Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Computational Complexity of Testing Proportional Justified
  Representation   We consider a committee voting setting in which each voter approves of a
subset of candidates and based on the approvals, a target number of candidates
are selected. Aziz et al. (2015) proposed two representation axioms called
justified representation and extended justified representation. Whereas the
former can be tested as well as achieved in polynomial time, the latter
property is coNP-complete to test and no polynomial-time algorithm is known to
achieve it. Interestingly, S{\'a}nchez-Fern{\'a}ndez et~al. (2016) proposed an
intermediate property called proportional justified representation that admits
a polynomial-time algorithm to achieve. The complexity of testing proportional
justified representation has remained an open problem. In this paper, we settle
the complexity by proving that testing proportional justified representation is
coNP-complete. We complement the complexity result by showing that the problem
admits efficient algorithms if any of the following parameters are bounded: (1)
number of voters (2) number of candidates (3) maximum number of candidates
approved by a voter (4) maximum number of voters approving a given candidate.

30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an