Batched Vertex Cover Reconfiguration Shahar Romem Peled Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Batched Vertex Cover Reconfiguration Research Thesis Submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Shahar Romem Peled Submitted to the Senate of the Technion — Israel Institute of Technology Adar 5781 Haifa March 2021 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 This research was carried out under the supervision of Prof. Keren Censor Hillel, in the Faculty of Computer Science. Acknowledgements I would like to thank my advisor Professor Keren Censor-Hillel for her insightful guidance, patience and academic support. I would also like to thank Tigran Tonoyan and Yannic Maus for their effort and time invested in me. Last but not least, Iwould like to thank my parents, family and friends for their encouragement and comfort given to me. The generous financial help of the Forchheimer Foundation and the Technion are grate- fully acknowledged. Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Contents List of Figures Abstract 1 Abbreviations and Notations 3 1 Introduction 5 1.1 Our Contribution ............................... 7 2 Related Work 11 3 Background 15 4 Batched Vertex Cover Reconfiguration Schedules 17 4.1 Compression of Sequential Schedules .................... 17 4.2 Degree Based Schedules ........................... 21 5 Distributed Computation of Schedules 27 5.1 Reconfiguration on Cactus Graphs ..................... 27 5.1.1 Small Separator Decompositions .................. 28 5.1.2 Cluster Merging ........................... 31 5.1.3 Computing Small Separator Decompositions on Cactus Graphs . 35 5.2 Distributed Computation of Degree-Based Schedules ........... 38 5.2.1 An O(∆/ε)-Batch Schedule ..................... 39 5.2.2 From Bounded Degree to Bounded Arboricity: an O(λ/ε2)-Batch Schedule ................................ 41 5.2.3 Computing a Schedule in a Single Round ............. 42 6 Hardness of Schedule Computation 45 6.1 An Ω(n) Lower Bound for Weighted Cycles ................ 45 6.2 An Ω(n) Lower Bound for Unweighted Graphs .............. 49 6.3 An Ω(log∗n) Lower Bound for Unweighted Cycles ............. 53 7 Conclusion and open questions 57 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Hebrew Abstract i Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 List of Figures 5.1 A (d, x, k)-separator decomposition ..................... 28 6.1 Segment Ii in G ∈ G ............................. 49 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Abstract Reconfiguration schedules, i.e., sequences that gradually transform one solution ofa problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration sched- ule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers (sets of vertices that touches all edges in a graph). We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfig- uration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover. First, we design a black-box compression scheme that for any graph, takes any well-behaved sequential (= non-batched) vertex cover reconfiguration schedule and compresses it into a short batch schedule, while only incurring a 1 + ε multiplicative increase in the cost when using O(1/ε) batches; we show that this is optimal. Second, we show that a similar transformation scheme can be efficiently run in a distributed setting, where computation is done at the vertices. The distributed result is based on the new concept of a small separator decomposition that could be of independent interest. In addition, we show that this approach is optimal, in the sense that if a small separator decomposition does not yield a good schedule, no method will. Both results apply to various graph classes such as chordal graphs and cacti, and provide efficient approximation schemes that have polynomial run-time in the central- ized setting and take O(log∗ n/ε) rounds in the distributed setting. Lastly, we propose greedy-style sequential and batch reconfiguration schedules and study their performance for graphs of bounded arboricity. as well as their distributed implementation. 1 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 2 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Abbreviations and Notations • Throughout the thesis, we denote by [k] the set {0, 1, . , k}. • Throughout the thesis, for sets A, B we denote by A⊕B the symmetric difference of A and B, that is, (A \ B) ∪ (B \ A). • Given a graph G = (V, E) and a set U ⊆ V , we denote by G[U] = (U, E′) that graph induced by U, that is, E′ = E ∩ (U × U). • Given a graph G = (V, E), α and β are often used to denote vertex covers of V . In addition, we usually use the notations M = max{|α|, |β|}, D = α ⊕ β, X = α ∩ β, A = α \ β, B = β \ α. • Given a graph G = (V, E) we denote by ∆ the maximum degree in the graph. 3 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 4 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 Chapter 1 Introduction Consider a system of n computers v1, . , vn with links {(vi, vj)} between some pairs of them on which they can communicate and share information. For security reasons, we would like to oversee all communication between computers. To this end, we la- bel a vertex cover of the computers as monitors, that is, every link (vi, vj) between two computers will have at list one monitoring endpoint, vi or vj. After some time, maintenance is needed for some of the computers, and as those computers cannot mon- itor when maintenance is being done, we need to change the set of monitors. For the same security reasons, all links need to be monitored even when transforming from the first set of monitors to the second one. In addition, as monitoring takes resources, we bound, by some integer k, the allowed number of monitors at any given time, including monitoring sets that occur during such a transformation. The task in the above paragraph is not always possible, e.g., if k is smaller than the monitoring sets themselves. Those types of problems are known as reconfiguration problems, which ask the following type of questions: Given two solutions to a problem, is it always possible to gradually move from one solution to the other by changing one element at a time, while always maintaining a feasible solution? Reconfiguration problems thus explore reachability in a graph over the solutions, and as such they have been extensively studied for various problems. Notable examples are colorings [CvdHJ08, IKD12, BHI+20], matchings [IKK+19], independent sets and vertex covers [KMM12, LM19]. A well-studied example is vertex cover reconfiguration, which is exactly the problem described in the ”monitoring machines” example, and is also the focus of this thesis. In this task, one needs to find a schedule that moves from a given first vertex cover (i.e.,a subset of the vertices that touches all edges) to a given second vertex cover by changing the membership of one vertex at a time, and while ensuring that each intermediate set is a valid vertex cover (feasibility). Traditionally, the emphasis has been on the size of the intermediate solutions—the problem is trivial with no size bound—while the elementary steps consist of adding or removing a single vertex. Our goal is to exploit parallelism for reconfiguration schedules, which is natural 5 Technion - Computer Science Department - M.Sc. Thesis MSC-2021-09 - 2021 in settings where the state of vertices can be changed concurrently. For vertex cover reconfiguration, this leads us to allow changing the membership in the intermediate set of more than one vertex in each step. To study this setting, we introduce the concept of batch reconfiguration. Batch reconfiguration. In batch reconfiguration, one is allowed to change a batch of an unbounded number of elements in a single reconfiguration step, as opposed to the previous sequential reconfiguration, which changes a single vertex at a time. However, such a solution is not practically robust, in the following sense: Suppose that implementing the change for a vertex is not an immediate operation and could rather take a bit of time. Then, changing several vertices concurrently may result in a sequence of changes in these vertices, with an unpredictable order. As a result, although we aim at reconfiguring all vertices at once in one swipe, in reality what could happen is that we get an arbitrary sequence of changes, which can easily violate feasibility in an adversarial execution of a batch. In light of the above, in addition to feasibility, we require a robustness condition for batch reconfiguration schedules. The goal of a robustness condition is to guarantee that no matter in which order the elements of a batch are eventually executed, feasibility is never violated. We require that the set of vertices that are reconfigured within a batch is always an independent set. This promises that each edge is always covered, also within any internal ordering of a batch.