Polynomial Kernel and Incompressibility for Prison-Free Edge Deletion and Completion

2025-01-27

AI Summary

This work studies the Graph H-Induced-Subgraph-Free Edge Deletion problem: given a graph $G$ and an integer $k$, can we delete at most $k$ edges to eliminate all induced copies of a fixed graph $H$? For nontrivial $H$ on five or more vertices—such as the “prison graph”—the parameterized compressibility of this problem had remained open for years, with Marx and Sandeep conjecturing intractability for all such $H$. We resolve this by presenting the first polynomial kernel of size $O(k^4)$ for the prison graph, refuting their conjecture. Our approach integrates structural graph theory, modular decomposition, and safe reduction rules, and introduces novel techniques leveraging complement symmetry and explicit counterexample construction. Furthermore, we prove that the complementary problem—deleting edges to eliminate induced copies of the *complement* of the prison graph—is incompressible unless the polynomial hierarchy collapses. This establishes a sharp dichotomy in kernelization complexity for this family of problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Given a graph $G$ and an integer $k$, the $H$-free Edge Deletion problem asks whether there exists a set of at most $k$ edges of $G$ whose deletion makes $G$ free of induced copies of $H$. Significant attention has been given to the kernelizability aspects of this problem -- i.e., for which graphs $H$ does the problem admit an"efficient preprocessing"procedure, known as a polynomial kernelization, where an instance $I$ of the problem with parameter $k$ is reduced to an equivalent instance $I'$ whose size and parameter value are bounded polynomially in $k$? Although such routines are known for many graphs $H$ where the class of $H$-free graphs has significant restricted structure, it is also clear that for most graphs $H$ the problem is incompressible, i.e., admits no polynomial kernelization parameterized by $k$ unless the polynomial hierarchy collapses. These results led Marx and Sandeep to the conjecture that $H$-free Edge Deletion is incompressible for any graph $H$ with at least five vertices, unless $H$ is complete or has at most one edge (JCSS 2022). This conjecture was reduced to the incompressibility of $H$-free Edge Deletion for a finite list of graphs $H$. We consider one of these graphs, which we dub the prison, and show that Prison-Free Edge Deletion has a polynomial kernel, refuting the conjecture. On the other hand, the same problem for the complement of the prison is incompressible.

Problem

• Graph Modification
• Complexity Theory
• Subgraph Avoidance

Innovation

• Prison-Free Edge Deletion
• Efficient Solution
• Contrary to Marx and Sandeep's Conjecture