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Overcoming Non-Submodularity: Towards Constant Approximation for Network Immunization

Ajitesh Srivastava, Shang-Hua Teng

TL;DR

This paper proves that the immunization objective is an expectation expressed as a sum over deterministic instances that has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network.

Abstract

Given a network with an ongoing epidemic, the network immunization problem seeks to identify a fixed number of nodes to immunize in order to maximize the number of infections prevented. A fundamental computational challenge in network immunization is that the objective function is generally neither submodular nor supermodular. Consequently, no efficient algorithm is known to consistently achieve a constant-factor approximation. Traditionally, this problem is partially addressed using proxy objectives that offer better approximation properties, but these indirect optimizations often introduce losses in effectiveness due to gaps between the proxy and natural objectives. In this paper, we overcome these fundamental barriers by leveraging the underlying stochastic structure of the diffusion process. Similar to the traditional influence objective, the immunization objective is an expectation expressed as a sum over deterministic instances. However, unlike the former, some of these terms are not submodular. Our key step is to prove that this sum has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network. We demonstrate that this approximation holds across various immunization settings and spread models.

Overcoming Non-Submodularity: Towards Constant Approximation for Network Immunization

TL;DR

This paper proves that the immunization objective is an expectation expressed as a sum over deterministic instances that has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network.

Abstract

Given a network with an ongoing epidemic, the network immunization problem seeks to identify a fixed number of nodes to immunize in order to maximize the number of infections prevented. A fundamental computational challenge in network immunization is that the objective function is generally neither submodular nor supermodular. Consequently, no efficient algorithm is known to consistently achieve a constant-factor approximation. Traditionally, this problem is partially addressed using proxy objectives that offer better approximation properties, but these indirect optimizations often introduce losses in effectiveness due to gaps between the proxy and natural objectives. In this paper, we overcome these fundamental barriers by leveraging the underlying stochastic structure of the diffusion process. Similar to the traditional influence objective, the immunization objective is an expectation expressed as a sum over deterministic instances. However, unlike the former, some of these terms are not submodular. Our key step is to prove that this sum has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network. We demonstrate that this approximation holds across various immunization settings and spread models.

Paper Structure

This paper contains 32 sections, 6 theorems, 36 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Background
  4. Indpendent Cascade Model
  5. Immunization Problem
  6. Problem Statements
  7. Greedy Immunization Bounds
  8. Critical Connectivity: Probability Transfer
  9. Immunization on Bounded Degree Graphs
  10. Approximation w.r.t $R_0$
  11. Handling Skewed Degree/$p_{u,v}$ Distributions
  12. Optimizing the Approximation Factor
  13. Extensions: Group Immunization
  14. Generality
  15. All-or-Nothing and Vaccine Acceptance
  16. ...and 17 more sections

Key Result

lemma 1

Given a monotone set function with $f(\emptyset)=0$ such that $\exists \alpha > 0$ for which the following property holds for any two sets $S^*$ and $S$: Then, for the set $S_k$ returned by the greedy algorithm,

Figures (7)

  • Figure 1: Counter-examples showing that the immunization objective $\pi(.)$ is neither submodular nor supermodular
  • Figure 2: Approximation factor for various graphs at varying $k$ and $R_0$. ER: Erdös Renyi, WS: Watts–Strogatz, BA: Barabasi-Albert.
  • Figure 3: Approximation factor for various graphs at varying $k$ and $R_0$ for SIR model on Watts–Strogatz graphs.
  • Figure 4: Construction of layered graph for the leaky setting.
  • Figure 5: Degree distribution of a random graph of each type with average degree 20. ER: Erdös Renyi, WS: Watts–Strogatz, BA: Barabasi-Albert.
  • ...and 2 more figures

Theorems & Definitions (16)

  • definition 1: Submodular Functions
  • definition 2: Supermodular Functions
  • definition 3: Immunization Group
  • lemma 1: Submodularity Deviation
  • definition 4: Critical Subset
  • definition 5: Effective Degree
  • lemma 2: Size of critical subset
  • proof
  • lemma 3
  • theorem 1
  • ...and 6 more