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Impartial Selection Under Combinatorial Constraints

Javier Cembrano, Max Klimm, Arturo Merino

TL;DR

This work generalizes impartial selection to weighted nominations under combinatorial feasibility constraints encoded as independence systems, and analyzes how the structure of these constraints affects performance guarantees. It adapts and extends partition-based randomized mechanisms to obtain constant-factor approximations across general independence systems, knapsack, and matroid constraints, with both randomized and deterministic options depending on sparsity and item sizes. The results include a $\frac{1}{4}$-approximation for general independence systems, girth-based upper bounds, a $\frac{1}{3}$-approximation for simple graphic matroids with binary scores, and a $\frac{1}{2}$-approximation for (general) matroids under $1$-sparsity, enriching the theory of strategyproof and fair selection. The findings have potential applications in peer review and resource allocation where strategyproofness and combinatorial feasibility must be balanced, and they connect to established impartiality literature through principled, order-based randomized mechanisms.

Abstract

Impartial selection problems are concerned with the selection of one or more agents from a set based on mutual nominations from within the set. To avoid strategic nominations of the agents, the axiom of impartiality requires that the selection of each agent is independent of the nominations cast by that agent. This paper initiates the study of impartial selection problems where the nominations are weighted and the set of agents that can be selected is restricted by a combinatorial constraint. We call a selection mechanism $α$-optimal if, for every instance, the ratio between the total sum of weighted nominations of the selected set and that of the best feasible set of agents is at least $α$. We show that a natural extension of a mechanism studied for the selection of a single agent remains impartial and $\frac{1}{4}$-optimal for general independence systems, and we generalize upper bounds from the selection of multiple agents by parameterizing them by the girth of the independence system. We then focus on independence systems defined by knapsack and matroid constraints, giving impartial mechanisms that exploit a greedy order of the agents and achieve approximation ratios of $\frac{1}{3}$ and $\frac{1}{2}$, respectively, when agents cast a single nomination. For graphic matroids, we further devise an impartial and $\frac{1}{3}$-optimal mechanism for an arbitrary number of unweighted nominations.

Impartial Selection Under Combinatorial Constraints

TL;DR

This work generalizes impartial selection to weighted nominations under combinatorial feasibility constraints encoded as independence systems, and analyzes how the structure of these constraints affects performance guarantees. It adapts and extends partition-based randomized mechanisms to obtain constant-factor approximations across general independence systems, knapsack, and matroid constraints, with both randomized and deterministic options depending on sparsity and item sizes. The results include a -approximation for general independence systems, girth-based upper bounds, a -approximation for simple graphic matroids with binary scores, and a -approximation for (general) matroids under -sparsity, enriching the theory of strategyproof and fair selection. The findings have potential applications in peer review and resource allocation where strategyproofness and combinatorial feasibility must be balanced, and they connect to established impartiality literature through principled, order-based randomized mechanisms.

Abstract

Impartial selection problems are concerned with the selection of one or more agents from a set based on mutual nominations from within the set. To avoid strategic nominations of the agents, the axiom of impartiality requires that the selection of each agent is independent of the nominations cast by that agent. This paper initiates the study of impartial selection problems where the nominations are weighted and the set of agents that can be selected is restricted by a combinatorial constraint. We call a selection mechanism -optimal if, for every instance, the ratio between the total sum of weighted nominations of the selected set and that of the best feasible set of agents is at least . We show that a natural extension of a mechanism studied for the selection of a single agent remains impartial and -optimal for general independence systems, and we generalize upper bounds from the selection of multiple agents by parameterizing them by the girth of the independence system. We then focus on independence systems defined by knapsack and matroid constraints, giving impartial mechanisms that exploit a greedy order of the agents and achieve approximation ratios of and , respectively, when agents cast a single nomination. For graphic matroids, we further devise an impartial and -optimal mechanism for an arbitrary number of unweighted nominations.
Paper Structure (7 sections, 19 theorems, 97 equations, 4 figures, 1 table, 6 algorithms)

This paper contains 7 sections, 19 theorems, 97 equations, 4 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution and Techniques
  3. Related Work
  4. Preliminaries
  5. General Independence Systems
  6. Knapsack Constraints
  7. Matroid Constraints

Key Result

Lemma 1

Let $(E,\mathcal{I})$ be an independence system with rank function $r$. If $x\in [0,1]^{2^E}$ is a probability distribution such that $\sum_{S\subseteq E} x_S = \sum_{S\in \mathcal{I}} x_S = 1$, then $p\in [0,1]^E$ defined as $p_i\coloneqq\sum_{S\subseteq E: i\in S} x_S$ for each $i\in E$ satisfies

Figures (4)

  • Figure 1: Illustration of the upper bounds stated in \ref{['thm:gral-indep-ub-plu', 'thm:gral-indep-ub-app']} for $g=7$, where votes are represented by arcs. The depicted agents---other than $i$ in part \ref{['fig:gral-indep-ub-b']}---form a dependent set, but removing any of them renders the set independent.
  • Figure 2: Two examples of KnapsackPlurality. The first agents according to the greedy order are depicted as rectangles, with sizes written above and total scores inside. Non-zero scores between depicted agents are represented as arcs along with their scores. Colored agents are selected with probability $\frac{1}{3}$; colors illustrate the feasibility of the probability assignment as agents of the same color fit into one knapsack. Note that the score assigned by some agent (other than the agent of size $1$) to the agent of size $10$ decreases by $7$ from \ref{['fig:knapsack-greedy-a']} to \ref{['fig:knapsack-greedy-b']}, thus keeping the total score of the agent of size 10 at 25.
  • Figure 3: Two examples of DetKnapsackPlurality for $1$-sparse instances with~$s_{\max}=4$. The notation follows that of \ref{['fig:knapsack-greedy']}, votes between depicted agents are represented as arcs along with their scores, and $\varepsilon>0$ is a small value. Colored agents are selected by the mechanism. Instance (b), where agents have unit density and are thus sorted by index (not shown in the figure), illustrates the tightness of the analysis, as the guarantee of $\frac{3}{5}$ implied by \ref{['thm:knapsack-lb-det']} is reached by taking $\varepsilon$ arbitrarily small.
  • Figure 4: Example of a graph $(V,E)$ (in black) and (binary) score matrix (represented by light blue arcs) such that VertexPartition is only $\frac{1}{3}$-optimal on the instance~$(E,V,W)$ when the number of vertices $\{v_j\}_{j\in [n-1]}$ tends to infinity.

Theorems & Definitions (36)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3: alon2011sum
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • ...and 26 more