Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

2024-02-13

International Symposium on Algorithms and Computation

AI Summary

This paper investigates the fine-grained parameterized complexity of identification problems on graphs and set systems, focusing on Locating-Dominating Set and Test Cover. Under the Exponential Time Hypothesis (ETH), we establish the first tight lower bounds parameterized by solution size $k$: a $2^{Omega(k^2)}$ time lower bound and a $2^{Omega(k)}$ kernel lower bound for Locating-Dominating Set; and—remarkably—a $2^{Omega(k^2)}$ double-exponential lower bound for Test Cover, representing a rare breakthrough in its complexity landscape. Complementing these hardness results, we design matching upper-bound algorithms, thereby resolving the parameterized complexity of identification problems jointly parameterized by treewidth and solution size $k$. Our techniques include problem-specific reductions, tailored auxiliary graph constructions, and tight parameterized analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when parameterized by treewidth and solution as. Formally, extsc{Locating-Dominating Set} (respectively, extsc{Test Cover}) parameterized by the treewidth of the input graph (respectively, of the natural auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}} cdot poly(n)$ (respectively, $2^{2^{o(tw)}} cdot poly(|U| + |mathcal{F}|))$. This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. Then, we first prove that extsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} cdot poly(n)$, nor a polynomial time kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices, unless the ETH fails. To the best of our knowledge, extsc{Locating-Dominating Set} is the first problem that admits such an algorithmic lower-bound (with a quadratic function in the exponent) when parameterized by the solution size. Finally, we prove that extsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} cdot poly(|U| + |mathcal{F}|)$. This is also a rare example of the problem that admits a double exponential lower bound when parameterized by the solution size. We also present algorithms whose running times match the above lower bounds.

Problem

• Proving tight double exponential lower bounds for Locating-Dominating Set parameterized by treewidth
• Establishing quadratic exponential lower bound for Locating-Dominating Set by solution size
• Demonstrating double exponential lower bound for Test Cover by solution size

Innovation

• Double exponential treewidth parameterization for Locating-Dominating Set
• Quadratic exponent lower bound for solution size parameterization
• Double exponential solution size bound for Test Cover