The submodular knapsack problem (SKP), which seeks to maximize a submodular set function by selecting a subset of elements within a given budget, is an important discrete optimization problem. The majority of existing approaches to solving the SKP are approximation algorithms. However, in domains such as health-care facility location and risk management, the need for optimal solutions is still critical, necessitating the use of exact algorithms over approximation methods. In this paper, we present an optimal branch-and-bound approach, featuring a novel upper bound with a worst-case tightness guarantee and an efficient dual branching method to minimize repeated computations. Experiments in applications such as facility location, weighted coverage, influence maximization, and so on show that the algorithms that implement the new ideas are far more efficient than conventional methods.

The maximum vertex-weighted clique problem (MVWCP) and the maximum edge-weighted clique problem (MEWCP) are two natural extensions of the fundamental maximum clique problem. In this paper, we systematically study MEWCP and make the following major contributions: (1) We show that MEWCP is NP-hard even when the minimum degree of the graph is n-2, in contrast to MVWCP which is polynomial-time solvable when the minimum degree of the graph is at least n-3. This result distinguishes the complexity of the two problems for the first time. (2) To address MEWCP, we develop an efficient branch-and-bound algorithm called MEWCat with both practical and theoretical performance guarantees. In practice, MEWCat utilizes a new upper bound tighter than existing ones, which allows for more efficient pruning of branches. In theory, we prove a running-time bound of O*(1.4423^n) for MEWCat, which breaks the trivial bound of O*(2^n) in the research line of practical exact MEWCP solvers for the first time. (3) Empirically, we evaluate the performance of MEWCat on various benchmark instances. The experiments demonstrate that MEWCat outperforms state-of-the-art exact solvers significantly. For instance, on 16 DIMACS graphs that the state-of-the-art solver BBEWC fails to solve within 7200 seconds, MEWCat solves all of them with an average time of less than 1000 seconds. On real-world graphs, MEWCat achieves an average speedup of over 36x.

A k-defective clique of an undirected graph G is a subset of its vertices that induces a nearly complete graph with a maximum of k missing edges. The maximum k-defective clique problem, which asks for the largest k-defective clique from the given graph, is important in many applications, such as social and biological network analysis. In the paper, we propose a new branching algorithm that takes advantage of the structural properties of the k-defective clique and uses the efficient maximum clique algorithm as a subroutine. As a result, the algorithm has a better asymptotic running time than the existing ones. We also investigate upper-bounding techniques and propose a new upper bound utilizing the conflict relationship between vertex pairs. Because the conflict relationship is common in many graph problems, we believe that this technique can be potentially generalized. Finally, experiments show that our algorithm outperforms state-of-the-art solvers on a wide range of open benchmarks. Our source code, as well as the experiment data, is open source and available https://github.com/cy-Luo000/Maximum-k-Defective-Clique.git.

Given a graph, the k-plex is a vertex set in which each vertex is not adjacent to at most k-1 other vertices in the set. The maximum k-plex problem, which asks for the largest k-plex from a given graph, is an important but computationally challenging problem in applications like graph search and community detection. So far, there is a number of empirical algorithms without sufficient theoretical explanations on the efficiency. We try to bridge this gap by defining a novel parameter of the input instance, g_k(G), the gap between the degeneracy bound and the size of maximum k-plex in the given graph, and presenting an exact algorithm parameterized by g_k(G). In other words, we design an algorithm with running time polynomial in the size of input graph and exponential in g_k(G) where k is a constant. Usually, g_k(G) is small and bounded by O(log(|V|)) in real-world graphs, indicating that the algorithm runs in polynomial time. We also carry out massive experiments and show that the algorithm is competitive with the state-of-the-art solvers. Additionally, for large k values such as 15 and 20, our algorithm has superior performance over existing algorithms.

