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

Abstract

Foucaud et al. [ICALP 2024] demonstrated that some problems in NP can admit (tight) double-exponential lower bounds when parameterized by treewidth or vertex cover number. They showed these first-of-their-kind results by proving conditional lower bounds for certain graph problems, in particular, the metric-based identification problems (Strong) Metric Dimension. We continue this line of research and highlight the usefulness of this type of problems, to prove relatively rare types of (tight) lower bounds. We investigate fine-grained algorithmic aspects of classical (non-metric based) identification problems in graphs, namely Locating-Dominating Set, and in set systems, namely Test Cover. In the first problem, an input is a graph G on n vertices and an integer k, and the objective is to decide whether there is a subset S of k vertices such that any two distinct vertices not in S are dominated by distinct subsets of S. In the second problem, an input is a set of items U, a collection of subsets ℱ of U called tests, and an integer k, and the objective is to select a set S of at most k tests such that any two distinct items are contained in a distinct subset of tests of S. 
For our first result, we adapt the techniques introduced by Foucaud et al. [ICALP 2024] to prove similar (tight) lower bounds for these two problems.  
- Locating-Dominating Set (respectively, Test Cover) parameterized by the treewidth of the input graph (respectively, the natural auxiliary graph) does not admit an algorithm running in time 2^{2^o(tw)} ⋅ poly(n) (respectively, 2^{2^o(tw)} ⋅ poly(|U| + |ℱ|))), unless the ETH fails.  This augments the short list of NP-Complete problems that admit tight double-exponential lower bounds when parameterized by treewidth, and shows that "local" (non-metric-based) problems can also admit such bounds. We show that these lower bounds are tight by designing treewidth-based dynamic programming schemes with matching running times.
Next, we prove that these two problems also admit "exotic" (and tight) lower bounds, when parameterized by the solution size k. We prove that unless the ETH fails,  
- Locating-Dominating Set does not admit an algorithm running in time 2^o(k²) ⋅ poly(n), nor a polynomial-time kernelization algorithm that reduces the solution size and outputs a kernel with 2^o(k) vertices, and 
- Test Cover does not admit an algorithm running in time 2^{2^o(k)} ⋅ poly(|U| + |ℱ|) nor a kernel with 2^{2^o(k)} vertices.  Again, we show that these lower bounds are tight by designing (kernelization) algorithms with matching running times. To the best of our knowledge, Locating-Dominating Set is the first known problem which is FPT when parameterized by solution size k, where the optimal running time has a quadratic function in the exponent. These results also extend the (very) small list of problems that admit an ETH-based lower bound on the number of vertices in a kernel, and (for Test Cover) a double-exponential lower bound when parameterized by the solution size. Whereas it is the first example, to the best of our knowledge, that admit a double exponential lower bound for the number of vertices.