Discrete OptimizationThe equitable dispersion problem
Introduction
Suppose, we are given a set N of n elements, and , the inter-element distance between any two elements i and j. The so-called Maximum Dispersion Problem (Max DP) is to select a subset of cardinality m, i.e., , such that some efficiency-based function of from the selected subset M is maximized. The problem is then formulated as , where is the set containing all subsets of N. Two important and widely addressed functions in the literature (see, e.g., [3]) are the sum and minimum of for all , respectively. The former and latter functions can thus be written as and , where it is usually assumed that for all , we have . Hence, the first selection criterion for the maximally disperse set is called Maxsum and the second criterion is called Maxmin. To differentiate the two resultant maximization problems, we name them accordingly as Maxsum DP and Maxmin DP.
Define a binary variable if element i belongs to M and otherwise, for . Then the problem with the former objective can be rewritten as a cardinality-constrained quadratic 0–1 programming problem:
Problems (1), (2), (3) sometimes is also referred to as maximum diversity/similarity problem [15], where measures the difference/similarity between elements i and j. Problems of type (1), (2), (3) can be tackled using algorithms designed for solving quadratic 0–1 programs [8], [25], or we can easily obtain a classical linear mixed 0–1 reformulation (see e.g., [17], [23]), which, in turn, can be solved using any standard mixed integer programming (MIP) solver (e.g., CPLEX, XPRESS-MP):In the literature, the maximum dispersion problem with the second objective is also referred to as the p-dispersion problem [12], [22], [24]. It is regarded as a maxmin problem among individual inter-element distances. A mixed-integer 0–1 reformulation of the problem is presented as:Note that is conveniently chosen as , where is an upper bound on the optimal value of y, e.g., . The above linearization is straightforward and thus the formulations may be tightened. For detailed development of the above formulations and further tightening, we refer the readers to [3].
As discussed in [15], [23], an element i may be characterized by a vector with possible attributes . Then for we can consider, for instance, a Euclidean distance measure, or a general norm:In the case of (8), all ’s are nonnegative. However, can generally take both positive and negative values.
The maximum dispersion problem primarily focuses on operational efficiency of locating facilities according to distance, accessibility, impacts, etc. (see [11], [12], [22], [29]). It also arises in various other contexts including maximally diverse/similar group selection (e.g., biological diversity, admissions policy formulation, committee formation, curriculum design, market planning, etc.) [1], [15], [16], [23], [34] and densest subgraph identification [21]. For a detailed literature survey on Maxsum DP and Maxmin DP, we refer the readers to [3], [11], [23]. For these two problems, exact methods [3], [15], [28] and heuristics [15], [18], [20], [30], [31] have been developed.
An equally important concern in facility location theory is equity. It is here referred to as the concept or idea of fairness among candidate facility sites. Furthermore, one particular interest is in balancing equity with efficiency in a model-building paradigm [33]. However, the majority of models in the operations research literature are efficiency-based and little work has been done for the equity concern [26]. This is especially critical when dealing with urban public facility location [33]. This paper explores the use of several equity measures in the framework of the dispersion problem. These equity measures may also arise in the contexts of diverse/similar group selection, and dense and regular subgraph identification. For example, one may address fair diversification or assimilation among members of a network. Somewhat related work on equity based measures in network flow problems can be found in [6], [7].
The remainder of this paper is organized as follows. Section 2 introduces a number of new measures in the context of the dispersion problem. Section 3 develops a nonlinear binary optimization framework with the introduced equity measures being objectives. Section 4 presents the corresponding linear mixed 0–1 reformulations. Section 5 shows the computational complexity of the resultant optimization problems. Section 6 provides several graph-theoretic results related to the measures. Section 7 discusses our preliminary computational experiments. Section 8 concludes the paper and points out future research directions.
