Fully Dynamic Approximate Maximum Independent Set on Massive Graphs Xiangyu Gao xz, Jianzhong Li xz, Dongjing Miao x xDepartment of Computer Science and Technology, Harbin Institute of Technology, Harbin, China zShenzhen Institute of Advanced Technology, Chinese Acadamy of Sciences, Shenzhen, China fgaoxy, lijzh, [email protected] Abstract—Computing a maximum independent set (MaxIS) is The MaxIS problem has a wide range of real-world ap- a fundamental NP-hard problem in graph theory, which has plications, such as indexing techniques for shortest path and important applications in a wide range of areas such as social net- distance queries [16], [24], collusion detection [4], automated work analysis, graphical information systems and coding theory. Since the underlying graphs of numerous applications are always map labeling [18], social network analysis [19], and associa- changing continuously, the problem of maintaining a MaxIS over tion rule mining [31]. Besides having these direct applications, dynamic graphs has received increasing attention in recent years. the MaxIS problem is also closely related to two well-known Due to the intractability of maintaining an exact MaxIS, this optimization problems, i.e., the minimum vertex cover problem paper studies the problem of maintaining an approximate MaxIS and the maximum clique problem. To find the maximum clique over fully dynamic graphs, where 4 graph update operations are allowed i.e., adding or deleting a vertex or an edge. Based on (the largest complete subgraph) of a graph G, it suffices to swap operation, we present a novel framework for maintaining find the maximum independent set of the complement of G. an approximate maximum independent set which contains no k- And, to find the minimum vertex cover (the smallest subset swaps and make a deep analysis of performance ratio achieved of vertices that contains at least one endpoint of each edge by it. We implement a dynamic ( ∆ + 1)-approximate algorithm 2 in the graph) of G = (V; E), one can compute the maximum and a more effective algorithm based on one-swap vertex and two-swap vertex set respectively and make a further analysis of independent set M of G and return V n M. their performance based on Power-Law Random graph model. Due to the importance of the MaxIS problem, it has been Extensive experiments are conducted over real graphs to confirm extensively studied in static graphs for decades of years [2], the effectiveness and efficiency of the proposed algorithms. [14], [15], [22], [29], [30]. Since it is NP-hard to compute an exact maximum independent set, all known exact algorithms I. INTRODUCTION have worst-case exponential time complexities regarding the number of vertices in the graph. The state-of-the-art algorithm Graph has been widely used to model many types of proposed by Xiao et al. [30] reduces the base of the exponent relationships among entities in a wide spectrum of applica- to 1.1996, i.e., with time complexity O(1:1996nnO(1)), which tions such as social networks, collaboration networks, com- still can not handle large graphs. Moreover, the MaxIS problem munication networks and biological networks. The maximum is also hard to approximate. It has been proved that the MaxIS independent set (MaxIS) problem is a classic NP-hard problem problem does not admit a constant approximation factor in in graph theory [17]. A subset M of vertices in a graph G is general graphs [28] and for any " > 0, there is no polynomial- an independent set if there is no edge between any two vertices time n1−" algorithm for the MaxIS problem unless NP = in M. A maximal independent set is an independent set such ZPP [23]. As a result, the approximation ratios of the existing that adding any other vertex to the set forces the set to contain techniques depend on either n or ∆, where n is the number arXiv:2009.11435v2 [cs.DS] 21 Jun 2021 an edge. The independent set with the largest size, measured of vertices in G and ∆ is the maximum vertex degree of G. by the number of vertices in it, among all independent sets Till now, the best approximation ratio known for the MaxIS of G is called the maximum independent set of G, which problem is O(n(log log n)2=(log n)3) [14]. In recent yeas, may not be unique. For example, in Figure 1, fv2; v6; v8g is many algorithms adopt heuristics techniques to compute high- a maximal independent set of size 3, while fv1; v4; v6; v8g is quality (large-size) independent sets [3], [10], [12], [20], [25], the maximum independent set of size 4. [27]. The state-of-the-art method is proposed by Chang et al., which iteratively applies reduction rules on vertices and remove the vertex with highest degree when no reduction rules can be applied [10]. Notice that if all vertices are removed from the graph according to reduction rules, the