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)."

Key Insights Distilled From

by Orestis Plev... at **arxiv.org** 04-24-2024

Deeper Inquiries

The query complexity results obtained for approximating max-cut in the cut query model can be extended to other graph optimization problems that can be formulated as submodular function maximization or minimization. Since max-cut is a special case of submodular function maximization, similar techniques and lower bounds can be applied to problems like minimum cut, connectivity, bipartiteness, and triangle detection. By understanding the query complexity required for approximating max-cut, we can gain insights into the efficiency of algorithms for these related graph optimization problems in the cut query model. The phase transitions and bounds observed in the context of max-cut can provide valuable guidance for developing efficient algorithms for a range of graph optimization tasks.

The techniques used to prove the deterministic lower bounds for approximating max-cut in the cut query model can indeed be applied to obtain stronger lower bounds for other submodular function optimization problems in the same model. By extending the concept of cut dimension and utilizing linear programming and strong duality theorems, it is possible to establish lower bounds on the query complexity required for achieving certain levels of approximation for various submodular functions. The approach of considering the structure of the function and the properties of the underlying graph can help in deriving more precise lower bounds for a broader class of optimization problems beyond max-cut. By adapting and extending these techniques, it is feasible to provide deeper insights into the query complexity of submodular function optimization in the cut query model.

The results obtained from the study of query complexity in approximating max-cut in the cut query model have significant implications for the practical design of graph algorithms with limited access to the input graph. Understanding the minimum number of queries required to achieve a certain level of approximation can guide the development of more efficient algorithms for various graph optimization tasks. By optimizing the query complexity, algorithm designers can focus on strategies that minimize the number of queries while still providing accurate solutions. Additionally, the insights gained from the lower bounds and phase transitions can inform the design of algorithms that make the most effective use of limited query access, leading to improved performance and scalability in real-world applications.

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