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, k-plex, – graph with each vertex adjacent to all but at most k vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected k-plexes with small k. In this paper, we continue the research line of listing all maximal k-plexes and maximal k-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal k-plexes in O*(γD) time for each constant k, where γ is a value related to k but strictly smaller than 2, and D is the degeneracy of the graph that is far less than the vertex number n in real-word graphs. Compared to the trivial bound of 2n, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing k-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

The clique partitioning problem (CPP) of an edge-weighted complete graph is to partition the vertex set V into k disjoint subsets such that the sum of the edge weights within all cliques induced by the subsets is as large as possible. The problem has a number of practical applications in areas, such as data mining, engineering, and bioinformatics, and is, however, computationally challenging. To solve this NP-hard problem, we propose the first evolutionary algorithm that combines a dedicated merge-divide crossover operator to generate offspring solutions and an effective simulated annealing-based local optimization procedure to find high-quality local optima. The extensive experiments on three sets of 94 benchmark instances (including two sets of 63 classical benchmark instances and one new set of 31 large benchmark) show a remarkable performance of the proposed approach compared to the state-of-the-art methods. We analyze the key algorithmic ingredients to shed light on their impacts on the performance of the algorithm. The algorithm and its available source code can benefit people working on practical problems related to CPP.

Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in power-law graphs than vertex partition and thereby has been more widely adopted as the default partition strategy by existing graph systems. The graph edge partition problem, which is to split the edge set into multiple balanced parts with the objective of minimizing the total number of copied vertices, has been widely studied from the view of optimization and algorithms. In this paper, we study local search algorithms for this problem to further improve the partition results from existing methods. More specifically, we propose two novel concepts, namely adjustable edges and blocks. Based on these, we develop a greedy heuristic as well as an improved search algorithm utilizing the property of max-flow model. To evaluate the performance of our algorithms, we first provide adequate theoretical analysis in terms of approximation quality. We significantly improve the previous known approximation ratio for this problem. Then we conduct extensive experiments on a large number of benchmark datasets and state-of-the-art edge partition strategies. The results show that our proposed local search framework can further improve the quality of graph partition by a wide margin.

In a graph, a k-plex is a vertex set in which every vertex is not adjacent to at most k vertices of this set. The maximum k-plex problem, which asks for the largest k-plex from the given graph, is a key primitive in a variety of real-world applications like community detection and so on. In the paper, we develop an exact algorithm, Maplex, for solving this problem in real world graphs practically. Based on the existing first-order and the novel second-order reduction rules, we design a powerful preprocessing method which efficiently removes redundant vertices and edges for Maplex. Also, the graph color heuristic is widely used for overestimating the maximum clique of a graph. For the first time, we generalize this technique for bounding the size of maximum k-plex in Maplex. Experiments are carried out to compare our algorithm with other state-of-the-art solvers on a wide range of publicly available graphs. Maplex outperforms all other algorithms on large real world graphs and is competitive with existing solvers on artificial dense graphs. Finally, we shed light on the effectiveness of each key component of Maplex.

The maximum independent set problem is one of the most fundamental problems in graph algorithms and has been widely studied in social networks. The weighted version of this problem, where each vertex is assigned a nonnegative weight, also receives a lot of attention due to its potential applications in many areas. However, many nice properties and fast algorithms for the unweighted version can not be extended to the weighted version. In this paper, we study the structural properties of this problem, giving some sufficient conditions for a vertex being or not being in a maximum weighted independent set. These properties provide a suite of reduction rules that includes and generalizes almost all frequently used reduction rules for this problem. These rules can efficiently find partial solutions and greatly reduce the instances, especially for sparse graphs. Based on them, we also propose a simple exact yet practical algorithm. To demonstrate the efficiency of our algorithm, we compare it with state-of-the-art algorithms on several well-known datasets from the real world. The experimental results reveal that our exact algorithm is not only faster than existing algorithms but also can exactly solve more hard instances with 1,000 seconds. For remaining infeasible instances, our reduction rules can also improve existing heuristic algorithms by producing higher-quality solutions using less time.

The problem of enumerating all maximal cliques in a graph is a key primitive in a variety of real-world applications such as community detection and so on. However, in practice, communities are rarely formed as cliques due to data noise. Hence, k-plex, a subgraph in which any vertex is adjacent to all but at most k vertices, is introduced as a relaxation of clique. In this paper, we investigate the problem of enumerating all maximal k-plexes and present FaPlexen, an enumeration algorithm which integrates the “pivot” heuristic and new branching schemes. To our best knowledge, for the first time, FaPlexen lists all maximal k-plexes with provably worst-case running time O(n2γn) in a graph with n vertices, where γ < 2. Then, we propose another algorithm CommuPlex which non-trivially extends FaPlexen to find all maximal k-plexes of prescribed size for community detection in massive real-life networks. We finally carry out experiments on both real and synthetic graphs and demonstrate that our algorithms run much faster than the state-of-the-art algorithms.