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Welfare Loss in Connected Resource Allocation

Xiaohui Bei, Alexander Lam, Xinhang Lu, Warut Suksompong

TL;DR

The paper defines egalitarian and utilitarian price of connectivity (PoC) to quantify welfare loss when demanding connected bundles in graph-structured indivisible allocations. It delivers tight or asymptotically tight bounds for two agents across dense and sparse graph classes (e.g., complete graphs with a matching removed, complete bipartite graphs, cycles, paths, stars, and trees) and extends to arbitrary numbers of agents with general bounds of O(m−n+1) for egalitarian PoC and O(n) for utilitarian PoC. Key results include exact values for several classes (e.g., Egal-PoC(K_m minus a matching) = (m−2)/(m−3) except L3/L5, and Util-PoC(K_{2, s}) = 4/3 when the smaller side has size 2) and a unified approach using structured instances and graph decompositions. The work provides theoretical insight into the trade-off between social welfare and connectivity constraints with implications for centralized decision-making in partitioning tasks and resource allocation under connectivity requirements.

Abstract

We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents -- including graphs with vertex connectivity $1$ or $2$ and complete bipartite graphs -- as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.

Welfare Loss in Connected Resource Allocation

TL;DR

The paper defines egalitarian and utilitarian price of connectivity (PoC) to quantify welfare loss when demanding connected bundles in graph-structured indivisible allocations. It delivers tight or asymptotically tight bounds for two agents across dense and sparse graph classes (e.g., complete graphs with a matching removed, complete bipartite graphs, cycles, paths, stars, and trees) and extends to arbitrary numbers of agents with general bounds of O(m−n+1) for egalitarian PoC and O(n) for utilitarian PoC. Key results include exact values for several classes (e.g., Egal-PoC(K_m minus a matching) = (m−2)/(m−3) except L3/L5, and Util-PoC(K_{2, s}) = 4/3 when the smaller side has size 2) and a unified approach using structured instances and graph decompositions. The work provides theoretical insight into the trade-off between social welfare and connectivity constraints with implications for centralized decision-making in partitioning tasks and resource allocation under connectivity requirements.

Abstract

We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents -- including graphs with vertex connectivity or and complete bipartite graphs -- as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.
Paper Structure (35 sections, 26 theorems, 37 equations, 4 figures, 2 tables, 4 algorithms)

This paper contains 35 sections, 26 theorems, 37 equations, 4 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Preliminaries
  5. Egalitarian Price of Connectivity
  6. Two Agents
  7. Lower Bound (for $m \geq 5$):
  8. Upper Bound (for $m \geq 5$):
  9. Lower Bound:
  10. Upper Bound:
  11. Three Agents
  12. Lower Bound:
  13. Upper Bound:
  14. Case 1: $\delta = \Delta_1(G) + \Delta_2(G) - 1$.
  15. Case 2: $\delta = \Delta_1(G) + \Delta_2(G) - 2$.
  16. ...and 20 more sections

Key Result

Proposition 3.1

For any graph $G$, it holds that $\textup{Egal-PoC}(G, n) \leq m - n + 1$.

Figures (4)

  • Figure 1: An instance with $n=2$ agents and $m=4$ items. The numbers in each vertex indicate the agents' utilities for the item. The items assigned to agent $2$ in the egalitarian-optimal (resp., utilitarian-optimal) connected allocation are those in the blue ellipse (resp., red dashed shape).
  • Figure 2: The graph $L_5$ and a utility profile showing that its egalitarian PoC is at least $2$. The number at each vertex indicates the corresponding agent's utility for the item. All utilities not indicated are zero.
  • Figure 3: An illustration for the proof of \ref{['thm:egal:2-agent-1connected']}. \ref{['fig:bd:graph']} displays a graph such that the maximum number of connected components after removing a vertex is $2$, i.e., $d = 2$; vertices $v_1, v_2, v_3$ are cut vertices. \ref{['fig:bd:decomposition']} demonstrates its block decomposition, which is not a path; black vertices correspond to blocks and white vertices correspond to cut vertices.
  • Figure 4: A cycle with $2n - 2 \leq m < n^2$ vertices. Agent $1$ values $n-1$ non-adjacent vertices (labeled $a_1$) at $\frac{1}{n-1}$ each, while each remaining agent $i\in\{2,\dots,n\}$ values her designated vertex $a_i$ at $1$.

Theorems & Definitions (54)

  • Definition 2.1: Egal-PoC
  • Definition 2.2: Util-PoC
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 44 more