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Multidimensional assignment problem

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The multidimensional assignment problem (MAP) is a fundamental <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> problem which was introduced by William Pierskalla.<a href="#cite_note-Pier68-1">[1]</a> This problem can be seen as a generalization of the linear <a href="Assignment_problem" title="Assignment problem">assignment problem</a>.<a href="#cite_note-Pasi21-2">[2]</a> In words, the problem can be described as follows:

<dl><dd>An instance of the problem has a number of agents (i.e., cardinality parameter) and a number of job characteristics (i.e., dimensionality parameter) such as task, machine, time interval, etc. For example, an agent can be assigned to perform task X, on machine Y, during time interval Z. Any agent can be assigned to perform a job with any combination of unique job characteristics at some cost. These costs may vary based on the assignment of agent to a combination of job characteristics - specific task, machine, time interval, etc. The problem is to minimize the total cost of assigning the agents so that the assignment of agents to each job characteristic is an <a href="Injective_function" title="Injective function">injective function</a>, or <a href="One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one function</a> from agents to a given job characteristic.</dd></dl>

Alternatively, describing the problem using graph theory:

<dl><dd>The multidimensional assignment problem consists of finding, in a <a href="Weighted_graph" class="mw-redirect" title="Weighted graph">weighted</a> <a href="Multipartite_graph" title="Multipartite graph">multipartite graph</a>, a <a href="Matching_(graph_theory)" title="Matching (graph theory)">matching</a> of a given size, in which the sum of weights of the edges is minimum.<a href="#cite_note-3">[3]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

Various formulations of this problem can be found in the literature. Using cost-functions, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">–dimensional assignment problem (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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<dl><dd>Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> sets, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1},\ldots J_{D-1}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{1},\ldots J_{D-1}}</annotation>

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</math><img src="./101a46f19008e860c14d6c79159d12443670568f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.472ex; height:2.509ex;" alt="{\displaystyle J_{1},\ldots J_{D-1}}" loading="lazy">, of equal size, together with a cost <a href="Array" title="Array">array</a> or multidimensional <a href="Weight_function" title="Weight function">weight function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times J_{1}\times \ldots \times J_{D-1}\rightarrow \mathbb {R} _{+}}">

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<annotation encoding="application/x-tex">{\displaystyle A\times J_{1}\times \ldots \times J_{D-1}\rightarrow \mathbb {R} _{+}}</annotation>

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</math><img src="./12c811b8308e611636dc5ca8eb06e94bc50e99ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.119ex; height:2.509ex;" alt="{\displaystyle A\times J_{1}\times \ldots \times J_{D-1}\rightarrow \mathbb {R} _{+}}" loading="lazy">, find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D-1}">

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<annotation encoding="application/x-tex">{\displaystyle D-1}</annotation>

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</math><img src="./a2882b444f937e7119403c8625aea8647b667747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.927ex; height:2.343ex;" alt="{\displaystyle D-1}" loading="lazy"> <a href="Permutation" title="Permutation">permutations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{d}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{d}}</annotation>

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</math><img src="./27be50fe8b354ac4913e85699af1b69a8af07c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.417ex; height:2.009ex;" alt="{\displaystyle \pi _{d}}" loading="lazy"> : A → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{d}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{d}}</annotation>

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</math><img src="./d409f8f94210ec359721cf70ebc383905a7bad24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.509ex;" alt="{\displaystyle J_{d}}" loading="lazy"> such that the total <a href="Loss_function" title="Loss function">cost function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{a\in A}C(a,\pi _{1}(a),\ldots ,\pi _{D-1}(a))}</annotation>

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</math><img src="./65ec3f009d9b21f050e62c7312d9e43b692c6ba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.236ex; height:5.676ex;" alt="{\displaystyle \sum _{a\in A}C(a,\pi _{1}(a),\ldots ,\pi _{D-1}(a))}" loading="lazy"></dd></dl></dd></dl>

is minimized.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Problem_parameters">Problem parameters</h3></div>

The multidimensional assignment problem (MAP) has two key parameters that determine the size of a problem instance:

<ol><li>The <a href="Dimension" title="Dimension">dimensionality</a> parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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<li>The <a href="Cardinality" title="Cardinality">cardinality</a> parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=|A|}">

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<annotation encoding="application/x-tex">{\displaystyle N=|A|}</annotation>

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</math><img src="./0aeacf7d80ff6fea4e01e6e9779a288ec891b4b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.199ex; height:2.843ex;" alt="{\displaystyle N=|A|}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|}">

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<annotation encoding="application/x-tex">{\displaystyle |A|}</annotation>

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</math><img src="./648fce92f29d925f04d39244ccfe435320dfc6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle |A|}" loading="lazy"> denotes the number of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Size_of_cost_array">Size of cost array</h3></div>

Any problem instance of the MAP with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D,N}">

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<annotation encoding="application/x-tex">{\displaystyle D,N}</annotation>

