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Linear Contracts for Supermodular Functions Based on Graphs

Kanstantsin Pashkovich, Jacob Skitsko

TL;DR

This work resolves the open question of obtaining an additive PTAS for graph-based supermodular contracts with heterogeneous agent costs. It develops a structured approach around a pseudo-core $S'$, builds a tailored LP over a four-way agent partition (A,B,C,D) that encodes edge-coverage and degree constraints, and uses a preprocessing step (fractional coring) plus randomized rounding to produce a near-optimal set $S''$ with $g(S'') \ge OPT - 5\sqrt{\varepsilon}$ w.h.p. The method leverages sampling to estimate high-degree vertices, concentration inequalities for rounding, and connects to dense-graph optimization via core-like structures. The results extend additive-PTAS techniques from identical-cost settings to general-costs, offering new tools for graph-structured contract design and potentially broader dense optimization problems.

Abstract

We study linear contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a linear contract with several agents, each of whom can decide to take a costly action or not. The principal observes only the outcome of the agents' collective actions, not the actions themselves, and obtains a reward from this outcome. Agents that take an action incur a cost, and so naturally agents require a fraction of the principal's reward as an incentive for taking their action. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. Our focus is on the case when the agents are vertices in a graph and the principal's reward corresponds to the number of edges between agents who take their costly action. This case represents the natural scenario when an action of each agent complements actions of other agents though collaborations. Recently, Deo-Campo Vuong et.al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless $\mathcal{P} = \mathcal{NP}$. On a positive note, the authors provided an additive PTAS for the case when all agents have the same cost. They asked whether an additive PTAS can be obtained for the general case, i.e when agents potentially have different costs. We answer this open question in positive.

Linear Contracts for Supermodular Functions Based on Graphs

TL;DR

This work resolves the open question of obtaining an additive PTAS for graph-based supermodular contracts with heterogeneous agent costs. It develops a structured approach around a pseudo-core , builds a tailored LP over a four-way agent partition (A,B,C,D) that encodes edge-coverage and degree constraints, and uses a preprocessing step (fractional coring) plus randomized rounding to produce a near-optimal set with w.h.p. The method leverages sampling to estimate high-degree vertices, concentration inequalities for rounding, and connects to dense-graph optimization via core-like structures. The results extend additive-PTAS techniques from identical-cost settings to general-costs, offering new tools for graph-structured contract design and potentially broader dense optimization problems.

Abstract

We study linear contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a linear contract with several agents, each of whom can decide to take a costly action or not. The principal observes only the outcome of the agents' collective actions, not the actions themselves, and obtains a reward from this outcome. Agents that take an action incur a cost, and so naturally agents require a fraction of the principal's reward as an incentive for taking their action. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. Our focus is on the case when the agents are vertices in a graph and the principal's reward corresponds to the number of edges between agents who take their costly action. This case represents the natural scenario when an action of each agent complements actions of other agents though collaborations. Recently, Deo-Campo Vuong et.al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless . On a positive note, the authors provided an additive PTAS for the case when all agents have the same cost. They asked whether an additive PTAS can be obtained for the general case, i.e when agents potentially have different costs. We answer this open question in positive.

Paper Structure

This paper contains 14 sections, 15 theorems, 68 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Contract Design Setting
  4. Overview and Intuition
  5. Formal Statements
  6. Appendix: Omitted Proofs
  7. Linear Relaxation and Rounding
  8. Proof of Lemma \ref{['lem:main_supermodular1']}
  9. Proof of Theorem \ref{['thm:main_supermodular2']}
  10. Proof of Theorem \ref{['thm:main_supermodular3']}
  11. Bound on $R(S")$.
  12. Bound on $L(S")$.
  13. Bound on $g(S")$.
  14. Structured Near Optimal Solutions

Key Result

theorem 1

Let us consider linear contracts in the multi-agent setting, where the reward function is a supermodular function based on graphs. There exists an additive PTAS in the case of general costs.

Theorems & Definitions (25)

  • theorem 1
  • lemma 1
  • definition 1
  • lemma 2
  • theorem 2
  • theorem 3
  • lemma 3: deo2024supermodulardaskalakis2011oblivious
  • lemma 4
  • proof
  • lemma 5
  • ...and 15 more