[论文解读] Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

Tanmay Inamdar, Daniel Lokshtanov+2 more|arXiv (Cornell University)|Jan 1, 2023

Advanced Graph Theory Research被引用 1

一句话总结

本文提出了公平二分（Fair Bisection），一个带有顶点颜色公平性约束的参数化聚类问题，并证明当以颜色数 c 参数化时，即使在 k=0 的情况下，该问题也是 W[1]-难的。主要贡献是一项 FPT-approximation 算法，其时间复杂度为 $2^{O(k\log k)} \cdot \left(\frac{c}{\epsilon}\right)^{O(c)} \cdot n^{O(1)}$，可找到一种平衡划分，使得跨边数不超过 k，且 A 中每种颜色类别的表示满足 $(1\pm\epsilon)$-公平性，该算法利用不可分割的树分解与 Lampis 风格的动态规划实现。

ABSTRACT

In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth.

研究动机与目标

• 形式化并研究在公平性约束下的最小二分问题，其中每种颜色类别在两个聚类中必须按比例表示。
• 研究公平二分的参数化复杂度，特别是针对参数 k（切割大小）、c（颜色数量）和 ϵ（公平性近似因子）的复杂度。
• 确定最小二分的精确 FPT 算法是否可在标准参数化下推广至公平二分问题。
• 开发一种新颖的 FPT-近似算法，实现在保持切割大小 k 有界的条件下，达到 $(1\pm\epsilon)$-公平性。

提出的方法

• 提出一种基于不可分割袋的树分解的参数化近似框架，利用 Lampis（ICALP'18）提出的有界宽度动态规划技术。
• 提出一种新颖的组合构造：对于任意图 G 和整数 k，可在时间 $2^{O(k\log k)}n^{O(1)}$ 内计算出深度为对数级、联结大小不超过 8k 的 (9k, k)-不可分割树分解。
• 通过在不可分割袋上使用动态规划，高效探索满足近似公平性约束的解。
• 通过使用星形图对每个向量进行参数保持的构造，将多维子集和问题归约至公平二分问题，从而在 k=0 时建立 W[1]-难性。
• 对精确公平性约束进行松弛：不再要求 A 中恰好有 $r_i$ 个颜色 $i$ 的顶点，而允许 $ (1 \pm \epsilon)r_i $ 个，从而实现高效的 FPT-近似。
• 将多维划分问题归约至公平二分问题，表明存在切割大小为 0 的公平二分当且仅当原始 MDSS 实例为“是”实例。

实验结果

研究问题

• RQ1 当以颜色数 c 参数化时，即使在 k=0 的情况下，公平二分是否为 W[1]-难？
• RQ2 能否将最小二分的 FPT 算法扩展至公平二分，时间复杂度为 $f(k,c) \cdot n^{O(1)}$？
• RQ3 是否存在一种 FPT-近似算法，可在保持切割大小 k 有界的前提下，实现 $(1\pm\epsilon)$-公平性？
• RQ4 Lampis（ICALP'18）在有界树宽图上用于 FPT-近似的技巧，能否通过使用不可分割袋推广至树宽无界的图？
• RQ5 树分解的何种结构性质使得公平二分的高效 FPT-近似成为可能？

主要发现

• 当以颜色数 c 参数化时，即使在 k=0 的情况下，公平二分也是 W[1]-难的，这意味着除非 FPT=W[1]，否则不存在形式为 $f(k,c) \cdot n^{O(1)}$ 的 FPT 算法。
• 存在一种 FPT-近似算法，其时间复杂度为 $2^{O(k\log k)} \cdot \left(\frac{c}{\epsilon}\right)^{O(c)} \cdot n^{O(1)}$，可实现 $(1\pm\epsilon)$-公平性且跨边数不超过 k。
• 该算法通过将 Lampis 风格的动态规划应用于不可分割袋的树分解获得，而这些树分解可在 $2^{O(k\log k)}n^{O(1)}$ 时间内构造。
• 对于任意图 G 和整数 k，可计算出深度为对数级、联结大小不超过 8k 的 (9k, k)-不可分割树分解。
• 该近似方案具有鲁棒性，可推广至顶点可能属于多个颜色类别的场景。
• 通过设置 $\epsilon = 1/(2n)$，FPT-近似算法可作为特殊情况恢复最小二分问题的 $2^{O(k\log k)} \cdot n^{O(1)}$ 算法。