发表机构
Queens College, City University of New York; Rutgers University（纽约市立大学皇后学院; 罗格斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Max-k-CSP和有界度k-SAT的多样化解生成问题，针对约束多样性和变量多样性两种度量，设计了双准则近似算法，并在Lovász局部引理机制下证明了精确多样性问题的NP困难性及给出了多项式时间近似算法。
AI 中文摘要
我们研究生成Max-$k$-CSP和有界度$k$-SAT的多样化解的问题，重点关注两个不同的度量：约束多样性和变量多样性。对于约束多样性，目标是输出$s \geq 2$个CSP的赋值，使得每个赋值满足$c$比例的约束，同时最大化满足约束的$0$-$1$指示向量在汉明度量下的多样性。通过将此归约为多准则优化问题，我们设计了$poly(n,s)$时间的近似算法，返回$s$个赋值，并在满足约束的比例和约束向量的多样性上同时提供可证明的双准则保证。对于变量多样性，目标是在最大化赋值之间的汉明距离的同时，最大化满足的约束数量。对于期望解数量为$s=2^{O(n)}$的Max-$k$-CSP实例，我们通过在解空间内构造线性码来隐式表示这些多样的近似解。最后，我们研究了Lovász局部引理机制下$k$-SAT的变量多样性。在此设置中，我们证明了精确多样性问题（计算解空间的直径）的NP困难性，并提供了多项式时间近似算法以高效生成多样的满足赋值。
英文摘要
We study the problem of generating diverse solutions to Max-$k$-CSP and bounded-degree $k$-SAT, focusing on two distinct metrics: constraint diversity and variable diversity. For constraint diversity, the goal is to output $s \geq 2$ assignments to the CSP such that each assignment satisfies a $c$-fraction of the constraints, while maximizing the diversity among the $0$-$1$ indicator vectors of satisfied constraints in the Hamming metric. By reducing this to a multi-criteria optimization problem, we design $poly(n,s)$ time approximation algorithms that return s assignments achieving provable bi-criteria guarantees on both the fraction of satisfied constraints and diversity of the constraint vectors. For variable diversity, the objective is to maximize the Hamming distance between the assignments, while also maximizing the number of constraints satisfied. For Max-$k$-CSP instances when the desired number of solutions is $s=2^{O(n)}$, we implicitly represent these diverse approximate solutions by constructing linear codes within the solution space. Finally, we investigate variable diversity for $k$-SAT in the Lovász Local Lemma regime. In this setting, we establish NP-hardness for the exact diversity problem (computing the diameter of the solution space) and provide a polynomial-time approximation algorithm to efficiently generate diverse satisfying assignments.
