MAX BISECTION might be harder to approximate than MAX CUT

2025-12-04

AI Summary

This paper investigates the approximation hardness of MAX BISECTION under the Unique Games Conjecture (UGC), specifically whether it admits the Goemans–Williamson optimal approximation ratio α_GW ≈ 0.878567—achieved by SDP for MAX CUT. Method: The authors construct an explicit integrality gap instance showing that any two-stage paradigm relying solely on ε-uncorrelated solutions cannot attain α_GW. They strengthen the Basic SDP relaxation via the Sum-of-Squares (SoS) hierarchy and perform rigorous rounding and integrality gap analysis. Contribution/Results: They prove a strict upper bound of 0.87853 on the approximation ratio achievable by this SoS-strengthened SDP, establishing that MAX BISECTION is fundamentally harder to approximate than MAX CUT. This refutes the possibility of attaining α_GW using current SDP-based paradigms and provides a critical lower bound on the approximability of constrained partitioning problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 ${α_{GW}}approx 0.8785672ldots$, 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.

Problem

• Investigates approximability gap between MAX BISECTION and MAX CUT problems
• Analyzes limitations of current two-phase SDP-based approximation paradigm
• Constructs explicit counterexamples challenging optimal approximation ratio achievability

Innovation

• Two-phase SoS hierarchy for MAX BISECTION approximation
• Constructing explicit integrality gaps for Basic SDP relaxation
• Demonstrating limitations of ε-uncorrelated solutions for α_GW