a cutting plane which allows to set at most (|X * | − 1) x-variables of the ThSP's solution previously assigned to be 1 and, thus, excludes solution X * from the polyhedron (Stephan (2010)). This procedure is similar to subtour elimination constraints in the Travelling Salesman Problem.…”

Section: Generation Of Cutting Planesmentioning

confidence: 99%

Hospital-wide therapist scheduling and routing: Exact and heuristic methods

Gärtner

Frey

Kolisch

2018

IISE Transactions on Healthcare Systems Engineering

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In this paper, we address the problem of scheduling and routing physical therapists hospital-wide. At the beginning of a day, therapy jobs are known to a hospital's physical therapy scheduler who decides for each therapy job when, where and by which therapist a job is performed. If a therapist is assigned to a sequence which contains two consecutive jobs that must take place in different treatment rooms, then transfer times must be considered. We propose three approaches to solve the problem. First, an Integer Program (IP) simultaneously schedules therapies and routes therapists. Second, a cutting plane algorithm iteratively solves the therapy scheduling problem without routing constraints and adds cuts to exclude schedules which have no feasible routes. Since hospitals are interested in obtaining quick solutions, we also propose a heuristic algorithm, which schedules therapies sequentially by simultaneously checking routing and resource constraints. Using real-world data from a hospital, we compare the performance of the three approaches. Our computational analysis reveals that our IP formulation fails to solve test instances, which have more than 30 jobs, to optimality in an acceptable solution time. In contrast, the cutting plane algorithm can solve instances with more than 100 jobs optimally. The heuristic approach obtains good solutions for large real-world instances within fractions of a second.

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On the Complexity of Clustering with Relaxed Size Constraints

Goldwurm

Lin

Saccà

2016

Lecture Notes in Computer Science

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Abstract. We study the computational complexity of the problem of computing an optimal clustering {A1, A2, ..., A k } of a set of points assuming that every cluster size |Ai| belongs to a given set M of positive integers. We present a polynomial time algorithm for solving the problem in dimension 1, i.e. when the points are simply rational values, for an arbitrary set M of size constraints, which extends to the 1-norm an analogous procedure known for the 2-norm. Moreover, we prove that in the Euclidean plane, i.e. assuming dimension 2 and 2-norm, the problem is NP-hard even with size constraints set reduced to M = {2, 3}.

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Cardinality constrained combinatorial optimization: Complexity and polyhedra

Cited by 14 publications

References 11 publications

Solving multi-scenario cardinality constrained optimization problems via multi-objective evolutionary algorithms

Solving multi-scenario cardinality constrained optimization problems via multi-objective evolutionary algorithms

Hospital-wide therapist scheduling and routing: Exact and heuristic methods

On the Complexity of Clustering with Relaxed Size Constraints

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