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
