Core Concepts
We provide tight bounds on the query complexity of achieving a c-approximation to the global max-cut in a weighted undirected graph, for various values of c. Our results demonstrate a phase transition at c = 1/2, with significantly different query complexities above and below this threshold.
Abstract
The paper considers the problem of finding the maximum cut (max-cut) in a weighted, undirected graph, in the cut query model. In this model, the algorithm has access to an oracle that, given a subset of vertices S, returns the total weight of edges crossing the cut defined by S.
The main results are as follows:
-
Deterministic Algorithms:
- For c = 1, the query complexity for a deterministic algorithm to achieve a c-approximation is Θ(n^2).
- For c ∈ (1/2, 1), the query complexity is between Ω(n) and O(n^2).
- For c ∈ (0, 1/2), the query complexity is Θ(log n).
-
Randomized Algorithms:
- For c = 1, the query complexity for a randomized algorithm is between ˜Ω(n) and O(n^2).
- For c ∈ (1/2, 1), the query complexity is ˜Θ(n).
- For c ∈ (0, 1/2), the query complexity is Θ(1).
The key technical contributions include:
- A novel extension of the "cut dimension" technique to show a deterministic Ω(n) lower bound for c > 1/2.
- A query-efficient sparsifier construction for weighted graphs, which is used to obtain the ˜O(n)-query randomized algorithm for c ∈ (1/2, 1).
- A simple geometric lemma about covering the Boolean hypercube, which is crucial for the deterministic lower bound.
Overall, the results demonstrate a phase transition in the query complexity of approximating max-cut, with a significant gap between the complexities above and below the 1/2 threshold, for both deterministic and randomized algorithms.
Stats
For any c > 1/2, a deterministic algorithm requires at least Ω(n) queries to achieve a c-approximation.
For any c < 1, there exists a randomized algorithm with query complexity ˜O(n) that achieves a c-approximation.
Quotes
"For c ∈ (1/2, 1), the query complexity for a deterministic algorithm to achieve a c-approximation is between Ω(n) and O(n^2) and the query complexity for a randomized algorithm to do the same is ˜Θ(n)."
"For c ∈ (0, 1/2), the query complexity for a deterministic algorithm to achieve a c-approximation is Θ(log n) and the query complexity for a randomized algorithm to do the same is Θ(1)."