This paper studies the quantum query complexity of computing the edge connectivity of a simple graph with $n$ vertices and $m$ edges in both the adjacency matrix and adjacency array models. The edge connectivity of a simple graph is defined as the least number of edges whose removal disconnects the graph. For classical randomized algorithms, lower bounds of $\Omega(n^2)$ and $\Omega(m)$ are known for these models, respectively, showing that the trivial algorithm is optimal in terms of query complexity for both cases. However, this paper presents upper bounds of $\tilde O(n^{3/2})$ and $\tilde O(\sqrt{mn})$ on the quantum query complexity of computing edge connectivity in the adjacency matrix and adjacency array models, respectively. These upper bounds follow by combining a lemma on the structure of near-minimum cuts with a quantum sparsification algorithm. The authors also consider weighted graphs where instead of edge connectivity, the relevant notion is the weight of a minimum cut, i.e., the minimum total weight of a set of edges whose removal disconnects the graph. For weighted graphs, they show that in both adjacency matrix and adjacency array models no quantum speedup is possible as $\Omega(n^2)$ queries can be needed to compute the weight of a minimum cut by a quantum algorithm in worst-case scenarios. The paper concludes with five open problems related to their work. One such problem asks if it's possible to give an algorithm that shows that their lower bound on computing minimum cut weights in adjacency matrix model is tight throughout all ranges 1 ≤ α < n when we have an upper bound α on maximum weight. Another open problem seeks similar time complexity bounds as their query complexity results for computing edge connectivity. Finally, they note that there remains a gap between their upper bound on query complexity for adjacency array model and existing lower bounds from previous works for simple graphs.
- The paper studies the quantum query complexity of computing edge connectivity in simple graphs.
- Edge connectivity is defined as the minimum number of edges whose removal disconnects the graph.
- Lower bounds of $\Omega(n^2)$ and $\Omega(m)$ are known for classical randomized algorithms in adjacency matrix and adjacency array models, respectively.
- Upper bounds of $\tilde O(n^{3/2})$ and $\tilde O(\sqrt{mn})$ are presented for quantum query complexity in adjacency matrix and adjacency array models, respectively.
- Quantum sparsification algorithm is used to obtain these upper bounds by combining a lemma on near-minimum cuts.
- For weighted graphs, no quantum speedup is possible for computing minimum cut weights in both models as $\Omega(n^2)$ queries can be needed in worst-case scenarios.
- Five open problems related to their work are presented, including finding tight lower bounds on computing minimum cut weights and similar time complexity bounds for computing edge connectivity.
The paper is about how long it takes to solve a problem in graphs using computers. A graph is a picture of things connected by lines. Edge connectivity means the smallest number of lines you need to remove to break the connections between things in the graph. The paper talks about how fast classical and quantum computers can solve this problem, with some models being faster than others. They used a special algorithm called quantum sparsification to make their calculations faster. There are still some problems that they haven't solved yet, like figuring out the fastest way to find the weight of minimum cuts in weighted graphs.
Definitions
- Quantum: a type of computing that uses principles from physics to solve problems
- Query complexity: how many questions (queries) a computer needs to ask before it can solve a problem
- Adjacency matrix/array: ways of representing graphs using numbers and tables
- Upper/lower bounds: limits on how fast or slow something can be done
- Lemma: a small idea or fact that helps prove a bigger idea
Quantum Query Complexity of Computing Edge Connectivity
This research paper studies the quantum query complexity of computing the edge connectivity of a simple graph with $n$ vertices and $m$ edges in both the adjacency matrix and adjacency array models. The authors present upper bounds on the quantum query complexity, as well as consider weighted graphs where instead of edge connectivity, the relevant notion is the weight of a minimum cut. In this article, we will discuss what is meant by edge connectivity, classical randomized algorithms for computing it, upper bounds on quantum query complexity for both models presented in this paper, considerations for weighted graphs, and five open problems related to their work.
What is Edge Connectivity?
Edge connectivity (also known as vertex-connectivity) is defined as the least number of edges whose removal disconnects a graph. A graph can be disconnected if there are two or more sets of vertices that have no edges connecting them; thus, removing enough edges to prevent any connection between these sets would result in a disconnected graph. For example, consider an undirected graph with 6 nodes connected by 7 edges:
In this case, removing any 3 out of 7 edges would result in two separate components (sets) with no connections between them; hence its edge connectivity is 3.
Classical Randomized Algorithms
For classical randomized algorithms (i.e., algorithms that use randomness), lower bounds are known for both models discussed in this paper: $\Omega(n^2)$ for adjacency matrix model and $\Omega(m)$ for adjacency array model respectively—showing that trivial algorithm is optimal in terms of query complexity for both cases. However these lower bounds do not apply to quantum algorithms which can exploit certain properties such as superposition and entanglement to achieve better performance than classical ones when solving certain problems efficiently.
Upper Bounds on Quantum Query Complexity
The authors present upper bounds on the quantum query complexity which follow from combining a lemma on structure near-minimum cuts with a quantum sparsification algorithm: $\tilde O(n^{3/2})$ queries are needed to compute edge connectivity using an adjacency matrix model while only $\tilde O(\sqrt{mn})$ queries are needed using an adjacency array model—both significantly better than their respective lower bound counterparts discussed above!
Weighted Graphs Considerations
When considering weighted graphs instead (where each edge has some associated cost or weight), instead of computing just its edge connectivity one needs to compute its minimum cut weight—the total weight associated with all removed edges whose removal disconnects the graph from itself—which can be done using either an adjacency matrix or an array model depending upon how data about weights is stored within our system memory structures . Unfortunately however no speedup over classical algorithms was found here since they showed that at least $\Omega(n^2)$ queries were needed even when using a quantum algorithm due to worst-case scenarios arising from certain types of input graphs considered during their analysis process .
Open Problems Related To This Work
The authors conclude their work by presenting five open problems related to it which include giving an algorithm showing whether their lower bound on computing minimum cut weights in adjacency matrix model holds true throughout all ranges 1 ≤ α < n when we have an upper bound α on maximum weight; finding similar time complexities as those obtained via their query complexity results but now applied towards computing actual values like edge connectivities rather than just determining whether they exist or not; finally closing gap between existing upper bound results obtained via their work versus previous works’ lower bound results regarding simple graphs —all three being important topics worthy further investigation into future research efforts!
