Search

The rooted Budgeted Cycle Cover (BCC) problem is a fundamental optimization problem arising in wireless sensor networks and vehicle routing. Given a metric space (V , w) with vertex set V consisting of two parts D (containing depots) and V ∖ D (containing nodes), and a budget B ≥ 0 , the rooted BCC problem asks to find a minimum number of cycles to cover all the nodes in V ∖ D , satisfying that each cycle has length at most B and each cycle must contain a depot in D. In this paper, we give new approximation algorithms for the rooted BCC problem. For the rooted BCC problem with single depot, we give an O (log ⁡ B μ) -approximation algorithm, where μ is the minimum difference of any two different distances between the vertices in V and the root. For the rooted BCC problem with multiple depots, we give an O (log ⁡ n) -approximation algorithm, where n is the number of vertices. We also test our algorithms on the randomly generated instances. The experimental results show that the algorithms have good performance in practice. • A log-factor approximation algorithm for the single depot budgeted cycle cover problem. • A log-factor approximation algorithm for the multi-depot budgeted cycle cover problem. • Experiments on randomly generated instances show good performance of the algorithms. [ABSTRACT FROM AUTHOR]

k -Clustering partitions a set of n points in a metric space into at most k clusters in a way that makes the resulted clusters satisfy both the homogeneity and the separation criteria. The diameter of a cluster is the maximum distance between pairs of points in the cluster, and the split of a cluster is the minimum distance between points of the cluster and ones outside the cluster. In this paper, we propose a new criterion for measuring the goodness of clusters: The ratio of the minimum cluster split to the maximum cluster diameter (abbr. RSD). We study the following two optimization problems: Maximizing RSD without and with cardinality constraints, i.e., each cluster should consist of at least N points. For both problems with k ≥ 3 , it is NP-hard to achieve a factor of (1 2 + ϵ) approximation for any ϵ > 0. For k = 2 , we solve the two problems optimally; For k ≥ 3 , we present a 1 2 -approximation and a 1 4 -approximation algorithm for the two problems, respectively. [ABSTRACT FROM AUTHOR]

Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization. In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283 − o (1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1 / 6 − ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints. [ABSTRACT FROM AUTHOR]

The grounded Laplacian matrix L − S of a graph G = (V , E) with n = | V | nodes and m = | E | edges is a (n − s) × (n − s) submatrix of its Laplacian matrix L , obtained from L by deleting rows and columns corresponding to s = | S | ≪ n ground nodes forming set S ⊂ V. The smallest eigenvalue of L − S plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding k ≪ n edges among all the nonexistent edges forming the candidate edge set Q = (V × V) ﹨ E , in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set Q to be (S × (V ﹨ S)) ﹨ E , and prove that it has the same optimal solution as the original problem, although the size of set Q is reduced from O (n 2) to O (n). Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio (1 − e − α γ) / α and time complexity O (k n 4) , where γ and α are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time O ˜ (k m) , where O ˜ (⋅) suppresses the poly (log ⁡ n) factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency. [ABSTRACT FROM AUTHOR]

The sheer size of modern datasets has led to an urgent need for summarization techniques that can identify representative elements of the data set. Fortunately, the vast majority of data summarization tasks satisfy an intuitive diminishing returns condition known as submodularity, which allows us to find nearly-optimal solutions in linear time. However, for many applications in practice, including experimental design and sparse Gaussian processes, the objective is in general not submodular. To solve these optimization problems, an important research method is to describe the characteristics of the non-submodular functions. The non-submodular function is a hot research topic in the study of nonlinear combinatorial optimizations. In this paper, we combine and generalize the curvature and the generic submodularity ratio to design an approximation algorithm for two-stage non-submodular maximization under a matroid constraint. [ABSTRACT FROM AUTHOR]

Partitioning a network into k pieces is a fundamental problem in network science. A simple measure of partitioning a network is provided by the Max k -Uncut problem. Given an n -vertex undirected graph G with nonnegative weights defined on edges, and a positive integer k , the Max k -Uncut problem asks to find a k -partition of the vertices of G to maximize the total weight of edges that are not in the cut. This problem is the complement of the classic Min k -Cut problem, and has close relation to many combinatorial optimization problems, including the famous Densest k -Subgraph problem. In this paper, we propose a greedy approximation algorithm for the Max k -Uncut problem with performance ratio 1 − 2 (k − 1) n. The algorithm is very simple, which consists of only k − 1 min cut computations. The algorithm has fast running time O (k n 2) and is hence implementable. The experimental results show that the algorithm has excellent practical performance. [ABSTRACT FROM AUTHOR]