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</math><img src="./c5a99f5710a6fab31bfcb532d04103f0ff640cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.022ex; height:2.509ex;" alt="{\displaystyle D,N}" loading="lazy"> has its specific cost array <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, which consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{D}}">

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<annotation encoding="application/x-tex">{\displaystyle N^{D}}</annotation>

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</math><img src="./b98a59faac52cd723a7eaabfa183d6dd0e9b5f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.716ex; height:2.676ex;" alt="{\displaystyle N^{D}}" loading="lazy"> instance-specific costs/weights parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(a,a_{1},\ldots ,a_{D-1})}">

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<annotation encoding="application/x-tex">{\displaystyle C(a,a_{1},\ldots ,a_{D-1})}</annotation>

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</math><img src="./6c09dab587e0c8f676f5f5033a009184ff6c3a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.225ex; height:2.843ex;" alt="{\displaystyle C(a,a_{1},\ldots ,a_{D-1})}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{D}}">

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<annotation encoding="application/x-tex">{\displaystyle N^{D}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Number_of_feasible_solutions">Number of feasible solutions</h3></div>

The <a href="Feasible_region" title="Feasible region">feasible region or solution space</a> of the MAP is very large. The number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> of feasible solutions (the size of the MAP instance) depends on the MAP parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D,N}">

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<annotation encoding="application/x-tex">{\displaystyle D,N}</annotation>

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</math><img src="./c5a99f5710a6fab31bfcb532d04103f0ff640cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.022ex; height:2.509ex;" alt="{\displaystyle D,N}" loading="lazy">. Specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=(N!)^{D-1}}">

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<annotation encoding="application/x-tex">{\displaystyle K=(N!)^{D-1}}</annotation>

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</math><img src="./f41e3fa2f36102ef82090159aaae921eba1f1bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.378ex; height:3.176ex;" alt="{\displaystyle K=(N!)^{D-1}}" loading="lazy">.<a href="#cite_note-Pasi21-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Computational_complexity">Computational complexity</h2></div>

The problem is generally <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>. In other words, there is no known <a href="Algorithm" title="Algorithm">algorithm</a> for solving this problem in polynomial time, and so a long computational time may be needed for solving problem instances of even moderate size (based on dimensionality and cardinality parameters).<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The problem found application in many domains:

<ul><li><a href="Scheduling_(production_processes)" title="Scheduling (production processes)">Scheduling (production processes)</a><a href="#cite_note-Pier68-1">[1]</a></li>

<li><a href="Data_fusion" title="Data fusion">Multi-sensor data fusion</a><a href="#cite_note-6">[6]</a></li>

<li><a href="Record_linkage" title="Record linkage">Record linkage or multipartite entity resolution</a><a href="#cite_note-Pasi21-2">[2]</a></li>

<li><a href="Particle_physics" title="Particle physics">Elementary particle physics</a><a href="#cite_note-7">[7]</a></li>

<li><a href="Medical_alarm" title="Medical alarm">Fall detection in elderly with small wearable devices</a><a href="#cite_note-8">[8]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFPierskalla1968" class="citation journal cs1">Pierskalla, William P. (1968). "Letter to the Editor—The Multidimensional Assignment Problem". Operations Research. 16 (2). INFORMS: 422–431. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.16.2.422">10.1287/opre.16.2.422</a>.</cite>

</li>

<li id="cite_note-Pasi21-2">^ <a href="#cite_ref-Pasi21_2-0">a</a> <a href="#cite_ref-Pasi21_2-1">b</a> <a href="#cite_ref-Pasi21_2-2">c</a> <cite id="CITEREFKammerdinerSemenovPasiliao2021" class="citation arxiv cs1">Kammerdiner, Alla; Semenov, Alexander; Pasiliao, Eduardo (2021). "Multidimensional Assignment Problem for multipartite entity resolution". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2112.03346">2112.03346</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.DM">cs.DM</a>].</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFNatuDateNagi2020" class="citation journal cs1">Natu, Shardul; Date, Ketan; Nagi, Rakesh (2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.parco.2020.102666">"GPU-accelerated Lagrangian heuristic for multidimensional assignment problems with decomposable costs"</a>. Parallel Computing. 97: 102666. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.parco.2020.102666">10.1016/j.parco.2020.102666</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0167-8191">0167-8191</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221667518">221667518</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFKarapetyanGutin2011" class="citation journal cs1">Karapetyan, Daniel; Gutin, Gregory (2011-06-01). <a rel="nofollow" class="external text" href="http://repository.essex.ac.uk/22073/1/0806.3258v6.pdf">"Local search heuristics for the multidimensional assignment problem"</a> (PDF). Journal of Heuristics. 17 (3): 201–249. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10732-010-9133-3">10.1007/s10732-010-9133-3</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9397">1572-9397</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3446729">3446729</a>.</cite>

</li>

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