# Committee selection with multimodal preferences

TL;DR: This work designs efficient algorithms for certain cases of committee selection with multimodal preferences and discusses applications of the model and the computational complexity of several generalizations of known committee scoring rules to this setting.

Abstract: We study committee selection with multimodal preferences: Assuming a set of candidatesA, a set of voters V , and ` layers, where each voter v ∈ V has ordinal preferences over the alternatives for each layer separately, the task is to select a committee S ⊆ A of size k. We discuss applications of our model and study the computational complexity of several generalizations of known committee scoring rules (specifically, k-Borda and Chamberlin–Courant) to our setting, as well as discuss domain restrictions for our model. While most problems we encounter are computationally intractable in general, we nevertheless design efficient algorithms for certain cases.

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2,033 citations

### "Committee selection with multimodal..." refers background in this paper

...A particularly popular domain restriction is the single-peaked domain, originally proposed by Black [6]....

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617 citations

### "Committee selection with multimodal..." refers methods in this paper

...We provide a polynomial time reduction from the W[1]hard problem Independent Set (IS) [12]....

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...Towards this, we give a reduction from the W[1]-hard problem Independent Set (IS) [12], in which given a graph G and an integer t; we shall decide the existence of a t-sized set X ⊆ V (G) containing only nonadjacent vertices....

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197 citations

### "Committee selection with multimodal..." refers background or methods in this paper

...We adapt the corresponding algorithm for CC [5]: We guess a clustering of the voters (i....

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...Due to Proposition 2, and NP-hardness and W[2]-hardness of EgalitarianCC with respect to the committee size k [5], we obtain:...

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...[5] for Max-CC and Min-CC under Global-SP; unfortunately,...

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...In particular, a polynomial time algorithm, computing winning committees under CC for single-peaked profiles is known [5]....

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...k [5], we obtain the following result....

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195 citations

### "Committee selection with multimodal..." refers background in this paper

...Due to Proposition 4 and the fact that Egalitarian-CC is NP-hard and W[2]-hard w.r.t. the committee size k [5], we have following result....

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...In particular, while finding winning committees under kBorda can be done in polynomial time (one has to select k candidates with the highest individual Borda scores), CC is NP-hard [26] but FPTfor certain parameters, admit approximation algorithms, and certain heuristics are known to be effective for it [5, 20, 28, 15]....

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...Due to Proposition 3 and the fact that CC is NP-hard and W[2]-hard w.r.t. k [5], we obtain the following result....

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...Corollary 8 Min-CC, Sum-CC, and Vector-CC are NP-hard and W[2]-hard w.r.t. k even for Local-SP, and n = 1 We next study the computational complexity of Max-CC for Global-SP profiles, and obtain intractability....

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...Due to Proposition 2, and NP-hardness and W[2]-hardness of EgalitarianCC with respect to the committee size k [5], we obtain: Corollary 4 Min-CC is NP-hard and W[2]-hard w.r.t. k even for n = 1....

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171 citations

### "Committee selection with multimodal..." refers methods in this paper

...Related Work In this paper we generalize several (ordinal) OWArules3 [27] – which is a subclass of the more general class of CSRs [19, 18] – to the setting of multimodal committee elections....

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...Related Work In this paper we generalize several (ordinal) OWArules(3) [27] – which is a subclass of the more general class of CSRs [19, 18] – to the setting of multimodal committee elections....

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