On the cut-query complexity of approximating max-cut

TL;DR

The paper studies query efficiency for global max-cut under the cut-query model. It proves a sharp divide: any deterministic c-approximation with c>1/2 requires Ω(n) queries, while a randomized approach achieves any c<1 with ṪO(n) queries by reducing the problem to a sparsified graph via edge-strength-aware sampling. It further characterizes the query complexity across regimes, including an exact max-cut lower bound of Ω(n^2) for deterministic algorithms and detailed bounds around the phase transition at c=1/2. The core methods combine an LP-based extension of the cut-dimension for hardness with a novel, query-efficient sparsifier based on edge strengths and maximum spanning trees, enabling near-optimal approximations from a sparse subgraph. These results illuminate the fundamental limits of restricted-query graph optimization and connect submodular maximization insights to graph-cut problems in restricted-access models.

Abstract

We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. Graph algorithms in this cut query model and other query models have recently been studied for various other problems such as min-cut, connectivity, bipartiteness, and triangle detection. Max-cut in the cut query model can also be viewed as a natural special case of submodular function maximization: on query $S \subseteq V$, the oracle returns the total weight of the cut between $S$ and $V \backslash S$. Our first main technical result is a lower bound stating that a deterministic algorithm achieving a $c$-approximation for any $c > 1/2$ requires $Ω(n)$ queries. This uses an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions). Secondly, we provide a randomized algorithm with $\tilde{O}(n)$ queries that finds a $c$-approximation for any $c < 1$. We achieve this using a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). To complement these results, for most constants $c \in (0,1]$, we nail down the query complexity of achieving a $c$-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at $c = 1/2$: we design a deterministic algorithm for global $c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that any randomized algorithm requires $Ω(n/\log n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $Ω(n^2)$ queries to find an exact max-cut (enough to learn the entire graph).

Figures (1)

• Figure 1: Summary of our results. For each range of $c$, we state the query complexity (up to constant and logarithmic factors) that we show for achieving a $c$-approximation in both the deterministic and randomized settings. $(n, n^2)$ indicates settings where we have a lower bound of $\tilde{\Omega}(n)$ and an upper bound of $O(n^2)$.

Theorems & Definitions (174)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 3.1
• Corollary 3.2
• proof
• Definition 3.3
• Lemma 3.4