Section snippets
Equity measures
As discussed in Section 1, the dispersion problem in the literature has mainly focused on efficiency-based objectives. In this section, we introduce several equity-based measures, each of which alternatively represents some function of dispersion with respect to individual elements. Therefore, unlike efficiency-based measures that consider some dispersion quantity for the entire selection M (e.g., total amount of dispersion and the minimum level of dispersion), these measures are used to
Mathematical programming models
After defining the equity measures in the previous section, we can next formulate respective mathematical programs. Except for the first mathematical program, we impose the cardinality restriction on the number of elements to be selected. Note that in the first model, the group size is part of the decision to make.
Basic linear mixed 0–1 reformulations
The development of exact linear reformulations facilitates the use of state-of-the-art mixed-integer linear programming solvers. In order to linearize the problems introduced in Section 3, we can apply techniques similar to those used in [3], [23], [32], [35]. Applying more specialized techniques may derive tighter mixed-integer programming formulations. Since the focus of this paper is to establish the equity measures in the context of the dispersion problem, we leave formulation tightening to
Computational complexity
It is known that both Maxsum DP and Maxmin DP are strongly NP-hard [11], [15], [23]. In this section we study the computational complexity of Max-Mean DP (problem (9)), Max-Minsum DP (problem (11)), and Min-Diff DP (problem (14)).
If , Max-Mean DP is polynomially solvable (see [27]). We next show that the problem becomes difficult when no constraints are imposed on the sign of . Proposition 6 Max-Mean DP is strongly NP-hard if coefficients can take both positive and negative values. Proof The proof of the
Graph-theoretic interpretation
In this section we will provide interesting interpretations of Maxsum DP, Maxmin DP, Max-Mean DP, Max-Minsum DP, and Min-Diff DP (see (1), (2), (3), (6), (7), (9), (11), (14)) in the graph-theoretic context. For graph-theoretic terms that appear in this section, we refer the readers to detailed discussions in [2], [4], [10].
Consider an undirected graph , where V denotes the set of nodes and E denotes the set of edges. For each of the problems, we consider two cases as follows:
- (a)
graph G is
Test instances
We randomly generated 4 sets of inter-element distance matrices, , given the number of elements . We generated 10 fully-dense matrices in each matrix set and used uniform distribution to generate each . For each value n, we varied the cardinality value m and thus dealt with 10 instances given each pair of n and m. All computational experiments were performed on a PC with Intel Core 2 CPU of 2.66 GHz and RAM of 3 GB.
Exact MIP solutions
The first experiment is to compare the exact
Conclusion
In this paper, we introduce three equity-based measures for the dispersion problem to address the imbalance of studying efficiency and equity in the optimization literature. We believe the main contribution of the paper is to enhance our understanding of equity in the optimization framework for the dispersion problems. We develop mathematical programs with proposed equity measures being objectives and present several mixed-integer linear reformulations. Computational complexity and
Acknowledgments
The authors would like to thank the anonymous referees for their helpful comments. The research of the first two authors is partially supported by the grant from AFOSR. The research of the first author is also supported by NSF grant CMMI-0825993.
References (36)
- et al.
The complexity of regular subgraph recognition
Discrete Applied Mathematics
(1990) - et al.
Analytical models for locating undesirable facilities
European Journal of Operational Research
(1989) The discrete p-dispersion problem
European Journal of Operational Research
(1990)Computational aspects of the maximum diversity problem
Operations Research Letters
(1996)- et al.
Approximation algorithms for maximum dispersion
Operations Research Letters
(1997) Upper bounds and exact algorithms for p-dispersion problems
Computers and Operations Research
(2006)Toward a theory of public facility location
Papers in Regional Science
(1968)A note on a global approach for general 0–1 fractional programming
European Journal of Operational Research
(1997)- et al.
Maximum diversity/similarity models with extension to part grouping
International Transactions in Operations Research
(2005) - et al.
Network Flows: Theory, Algorithms, and Applications
(1993)
Lagrangian solution of maximum dispersion problems
Naval Research Logistics
Modern Graph Theory
The maximum clique problem
The knaspack sharing problem
Operations Research
The sharing problem
Operations Research
Exact solution of the quadratic knapsack problem
INFORMS Journal on Computing
Graph Theory
Greedy randomized adaptive search procedures
Journal of Global Optimization
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