Graph Packing and Covering Problems -Computational Complexity and Algorithms Design - グラフ上の分割問題と被覆問題：計算量解析とアルゴリズム設計

This dissertation studies four combinatorial optimization problems on graphs: （1） Minimum Block Transfer problem （MBT for short）, （2） Maximum k-Path Vertex Cover problem （MaxPkVC for short）, （3） k-Path Vertex Cover Reconfiguration problem （k- PVCR for short）, and （4） Minimum （Maximum） Weighted Path Cover problem （MinPC （MaxPC） for short）. This dissertation provides hardness results, such as NP-hardness and inapproximabilities, and polynomial-time algorithms for each problem. In Chapter 2, we study MBT. Let G = （V, A） be a simple directed acyclic graph, i.e., G does not include any cycles, any multiple arcs, or any self-loops, with a node set V and an arc set A. Given a DAG G and a block size B, the objective of MBT is to find a partition of its node set such that it satisfies the following two conditions: （i） Each element （called a block） of the partition has a size which is at most B, and （ii） the maximum number of external arcs among directed paths from the roots to the leaves is minimized. The number of external arcs is defined as the number of arcs connecting two distinct blocks, that is, the number denotes the number of block transfers. The height of a DAG is defined as the length of the longest directed paths from its roots to the leaves. Let us consider the two-level I/O model for data transfers between an external memory with a large space and an internal memory with a limited space. Assume that the external memory is divided into fixed contiguous blocks of size B, and one query or modification transfers one block of B objects from the external memory to the internal one. Then, with our MBT problem, we can consider the efficient way to store data in the external memory such that the maximum number of data transfers between the external memory and the internal one is minimized. We first revisit the previous, naive bottom-up packing algorithm for MBT and show that its approximation ratio is 2 if B = 2. Additionally, we show that the approximation ratio of that algorithm is at least B if B gets larger. Next, we explicitly show that MBT is NP-hard even if each block size B is at most two and the height of DAGs is three, and maximum indegree and outdegree of a node are two and three, respectively. Our proof of the NP-hardness also shows that, if B = 2 and P 6= NP, MBT does not admit any polynomial-time （3=2 - ε）- approximation （（4/3 - ε）-approximation, resp.） algorithm for any ε > 0 even if the input is restricted to DAGs of height at most five （at least six, resp.）. Fortunately, however, we can obtain a linear time exact algorithm if the height of DAGs is bounded above by two. Also, for MBT with B = 2, we provide the following linear-time algorithms: A simple 2-approximation algorithm and improved （2 - ε）-approximation algorithms, where ε = 2/h and ε = 2/（h + 1） for the case where the height of the input DAGs is even and odd, respectively. If h = 3, the last algorithm achieves a 3/2-approximation ratio, matching the inapproximability. In Chapter 3, we study MaxPkVC. Let G = （V, E） be a simple undirected graph, where V and E denote the set of vertices and the set of edges, respectively. A path of length k - 1 is called a k-path. If a k-path Pk contains a vertex v in a vertex set S, then we say that the vertex v or the set S covers Pk. Given a graph G and an integer s, the goal of MaxPkVC is to find a vertex subset S of size at most s such that the number of k-paths covered by S is maximized. Given a graph G, MinPkVC problem, a minimization version of MaxPkVC, is to find a minimum vertex subset of G such that it covers all the k-paths of G. A great focus has been on MinPkVC since it was introduced in 2011, and it is known that MinPkVC has an application for maintaining the security of a network. MinVC is a classical, very famous problem in this field such that it seeks to find a minimum vertex subset to cover all the 2-paths, i.e., the edges of the graph. Also, its maximization version, MaxVC, is well studied. One can see that MaxPkVC is a generalized problem of MaxVC since MaxVC is a special case of MaxPkVC, in the case where k = 2. MaxPkVC, for example, has an application when we would like to cover as many areas as possible with a restricted amount of budget. First, we show that MaxP3VC （MaxP4VC, resp.） is NP-hard on split graphs （chordal graphs, resp.）