A Reduction-Driven Local Search for the Generalized Independent Set Problem

TL;DR

The paper tackles Generalized Independent Set (GIS), an extension of MIS with vertex profits and edge penalties, formulated on $G=(V,E_p\cup E_r)$ and objective $\alpha(G)=\max_S nb(S)$. It introduces 14 data-reduction rules with optimality guarantees and a reduction-driven local search (RLS) that tightly integrates reductions into preprocessing, initialization, and neighborhood search. Empirical evaluation on 278 graphs shows that RLS delivers superior solution quality and scalability, solving instances with up to $2.6\times 10^8$ edges where prior methods fail, and the data reductions are a key factor in achieving this performance. The approach demonstrates the practicality of kernelization-style reductions for GIS and highlights directions for further enhancement and application to dynamic GIS settings.

Abstract

The Generalized Independent Set (GIS) problem extends the classical maximum independent set problem by incorporating profits for vertices and penalties for edges. This generalized problem has been identified in diverse applications in fields such as forest harvest planning, competitive facility location, social network analysis, and even machine learning. However, solving the GIS problem in large-scale, real-world networks remains computationally challenging. In this paper, we explore data reduction techniques to address this challenge. We first propose 14 reduction rules that can reduce the input graph with rigorous optimality guarantees. We then present a reduction-driven local search (RLS) algorithm that integrates these reduction rules into the pre-processing, the initial solution generation, and the local search components in a computationally efficient way. The RLS is empirically evaluated on 278 graphs arising from different application scenarios. The results indicates that the RLS is highly competitive -- For most graphs, it achieves significantly superior solutions compared to other known solvers, and it effectively provides solutions for graphs exceeding 260 million edges, a task at which every other known method fails. Analysis also reveals that the data reduction plays a key role in achieving such a competitive performance.

Figures (6)

• Figure 1: A toy example of the GIS problem. The set of red vertices is a generalized independent set of net benefit of $2-1+6+5=12$.
• Figure 2: Examples of R8. Suppose that vertex $u$ is adjacent to $x$ and $y$ and, $(x,y)\in E_p$. $G_R$ represents the residual graph $G\setminus \{u,x,y\}$. $\alpha$ and $\alpha'$ indicate the optimal objective value in the original graph and the graph after reduction, respectively.
• Figure 3: Examples of R9. Suppose that vertex $u$ is adjacent to vertices $x$ and $y$, and $(x,y)\notin E_p$.
• Figure 4: The dependency graph of the reduction rules. A directed edge on the left side indicates how the reduction rule changes the graph $G$ to $G'$. A directed edge on the right side indicates what reduction rules can be satisfied due to this change.
• Figure 5: The variation of kernel sizes when ablating different reductions. A cell value indicates the difference in kernel size between without using reductions and with using reductions. A higher cell value indicates a larger effect.

Theorems & Definitions (13)

• proof : Proof of R\ref{['lem: reduction 1']}
• proof : Proof of R\ref{['lem: reduction 2']}
• proof : Proof of R\ref{['lem: reduction 4']}
• proof : Proof of R\ref{['lem: reduction 5']}
• proof : Proof of R\ref{['lem: reduction 6']}
• proof : Proof of R\ref{['lem: reduction 7']}
• proof : Proof of R\ref{['lem: reduction 8']}
• proof : Proof of R\ref{['lem: reduction 9']}
• proof : Proof of R\ref{['lem: reduction 10']}
• proof : Proof of R\ref{['lem: reduction 11']}