solution computed by the algorithm is certain to be maximum. However, the underlying graphs of many real-world ap- plications are changing continuously. For instance, when a (a) Maximal independent set. (b) Maximum independent set. user follows the other in Facebook, a directed edge will be Fig. 1. An example graph to illustrate independent sets. added between the two vertices corresponding to these two and O(log2 ∆ · log2 n). However, the solution maintained users. Similarly, a user can also remove the directed edges by these algorithms is only a maximal independent set that between himself and his out-neighbors by unfollowing them. may be not good enough to be used in many real-world Unfortunately, all the existing algorithms can not be used applications. Furthermore, these algorithms may be inefficient directly over dynamic graphs, as it is costly to compute in practice since they are complex and some of them are a solution from scratch each time, especially in large-scale analyzed against a non-adaptive oblivious adversary for an frequently updated graphs. Hence, the problem of maintaining expected time complexity. a MaxIS over dynamic graphs has received increasing attention In summary, the existing heuristic algorithms suffer from over the last few years. Because it is NP-hard to compute high time consumption and can not guarantee the solution an exact MaxIS over dynamic graphs [33], all the existing quality with the increasing of the amount of updates. And, methods resort to heuristic techniques to maintain a high- the theoretical algorithms with lower time complexities only quality solution without theoretical guarantee. provide a solution with limited accuracy in practice. To The first non-trivial algorithm is given by Zheng et al. in address this issue, in this paper, we study the problem of 2018 [33]. They propose a lazy search strategy to enable the maintaining an approximation maximum independent set over MaxIS computation over dynamic graphs. The main idea is dynamic graphs. The main challenge is to achieve a non-trivial that an exact MaxIS is obtained if a solution is found in the approximation ratio without sacrificing the time efficiency. search; Otherwise, some visited vertices will not be explored We develop a swap-based update framework for maintaining further. However, the quality of the maintained independent an independent set containing no k-swaps all the time and set is not satisfying after a few rounds of updates when the this structural feature also enables a non-trivial performance initial solution is not optimal. After that, to further improve ratio of our algorithms. Although swap-operations have been the quality of the maintained solution, they devise a directed successfully used to enlarge the independent set in static graph index DG, named as dependency graph, to guide the graphs [12], [27], none of them make an analysis of the search [32]. They category the vertices into reducing vertices performance ratio achieved by their algorithms. We also show and dependent vertices according to the procedure of applying that taking additional kinds of swap operations into consider reduction rules to vertices introduced in [10]. When a reducing will not improve the approximation ratio. This points out the vertex u (the vertex where a reduction rule is applied) and its limitation of the swap-based methods for the MaxIS problem. ∆ dependent vertices (the neighbors of u in current graph) are We implement a dynamic ( 2 + 1)-approximation algorithm removed from G, directed edges are added from u to each of and a practically more effective algorithm based on one-swap its dependent vertices in DG. Finally, directed edges will also vertex and two-swap vertex set respectively. Moreover, based be added from each dependent vertex to its reducing neighbors, on the observation that real networks are usually power-law excluding its current in-neighbors in DG. Assuming a vertex graphs with many lower-degree vertices, we make a further v is to be removed from the solution, they try to find two analysis of the performance of our algorithms based on Power- set of vertices Vin and Vout starting from v along with the law Random graph model. The main contributions of this directed edges in DG such that jVinj ≥ jVoutj. They design a paper are as follows. bottom-up dynamic searching algorithm with time complexity • A swap-based update framework is proposed for main- O(m) in the worst case, where m is the number of edges in taining an approximate maximum independent set over the graph. It is notice that as the number of updates increases, dynamic graphs, and the performance ratio achieved by the dependency graph will soonly become non-instructive due it is deeply analyzed. We also derive the lower bound of to the change in the vertex categories.