最大割问题的SDP精确性识别的复杂性
The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem
- Indian Institute of Technology Bombay（印度理工学院孟买分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决最大割问题SDP精确性的计算复杂性，证明加权图的SDP精确性判定为强NP难，还将该难解性扩展到简单无权图，所用几何锁定机制解耦了连续界与组合难解性。
AI中文摘要：
最大割（Max-Cut）问题的半定规划（SDP）松弛精确当且仅当其最优值等于整数最大割，几何上对应秩为1的最优解。Delorme和Poljak的开创性工作已证明识别精确性是NP难的，但他们的归约依赖指数级缩放的边权重，仅建立了弱NP难性，且明确留下了简单无权图的复杂性问题未解决。本文解决了精确性性质的计算复杂性：首先，通过构造具有多项式有界整数权重的平方和对偶证书，证明加权图的SDP精确性判定是强NP难的；其次，通过将受限非全相等4-SAT几何嵌入边不交团结构，将该难解性扩展到简单无权图。两种证明均利用几何锁定机制迫使连续SDP松弛达到绝对全局最小值，使连续界与底层组合难解性解耦。
英文摘要：
The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak resolved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We show that recognition is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for nonnegative integer edge weights. The reduction provides an explicit SDP optimum and makes the additive integrality gap equal to the minimum number of unsatisfied clauses in the source formula. Recognition remains NP-complete even when an exact rational optimal primal--dual pair is supplied. We also establish strong NP-hardness of recognizing exactness of the Frieze--Jerrum Max-$k$-Cut relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. An independent bounded-weight construction gives a second proof for Max-Cut. Finally, reductions preserving the additive gap up to explicit factors establish strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.
