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On Efficient Computation of DiRe Committees

Kunal Relia

TL;DR

This work studies the DiReCF problem, which is equivalent to Minimum Vertex Cover ($MVC$) on unweighted graphs. It proposes a four-phase, polynomial-time approach that combines Maximum Matching, BFS-level information, a tailored Maximal Matching with a Represent data structure, and a novel Local Minimization step to extract a small vertex cover. The authors formalize the Vertex Cover framework, provide a detailed algorithm (Vertex_Cover) with complexity analysis, and conjecture unconditional polynomial-time solvability for $MVC$ on unweighted simple connected graphs. If true, this implies efficient computation of DiRe committees that satisfy both diversity and representation constraints, with potential practical impact for committee design. The paper also grounds the reduction between DiReCF and $VC$, and presents a comprehensive complexity analysis highlighting the dominant terms in the procedure.

Abstract

Consider a committee election consisting of (i) a set of candidates who are divided into arbitrary groups each of size ${at~most}$ two and a diversity constraint that stipulates the selection of ${at~least}$ one candidate from each group and (ii) a set of voters who are divided into arbitrary populations each approving ${at~most}$ two candidates and a representation constraint that stipulates the selection of ${at~least}$ one candidate from each population who has a non-null set of approved candidates. The DiRe (Diverse + Representative) committee feasibility problem (a.k.a. the minimum vertex cover problem on unweighted undirected graphs) concerns the determination of the smallest size committee that satisfies the given constraints. Here, for this problem, we propose an algorithm that is an amalgamation of maximum matching, breadth-first search, maximal matching, and local minimization. We prove the algorithm terminates in polynomial-time. We conjecture the algorithm is an unconditional deterministic polynomial-time algorithm.

On Efficient Computation of DiRe Committees

TL;DR

This work studies the DiReCF problem, which is equivalent to Minimum Vertex Cover () on unweighted graphs. It proposes a four-phase, polynomial-time approach that combines Maximum Matching, BFS-level information, a tailored Maximal Matching with a Represent data structure, and a novel Local Minimization step to extract a small vertex cover. The authors formalize the Vertex Cover framework, provide a detailed algorithm (Vertex_Cover) with complexity analysis, and conjecture unconditional polynomial-time solvability for on unweighted simple connected graphs. If true, this implies efficient computation of DiRe committees that satisfy both diversity and representation constraints, with potential practical impact for committee design. The paper also grounds the reduction between DiReCF and , and presents a comprehensive complexity analysis highlighting the dominant terms in the procedure.

Abstract

Consider a committee election consisting of (i) a set of candidates who are divided into arbitrary groups each of size two and a diversity constraint that stipulates the selection of one candidate from each group and (ii) a set of voters who are divided into arbitrary populations each approving two candidates and a representation constraint that stipulates the selection of one candidate from each population who has a non-null set of approved candidates. The DiRe (Diverse + Representative) committee feasibility problem (a.k.a. the minimum vertex cover problem on unweighted undirected graphs) concerns the determination of the smallest size committee that satisfies the given constraints. Here, for this problem, we propose an algorithm that is an amalgamation of maximum matching, breadth-first search, maximal matching, and local minimization. We prove the algorithm terminates in polynomial-time. We conjecture the algorithm is an unconditional deterministic polynomial-time algorithm.
Paper Structure (23 sections, 6 theorems, 26 figures, 6 tables, 4 algorithms)

This paper contains 23 sections, 6 theorems, 26 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Algorithm Overview
  4. Maximum Matching
  5. Breadth-first Search
  6. Maximal Matching
  7. Local Minimization
  8. Summary
  9. Algorithm
  10. Time Complexity Analysis
  11. Time complexity of Algorithm \ref{['alg:FnR']}:
  12. Conclusion
  13. DiRe Committee and Vertex Cover
  14. VC $\leq_P$ DiReCF:
  15. DiReCF $\leq_P$ VC:
  16. ...and 8 more sections

Key Result

Lemma 1

In a given graph $G$, if $M$ is a maximum matching and $S$ is a minimum vertex cover, then $|S|\geq|M|$.

Figures (26)

  • Figure 1: VC1 denotes an instance of the vertex cover problem on unweighted, undirected graph. VC2 denotes an instance of the vertex cover problem on unweighted simple connected graph. The multi-edges connecting vertices 0 and 1 in VC1 are removed in VC2. The loop connecting vertex 1 to itself is replaced by an edge in VC2 that connects vertex 1 to a dummy vertex $d'$. Another dummy vertex $u'$ (yellow vertex) is added to VC2 and is connected to all existing vertices to make the graph connected.
  • Figure 2: Example Graph $G$.
  • Figure 3: Bold edges {(0, 1), (2, 7), (3, 5), (4, 6)} form a maximum matching of graph $G$.
  • Figure 4: The "BFS" table lists the vertices at each level of the BFS (seeded on vertex '0').
  • Figure 5: "Maximal Matching" table lists vertices 0 and 1 (orange vertices in graph $G$), which are the endpoints of the first edge selected during maximal matching. Each endpoint is marked as visited (orange font; BFS table).
  • ...and 21 more figures

Theorems & Definitions (26)

  • Definition 1: Minimum Vertex Cover Problem (MVC)
  • Definition 2: Vertex Cover Problem (VC)
  • Definition 3: Matching
  • Definition 4: Maximum Matching
  • Lemma 1
  • Definition 5: Breadth-First Search
  • Definition 6: Maximal Matching
  • Definition 7: Represents
  • Definition 8: Represents Table
  • Definition 9: Represents List
  • ...and 16 more