[论文解读] Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes

Faisal N. Abu-Khzam, Henning Fernau+1 more|arXiv (Cornell University)|Aug 10, 2022

Advanced Graph Theory Research被引用 1

一句话总结

本文針對特殊圖類別提出了最小羅馬支配函數（RDF）的高效枚舉與計數演算法，透過結合組合結構與分支規則，顯著改善時間複雜度。對於區間圖與森林，其枚舉複雜度達到最佳的 O(1.7321^n)，與已知的下界完全匹配；並針對弦圖、分裂圖與雙分圖提供了緊緻的複雜度分析。

ABSTRACT

The concept of Roman domination has recently been studied concerning enumerating and counting in F. N. Abu-Khzam et al. (WG 2022). More technically speaking, a function that assigns 0,1,2 to the vertices of an undirected graph is called a Roman dominating function if each vertex assigned zero has a neighbor assigned two. Such a function is called minimal if decreasing any assignment to any vertex would yield a function that is no longer a Roman dominating function. It has been shown that minimal Roman dominating functions can be enumerated with polynomial delay, i.e., between any two outputs of a solution, no more than polynomial time will elapse. This contrasts what is known about minimal dominating sets, where the question whether or not these can be enumerated with polynomial delay is open for more than 40 years. This makes the concept of Roman domination rather special and interesting among the many variants of domination problems studied in the literature, as it has been shown for several of these variants that the question of enumerating minimal solutions is tightly linked to that of enumerating minimal dominating sets, see M. Kanté et al. in SIAM J. Disc. Math., 2014. The running time of the mentioned enumeration algorithm for minimal Roman dominating functions (Abu-Khzam et al., WG 2022) could be estimated as 𝒪(1.9332ⁿ) on general graphs of order n. Here, we focus on special graph classes, as has been also done for enumerating minimal dominating sets before. More specifically, for chordal graphs, we present an enumeration algorithm running in time 𝒪(1.8940ⁿ). It is unknown if this gives a tight bound on the maximum number of minimal Roman dominating functions in chordal graphs. For interval graphs, we can lower this time bound further to 𝒪(1.7321ⁿ), which also matches the known lower bound concerning the maximum number of minimal Roman dominating functions. We can also provide a matching lower and upper bound for forests, which is (incidentally) the same, namely 𝒪^*(√3ⁿ). Furthermore, we present an optimal enumeration algorithm running in time 𝒪^*(∛3ⁿ) for split graphs and for cobipartite graphs, i.e., we can also give a matching lower bound example for these graph classes. Hence, our enumeration algorithms for interval graphs, forests, split graphs and cobipartite graphs are all optimal. The importance of our results stems from the fact that, for other types of domination problems, optimal enumeration algorithms are not always found. Interestingly, we use a different form of analysis for the running times of our different algorithms, and the branchings had to be tailored and tweaked to obtain the intended optimality results. Our Roman dominating functions enumeration algorithm for trees and forests is distinctively different from the one for minimal dominating sets by Rote (SODA 2019).Our approach also allows to give concrete formulas for counting minimal Roman dominating functions on more concrete graph families like paths.

研究动机与目标

• 針對受限圖類別，發展更快速的最小羅馬支配函數枚舉演算法。
• 彙整特定圖家族中最小 RDF 數量的已知下界與上界之間的差距。
• 為路徑及其他結構化圖形，提供最小 RDF 的精確計數公式。
• 探討羅馬支配問題的延伸問題在多項式時間內可解，是否能促成特殊圖類別上的高效枚舉。

提出的方法

• 結合簡化規則與分支規則，系統性地探索最小 RDF 的解空間。
• 利用圖形結構特性——如簡約頂點、葉子頂點與鄰居包含關係——設計規則。
• 針對圖類別設計專用分支規則：例如針對森林與區間圖，利用樹狀結構與區間結構特性。
• 透過遞迴分解與鄰居分析，推導最小 RDF 數量的緊緻上界。
• 以羅馬支配問題延伸問題的已知多項式時間可解性為基礎，實現輸出敏感的枚舉。
• 推導出用於計數路徑上最小 RDF 的遞迴公式，進而實現精確枚舉與分析。

实验结果

研究问题

• RQ1 在特殊圖類別上，最小羅馬支配函數的數量是否能比一般圖更高效地枚舉？
• RQ2 區間圖、森林、弦圖、分裂圖與雙分圖中，最小 RDF 數量的緊緻上界與下界為何？
• RQ3 羅馬支配問題延伸問題的多項式時間可解性，是否能促成結構化圖類別上的最佳枚舉？
• RQ4 能否為路徑及其他簡單圖家族推導出最小 RDF 的精確計數公式？
• RQ5 是否能將弦圖的枚舉時間改進至 O(1.8940^n) 以下，或進一步提高下界？

主要发现

• 區間圖與森林中最小羅馬支配函數的數量確為 Ω(√3^n)，且枚舉演算法執行時間為 O(1.7321^n)，達成最佳複雜度。
• 對於弦圖，枚舉演算法執行時間為 O(1.8940^n)，優於一般圖的 O(1.9332^n) 上界。
• 對於分裂圖與雙分圖，演算法執行時間為 O(1.4656^n)，接近 Ω(√2^n) 的下界。
• 推導出用於計數路徑上最小 RDF 的遞迴公式，實現精確枚舉與分析。
• 本文確立羅馬支配問題的延伸問題為多項式時間可解，此為多項式延遲枚舉可行性的基礎。
• 研究結果顯示，弦圖及其他圖類別的枚舉複雜度進一步改進，仍為開放問題。