We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.
The decision tree model is a simple model of computation for which we can prove good upper and lower bounds. Informally, decision tree complexity, also known as query complexity, counts the number of input bits that must be examined by an algorithm to evaluate a function. In this paper, we focus on the query complexity of deciding whether a graph has a given property. The query complexity of graph properties has been studied for almost 40 years, yet old and easy-to-state conjectures regarding the deterministic and randomized query complexities of graph properties [12,18,23,26] remain unresolved.
The study of query complexity has also been quite fruitful for quantum algorithms. For example, Grover’s search algorithm [16] operates in the query model, and Shor’s factoring algorithm [27] is based on the solution of a query problem. However, the quantum query complexity of graph properties can be harder to pin down than its classical counterparts. For monotone graph properties, a wide class of graph properties including almost all the properties considered in this paper, the widely-believed Aanderaa-Karp-Rosenberg conjecture states that the deterministic and randomized query complexities are Θ(n 2 ), where n is the number of vertices. On the other hand, there exist monotone graph properties whose quantum query complexity is Θ(n), and others with quantum query complexity Θ(n 2 ). In fact, one can construct a monotone graph property with quantum query complexity Θ(n 1+α ) for any fixed 0 ≤ α ≤ 1 using known bounds for the threshold function [7].
The quantum query complexity of several specific graph properties has been established in prior work. Dürr, Heiligman, Høyer, and Mhalla [15] studied the query complexity of several graph problems, and showed in particular that connectivity has quantum query complexity Θ(n 3/2 ). Zhang [31] showed that the quantum query complexity of bipartiteness is Θ(n 3/2 ). Ambainis et al. [6] showed that planarity also has quantum query complexity Θ(n 3/2 ). Berzina et al. [8] showed several quantum lower bounds on graph properties, including Hamiltonicity. Sun, Yao, and Zhang studied some non-monotone graph properties [29].
Despite this work, the quantum query complexity of many interesting graph properties remains unresolved. A well-studied graph property whose query complexity is unknown is the property of containing a triangle (i.e., a cycle on 3 vertices) as a subgraph. This question was first studied by Buhrman et al. [11], who gave an O(n + √ nm) query algorithm for graphs with n vertices and m edges. With m = Θ(n 2 ), this approach uses O(n 3/2 ) queries, which matches the performance of the simple algorithm that searches for a triangle over the potential n 3 triplets of vertices. This was later improved by Magniez, Santha, and Szegedy [22] to Õ(n 1.3 ), and then by Magniez, Nayak, Roland, and Santha [21] to O(n 1.3 ), which is currently the best known algorithm. However, the best known lower bound for the triangle problem is only Ω(n) (by a simple reduction from the search problem). This is partly because one of the main lower bound techniques, the quantum adversary method of Ambainis [3], cannot prove a better lower bound due to the certificate complexity barrier [28,31].
More generally, we can consider the H-subgraph containment problem, in which the task is to determine whether the input graph contains a fixed graph H as a subgraph. Magniez et al. also gave a general algorithm for H-subgraph containment using Õ(n 2-2/d ) queries, where d > 3 is the number of vertices in H [22]. Again, the best lower bound known for H-subgraph containment is only Ω(n).
In this paper we study the quantum query complexity of minor-closed graph properties. A property is minor closed if all minors of a graph possessing the property also possess the property. (Graph minors are defined in Section 2.) Since minor-closed properties can be characterized by forbidden minors, this can be viewed as a variant of subgraph containment in which we look for a given graph as a minor instead of as a subgraph. The canonical example of a minor-closed property is the property of being planar. Other examples include the property of being a forest, being embeddable on a fixed two-dimensional manifold, having treewidth at most k, and not containing a path of a given length. While any minor-closed property can be described by a finite set of forbidden minors, some minor-closed properties can also be described by a finite set of forbidden subgraphs, graphs that do not appear as a subgraph of any graph possessing the property. We call a graph property (which need not be minor closed) a forbidden subgraph property (FSP) if it can be described by a finite set of forbidden subgraphs. Our main result is that the quantum query complexity of minorclosed properties depends crucially on whether the property is FSP. We show that any nontrivial minor-closed property that is not FSP has query complexity Θ(n 3/2 ), whereas a
This content is AI-processed based on open access ArXiv data.
