OMP6741处理问题的算法

对于这个赋值的解，你可以依赖于所有的定理，lemas，以及课堂笔记中的结果。如果有其他作品（文章、维基百科条目、其他课程等）启发了您的解决方案，此任务的目标是改进最著名的Nonblocker内核。定义1。设G＝（V，E）是一个图。顶点的子集S？V是G的支配集，如果每个顶点有一组7个图，最多有7个顶点的支配数大于2n/5。以下内容非阻塞器的简化规则处理最多有7个顶点的连通分量。简化规则1（小组件）。如果G有一个最多有7个顶点的连通分量C，则计算C的最大非块体的大小，从G中去除C并将k减少C的最大不块体的尺寸。观察C的最大非块的大小可以在O（1）时间内计算出来，因为C有O（1的大小。有时，我们的非阻塞简化规则可以确定，对于某个顶点v。Assignment 2COMP6741: Algorithms for Intractable ProblemsName: insert your name hereStudent number: insert your student number here

1 Instructions

This assignment is an individual assignment. For the solutions to this assignment, you may rely on all theorems,lemmas, and results from the lecture notes. If any other works (articles, Wikipedia entries, lecture notes from othercourses, etc.) inspired your solutions, please cite them and give a list of references at the end.If you have questions about this assignment, please post them to the forum.Due date. This assignment is due on Friday, 10 March 2023, at 5 pm, Sydney time. Submitting x hours afterthe deadline, with x > 0, reduces the obtained mark by 5 · ?x/24? marks. No submissions will be accepted 5 days(120 hours) or more after the deadline.How to submit. Submit a PDF with your solutions to the exercises in Moodle. The first page of the PDF mustcontain your name and student number.

2 Background

The goal of this assignment is to improve on the best known kernel for Nonblocker.Definition 1. Let G = (V,E) be a graph. A subset of vertices S ? V is a dominating set of G if each vertexv ∈ V \ S has a neighbor in S. A subset of vertices S ? V is a nonblocker of G if V \ S is a dominating set of G.Example:ab cd efThis graph has a nonblocker of size 4 (highlighted) and {c, d} form a dominatingset of size 2.If we think of the graph as the state space of a system, and we are in a statethat belongs to the nonblocker S = {a, b, e, f}, then we can escape from S bytransitioning via one edge to a state that is not in S.Let us now define the Nonblocker problem.NonblockerInput: A graph G = (V,E) and an integer k ≥ 0Question: Does G have a nonblocker of size at least k?The current smallest kernel for Nonblocker relies heavily on an upper bound on the domination number ofgraphs of minimum degree at least 2.Definition 2. The domination number of a graph G is the size of a smallest dominating set of G.Theorem
3 ([1, 4]). Every connected graph on n ≥ 8 vertices with minimum degree at least 2 has

