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Committee Elections with Candidate Attribute Constraints

Aizhong Zhou, Fengbo Wang, Jiong Guo

TL;DR

CECAC studies selecting a $k$-member committee from candidates with attributes under propositional constraints over attributes, aiming to maximize total profit while achieving at least $p$; it establishes a classical dichotomy: polynomial-time solvable when each candidate has at most one attribute and each attribute occurs at most once, and NP-hard otherwise, with broader parameterized results across five parameters. The paper shows NP-hardness via reductions from Clique and Independent Set and provides a bottom-up DP approach for the tractable case, using a combined constraint formula $D(R)$ and a binary tree $T(D(R))$; it also explores a Simple Constraints variant yielding further tractability under tight per-constraint limits. Five-parameter analyses reveal W[1]-hardness for $k$ and $p$ but fixed-parameter tractability when parameterized by the number of attributes $\ell$ (or by the number of candidates $m$ in certain regimes). Collectively, these results advance understanding of constrained multiwinner election problems and guide algorithm design for practical settings with attribute-based constraints.

Abstract

In many real-world applications of committee elections, the candidates are associated with certain attributes and the chosen committee is required to satisfy some constraints posed on the candidate attributes. For instance, when dress collocation, it is generally acknowledged that when wearing a tie, you'd better wear a shirt, and wearing a suit, you'd better wear leather shoes. Here, dresses are categorized by upper garment, lower garment, shoes et.al, and upper garment is with the attribute tie and shirt, lower garment is with the attribute suit, and shoes is with the attribute leather. And two constraints "tie infers shirt" and "suit infers leather shoes" are proposed. We study this variant of committee elections from the computational complexity viewpoint. Given a set of candidates, each with some attributes and a profit, and a set of constraints, given as propositional logical expressions of the attributes, the task is to compute a set of k candidates, whose attributes satisfy all constraints and whose total profit achieves a given bound. We achieve a dichotomy concerning classical complexity with no length limit on constraints: the problem is polynomial-time solvable, if the following two conditions are fulfilled: 1) each candidate has only one attribute and 2) each attribute occurs at most once in the constraints. It becomes NP-hard if one of the two conditions is violated. Moreover, we examine its parameterized complexity. The parameterization with the number of constraints, the size of the committee, or the total profit bound as parameter leads to para-NP-hardness or W[1]-hardness, while with the number of attributes or the number of candidates as parameter, the problem turns out to be fixed-parameter tractable.

Committee Elections with Candidate Attribute Constraints

TL;DR

CECAC studies selecting a -member committee from candidates with attributes under propositional constraints over attributes, aiming to maximize total profit while achieving at least ; it establishes a classical dichotomy: polynomial-time solvable when each candidate has at most one attribute and each attribute occurs at most once, and NP-hard otherwise, with broader parameterized results across five parameters. The paper shows NP-hardness via reductions from Clique and Independent Set and provides a bottom-up DP approach for the tractable case, using a combined constraint formula and a binary tree ; it also explores a Simple Constraints variant yielding further tractability under tight per-constraint limits. Five-parameter analyses reveal W[1]-hardness for and but fixed-parameter tractability when parameterized by the number of attributes (or by the number of candidates in certain regimes). Collectively, these results advance understanding of constrained multiwinner election problems and guide algorithm design for practical settings with attribute-based constraints.

Abstract

In many real-world applications of committee elections, the candidates are associated with certain attributes and the chosen committee is required to satisfy some constraints posed on the candidate attributes. For instance, when dress collocation, it is generally acknowledged that when wearing a tie, you'd better wear a shirt, and wearing a suit, you'd better wear leather shoes. Here, dresses are categorized by upper garment, lower garment, shoes et.al, and upper garment is with the attribute tie and shirt, lower garment is with the attribute suit, and shoes is with the attribute leather. And two constraints "tie infers shirt" and "suit infers leather shoes" are proposed. We study this variant of committee elections from the computational complexity viewpoint. Given a set of candidates, each with some attributes and a profit, and a set of constraints, given as propositional logical expressions of the attributes, the task is to compute a set of k candidates, whose attributes satisfy all constraints and whose total profit achieves a given bound. We achieve a dichotomy concerning classical complexity with no length limit on constraints: the problem is polynomial-time solvable, if the following two conditions are fulfilled: 1) each candidate has only one attribute and 2) each attribute occurs at most once in the constraints. It becomes NP-hard if one of the two conditions is violated. Moreover, we examine its parameterized complexity. The parameterization with the number of constraints, the size of the committee, or the total profit bound as parameter leads to para-NP-hardness or W[1]-hardness, while with the number of attributes or the number of candidates as parameter, the problem turns out to be fixed-parameter tractable.
Paper Structure (10 sections, 10 theorems, 4 equations, 1 figure, 1 table)

This paper contains 10 sections, 10 theorems, 4 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Definition
  4. Parameterized Complexity
  5. Classical Complexity
  6. Polynomial-time solvable case
  7. NP-hard cases
  8. Parameterized Complexity
  9. Election with simple attribute constraints
  10. Conclusion

Key Result

theorem thmcountertheorem

When the following two conditions are fulfilled, CECAC is solvable in polynomial time: 1) each candidate has at most one attribute, $\forall\ c \in C: |\alpha(c)|\leq 1$; 2) each attribute occurs at most once, $\forall\ a\in A: N(a)\leq 1$.

Figures (1)

  • Figure 1: The root of the binary tree is set to $D(R)=(\overline{a_1}\vee a_2) \wedge (\overline{a_3} \vee a_4 \vee a_5) \wedge (a_6 \vee a_7)$, and each leaves is set to an attribute occuring in $D(R)$. In this binary tree, the subformulas $q$ and $q'$ of two children nodes can constitute the subformula of their father node by $q \vee q'$ or $q \wedge q'$.

Theorems & Definitions (16)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • ...and 6 more