. Then, we show that MaxP3VC is in FPT with respect to the combined parameter s + tw, where s and tw are the prescribed size of 3-path vertex cover and treewidth parameter, respectively. Treewidth is a well-known graph parameter, and it defines a tree-likeness of a graph; see Chapter 3. Our algorithm runs in O（（s + 1）2tw+4 ・ 4tw・n）-time, where |V| = n. In Chapter 4, we discuss k-PVCR. Let G = （V, E） be a simple graph. In a reconfiguration setting, two feasible solutions of a computational problem are given, along with a reconfiguration rule that describes an adjacency relation between solutions. A reconfiguration problem asks if one feasible solution can be transformed into the other via a sequence of adjacent feasible solutions where each intermediate member is obtained from its predecessor by applying the given reconfiguration rule exactly once. Such a sequence is called a reconfiguration sequence, if it exists. For any fixed integers k ≥ 2, given two distinct k-path vertex covers I and J of a graph G and a single reconfiguration rule, the goal of k-PVCR is to determine if there is a reconfiguration sequence between I and J. For the reconfiguration rule, we consider the following three well-known rules: Token Sliding （TS）, Token Jumping （TJ）, and Token Addition or Removal （TAR）. For the precise descriptions of each rule, refer to Chapter 4. The reconfiguration variant of MinVC （called VCR） has been well studied; the goal of our study is to find the difference between VCR and k-PVCR, such as the difference of the computational complexity on graph subclasses, and to design polynomial-time algorithms. We can again see that k-PVCR is a generalized problem of VCR, since VCR is a special case of k-PVCR if k = 2. First, we confirm that several hardness results for VCR remain true for k-PVCR; we show the PSPACE-completeness of k-PVCR on general graphs under each rule TS, TJ, and TAR using a reduction from a variant of VCR. As our reduction preserves some nice graph properties, we claim that the hardness results for VCR on several graphs （planar graphs, bounded bandwidth graphs, chordal graphs, bipartite graphs） can be converted into those for k-PVCR. Using another reduction, we moreover show that k-PVCR remains PSPACE-complete even on planar graphs of bounded bandwith and maximum degree 3. On the other hand, we design polynomial-time algorithms for k-PVCR on trees （under each of TJ and TAR）, paths and cycles （under each reconfiguration rule）. Furthermore, on paths, our algorithm constructs a shortest reconfiguration sequence. In Chapter 5, we investigate MinPC （MaxPC）, especially the （in）tractabilities of MinPC. Given a graph G = （V, E）, a collection P of vertex disjoint paths is called a path cover on G if every vertex v ⋲ V is in exactly one path of P. The goal of path cover problem （PC for short） is to find a path cover with the minimum number of paths on G. As a generalized variant of PC, we introduce MinPC （MaxPC） as follows: Let U = {0, 1,...,n-1} denote a set of path lengths. Given a graph G = （V, E） and a cost （profit） function f : U → R ⋃ {+∞, -∞}, which defines a cost （profit） for each path in its length, find a path cover P of G such that the total cost （profit） of the paths in P is minimized （maximized）. Let L be a subset of U. We denote the set of paths of length l ⋲ L as PL. We, especially, consider MinPC whose cost function is f（l） = 1 if l ⋲ L; otherwise f（l） = 0. The problem is denoted by MinPLPC and is to find a path cover with the minimum number of paths with length l ⋲ L. We can also define the problem MaxPLPC with f（l） = l + 1, if l ⋲ L, and f（l） = 0, otherwise. Note that several classical problems can be seen as special cases of MinPC or MaxPC. For example, Hamiltonian Path Problem （to seek a single path visiting every vertex exactly once） and Maximum Matching Problem are equivalent to MinP{n-1}PC and MaxP{1}PC, respectively. It is known that MinP{0}PC and MinP{0, 1}PC with the same cost function as ours can be solved in polynomial time. First, we show that MinP{0, 1, 2}PC is NP-hard on planar bipartite graphs with maximum degree three, reduced from Planar 3-SAT. Our reduction also shows that there exist no approximation algorithms for MinP{0, 1, 2}PC unless P = NP. As a positive result, we show that MinP{0,...,k}PC for any fixed integers k can be solved in polynomial time on graphs with bounded treewidth. Specifically, our algorithm runs in O（42W ・W2W+2 ・ （k + 2）2W+2 ・ n）-time, assuming we are given an n-vertex graph of width at most W with its tree decomposition. Finally, a conclusion of this dissertation and open problems are given in Chapter 6.

九州工業大学博士学位論文 学位記番号：情工博甲第355号 学位授与年月日：令和3年3月25日

令和2年度

九州工業大学博士学位論文（要旨）学位記番号：情工博甲第355号 学位授与年月日：令和3年3月25日

1 Introduction||2 Minimum Block Transfer problem||3 Maximum k-Path Vertex Cover problem||4 k-Path Vertex Cover Reconfiguration problem||5 Minimum （Maximum） Weighted Path Cover problem||6 Conclusion and Open Problems