3.Dominationnumberat most

2n/5.1There is a set of 7 graphs with at most 7 vertices that have domination number greater than 2n/5. The followingsimplification rule for Nonblocker deals with connected components with at most 7 vertices.Simplification rule 1 (small component). If G has a connected component C with at most 7 vertices, then computethe size of a largest nonblocker of C, remove C from G and decrease k by the size of a largest nonblocker of C.Observe that the size of a largest nonblocker of C can be computed in O(1) time, since C has O(1) size.Sometimes our simplification rules forNonblocker may be able to determine that for some vertex v there is alargest nonblocker that does not contain v. It would be nice to be able to annotate vertices with this information.However, our kernelization algorithm needs to return an instance for Nonblocker, which contains a graph (withno annotations) and an integer. Since such annotations would be lost, Dehne et al. [2] introduce a variant of theproblem whith one special vertex that is not allowed to be in the nonblocker. If we had more than one vertex thatis not allowed to belong to the nonblocker, we could just merge them into onevertex.Nonblocker with Catalytic Vertex (NBcat)Input: A graph G = (V,E), a vertex c ∈ V , and an integer k ≥ 0Question: Does G have a nonblocker S ? V \ {c} of size at least k?We refer to the vertex c as the catalytic vertex.Dehne et al. [2] reduce Nonblocker to NBcat by simply adding the catalytic vertex c as an isolated vertex(Catalyzation rule). The instance for NBcat is then preprocessed, making sure to handle all vertices in V {c} withdegree at most 1, and the instance is finally reduced back to an instance for Nonblocker by a De-catalyzationrule. The De-catalyzation rule adds a path on three vertices, makes these three vertices adjacent to c and increasesk by 3. The resulting graph has minimum degree at least 2 and the vertices added by the De-catalyzation rulemakes sure that c belongs to a largest nonblocker without any need for an annotation. After exhaustively applyingSimplification rule 1, Theorem
3 guarantees that the graph has a dominating set of size at most 2n/5, and thereforea nonblocker of size at least 3n/5. Therefore, if k ≤ 3n/5, or, equivalently, if n ≥ 5k/3, then the instance is aYes-instance, and otherwise the algorithm by Dehne et al. [2] outputs the current instance with less than 5k/3vertices.The simplification rules from [2] include the following (they areslightlyrephrased):Simplification rule 2 (isolated vertex). If G contains a vertex v ∈ V \ {c} of degree 0, then remove v from G.Simplification rule 3 (degree 1). If G contains a vertex v ∈ V {c} of degree 1 with neighbor u, then set k := k?1,remove v from G, and merge the vertex u into c. In the special case where u = c, the merge operation is void.Simplification rule 4 (good cut vertex). If G contains a vertex v ∈ V whose neighborhood contains a non-emptysubset U ? N(v) such that N(U) = {v} and c /∈ U , then set k := k ? |U |, remove U from G, and merge v into c.Simplification rule 5 (neighboring non-catalytic degree 2). If G contains neighboring degree-2 vertices u, v ∈V \ {c} that do not share a common neighbor, then set k := k? 2, merge the two vertices in N({u, v}), and removeu and v from G. If c was in N({u, v}), then the merged vertex is the new catalytic vertex.Simplification rule 6 (neighboring catalytic degree 2). If G contains a degree-2 vertex v whose neighborhoodN(v) = {c, v′} contains c, then remove the edge vv′ from G.This results is a 5k/3+O(1) vertex kernel for Nonblocker, i.e., a kernelization algorithm returning an equiv-alent instance where the graph has at most 5k/3 + O(1) vertices. Here, k refers to the parameter in the originalinstance. This value of k may increase by up to 3 due to the de-catalyzation rule, leading to an additionalconstantof at most 5 in the number of vertices in the kernel beyond 5k/3.Our improvement to the kernel size will rely on a better bound on the domination number in graphs withminimum degree at least 2 where there are additional restrictions on degree-2 vertices.Definition 4. A bad cut vertex in a graph G = (V,E) is a vertex x ∈ V such that G ? x contains a connectedcomponent C that is a cycle on four vertices and x is adjacent to at least one and at most three vertices from C.Definition 5. A special cycle C in a graph G = (V,E) is a subgraph of G that is a cycle on five vertices such thatamong every two consecutive vertices on C there is at least one vertex that has degree 2 in G.Theorem 6 ([3]). If G = (V,E) is a connected graph on n ≥ 14 vertices with minimum degree at least 2 thatcontains no bad cut vertex and no special cycle, then G has domination number at most 3n8 .23 ExercisesExercise 1. [30 points]Assume that (G, c, k) is an instance for NBcat such that G has a bad cut vertex x and G ? x contains aconnected component C that is a cycle on four vertices and x is adjacent to i vertices from C with 1 ≤ i ≤ 3.We assume that previously stated simplification rules do not apply (in particular Simplification rule 5 and 6),so i ≥ 2.We will have three simplification rules to handle the bad cut vertex x depending on whether the catalytic vertexbelongs to C or {x}, namely for the cases where c = x, c ∈ V (C), and c /∈ {x} ∪ V (C).Show that the following simplification rule is sound.Simplification rule 7 (cycle-catalytic bad cut). If c ∈ V (C), thenif c has degree 2 and i = 2, then merge c with one of its neighbors; the merged vertex is the new catalyticvertex;otherwise, if there is a vertex y ∈ V (C) \ (N [c] ∪N(x)), then merge y into c; otherwise, merge x into c.Exercise 2. [30 points]Based on the statements of the previous exercise, design simplification rules for the cases where x = c and wherex /∈ {x} ∪ V (C).Note. If you are asked to design a rule or an algorithm, it is not necessary to prove its correctness or runningtime. However, in case the rule or algorithm is not correct, it is sometimes difficult to give partial marks if you didnot exhibit additional (partially) correct reasoning that led you to the design of the rule or algorithm or that isclose to a correctness argument.Exercise 3. [10 points]Handle special cycles via simplification rules forNBcat.Exercise 4. [20 points]Argue that Nonblocker has a8k/5+O(1)vertex kernel based on the work by Dehne et al. [2] and the previousexercises.Exercise 5. [10 points]Design a constant-factor approximation algorithm for Nonblocker.References[1] M. Blank. “An Estimate of the External Stability Number of a Graph without Suspended Vertices (in Russian)”.In: Prikl. Math. i Programmirovanie Vyp. 10 (1973), pp. 3–11.[2] Frank K. H. A. Dehne et al. “NONBLOCKER: Parameterized Algorithmics for minimum dominating set”.In: SOFSEM 2006: Theory and Practice of Computer Science, 32nd Conference on Current Trends in Theoryand Practice of Computer Science, Mer′?n, Czech Republic, January 21-27, 2006, Proceedings. Ed. by Jir′?Wiedermann et al. Vol. 3831. Lecture Notes in Computer Science. Springer, 2006, pp. 237–245. doi: 10.1007/11611257_21. url:
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