MAX BISECTION might be harder to approximate than MAX CUT

TL;DR

The paper investigates whether MAX BISECTION can match the Goemans–Williamson ratio α_GW known for MAX CUT under UGC. It shows that the standard two-phase SoS-based paradigm, relying on ε-uncorrelated SDP solutions and THRESH rounding to achieve near-balance, cannot attain α_GW for MAX BISECTION by constructing an explicit integrality-gap instance with ratio at most 0.87853, strictly below α_GW by at least 3×10^-5. A new blueprint-based approach is developed, including a carefully chosen distribution of configurations and a Gaussian-mixture construction, verified via interval arithmetic to certify the gap. The results imply that surpassing α_GW for MAX BISECTION will require fundamentally different algorithmic ideas and provide a framework connecting blueprint-based integrality gaps to dictatorship tests, with broader implications for CSPs under global cardinality constraints.

Abstract

The MAX BISECTION problem seeks a maximum-size cut that evenly divides the vertices of a given undirected graph. An open problem raised by Austrin, Benabbas, and Georgiou is whether MAX BISECTION can be approximated as well as MAX CUT, i.e., to within ${α_{GW}}\approx 0.8785672\ldots$, which is the approximation ratio achieved by the celebrated Goemans-Williamson algorithm for MAX CUT, which is best possible assuming the Unique Games Conjecture (UGC). They conjectured that the answer is yes. The current paradigm for obtaining approximation algorithms for MAX BISECTION, due to Raghavendra and Tan and Austrin, Benabbas, and Georgiou, follows a two-phase approach. First, a large number of rounds of the Sum-of-Squares (SoS) hierarchy is used to find a solution to the ``Basic SDP'' relaxation of MAX CUT which is $\varepsilon$-uncorrelated, for an arbitrarily small $\varepsilon > 0$. Second, standard SDP rounding techniques (such as ${\cal THRESH}$) are used to round this $\varepsilon$-uncorrelated solution, producing with high probability a cut that is almost balanced, i.e., a cut that has at most $\frac12+\varepsilon$ fraction of the vertices on each side. This cut is then converted into an exact bisection of the graph with only a small loss. In this paper, we show that this two-stage paradigm cannot be used to obtain an $α_{GW}$-approximation algorithm for MAX BISECTION if one relies only on the $\varepsilon$-uncorrelatedness property of the solution produced by the first phase. More precisely, for any $\varepsilon > 0$, we construct an explicit instance of MAX BISECTION for which the ratio between the value of the optimal integral solution and the value of some $\varepsilon$-uncorrelated solution of the Basic SDP relaxation is less than $0.87853 < {α_{GW}}$. Our instances are also integrality gaps for the Basic SDP relaxation of MAX BISECTION.

Figures (2)

• Figure 1: A graph representation for the blueprint $\mathcal{D}^*$. A solid line between two biases means there is a configuration with these two biases. The dotted line connects the two biases with nonzero bias weights.
• Figure 2: A contour plot for $s(t_1, t_2) / c_{GW}$. Non-rigorous numerical experiments suggest that this function is at most $0.8785231...$, and we verified $s(t_1, t_2) / c_{GW} < 0.87853$ rigorously using interval arithmetic.

Theorems & Definitions (50)

• Theorem 1.1: Main Theorem, Informal
• Corollary 1.2: Dictatorship Test, informal
• Lemma 1.3
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: borell1985geometric
• Theorem 2.4
• Corollary 2.5
• Definition 2.6: Cf. Definition 2.2 in ABG16
• Remark 2.7