Structurally Parameterized d-Scattered Set

In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(\textrm{td}^2)}$-time algorithm parameterized by tree-depth ($\textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $\epsilon > 0$, runs in time $O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})})$ and returns a $d/(1+\epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists

距離独立集合問題および誘導マッチング問題に対する近似アルゴリズム

This thesis deals with the following two problems, the Maximum Distance-d Independent Set problem (MaxDdIS for short) and the Maximum Induced Matching problem (MaxIM for short), where d ≥ 3. We design some approximation algorithms to solve MaxDdIS and MaxIM. (1) We first study MaxDdIS. Our main results for MaxDdIS are as follows: (i) It is NP-hard to approximate MaxD3IS on 3-regular graphs within 1.00105 unless P=NP. (ii) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard, and show the inapproximability of MaxDdIS on r-regular graphs. (iii) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)- approximation algorithms for MaxDdIS on r-regular graphs. (iv) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (v) Furthermore, we design a polynomial-time 1.875-approximation algorithm for MaxD3IS on cubic graphs. (vi) Finally, we consider planar graphs and obtain that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs. (2) We then investigate MaxIM on r-regular graphs. For subclasses of r-regular graphs, several better approximation algorithms are known. The previously known best approximation ratios for MaxIM on C5-free r-regular graphs and C3, C5 -free r-regular graphs are (3r/4 - 1/8 + 3/16r - 8) and (0.7084r + 0.425), respectively. We design a (2r/3 + 1/3) -approximation algorithm, whose approximation ratio is strictly smaller/better than the previous one for C5-free r-regular graphs when r ≥ 6, and for {C3, C5 }-free r-regular graphs when r ≥ 3.九州工業大学博士学位論文 学位記番号：情工博甲第339号 学位授与年月日：平成31年3月25日1 Introduction|2 Preliminaries|3 Maximum Distance-d Independent Set problem|4 Maximum Induced Matching Problem|5 Conclusion九州工業大学平成30年