Algorithms and Complexity for a Class of Combinatorial Optimization Problems with Labelling

A 2-approximation NC algorithm for connected vertex cover and tree cover

The minimum sensor cover is an important optimization problem in wireless sensor networks, which can be seen as a special case of the minimum set cover problem. The minimum set cover problem is a well-known problem in combinatorial optimization, which has no polynomial-time (ρlnδ)-approximation for 0<ρ<1 unless NP⊆DTIME(n O(loglogn)) where δ is the maximum cardinality of a subset in the input collection. However, the minimum sensor cover problem has polynomial-time constant-approximations because this special case has a geometric structure. The design technique, called partition, is employed to take the advantage of this geometric structure. Especially, the double partition plays an important role. In this article, we would like to give an exploratory essay for the double partition technique together with research progress on approximations for the minimum sensor cover problem.

The minimum vertex cover (MVC) problem is a classic NP-hard combinatorial optimization problem with extensive real-world applications. In this paper, we propose an efficient local search algorithm, InfVC, to solve the MVC in massive graphs, which comprises three ideas. First, we introduce an inference-driven optimization strategy that explores better feasible solutions through inference rules. Second, we develop a structural-determined perturbation strategy that is motivated by the structure features of high-quality solutions, prioritizing high-degree vertices into the candidate solution to guide the search process to some potential high-quality search area. Third, we design a self-adaptive local search framework that dynamically balances exploration and exploitation through a perturbation management mechanism. Extensive experiments demonstrate that InfVC outperforms all the state-of-the-art algorithms on almost massive instances.

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

The tile assembly model is a novel biological computing model where information is encoded in DNA tiles. It is an efficient way to solve NP-complete problems due to its scalability and parallelism. In this paper, we apply the tile assembly model to solve the minimum and exact set cover problems, which are well-known NP-complete problems. To solve the minimum set cover problem, we design a MinSetCover system composed of three parts, i.e., the seed configuration subsystem, the nondeterministic choice subsystem, and the detection subsystem. Moreover, we improve the MinSetCover system and propose a MinExactSetCover system for solving the problem of exact cover by 3-sets. Finally we analyze the computation complexity and perform a simulation experiment to verify the effectiveness and correctness of the proposed systems.

We present the first study on the approximability of geometric versions of the unique coverage problem and the minimum membership set cover problem. In the former problem, one is given a family of sets of elements from some universe and aims to select sets that maximize the number of elements contained in precisely one selected set. Unique Coverage has important applications in wireless networks and it is thus natural to consider it in a geometric setting. We use the well-known (unit) disk model for wireless networks and show that Unique Coverage remains NP-hard in this case and present a polynomial-time 1/18-approximation algorithm. This algorithm is extended to the budgeted low-coverage problem, where covering can element multiple times yields less profit and we have a fixed budget to 'buy' sets. We give an asymptotic FPTAS in case the disks have arbitrary size, but bounded ply. For the case that the geometric objects are arbitrary fat objects, we show that these problems are as hard to approximate as in the general case. In the minimum membership set cover problem, the goal is to cover all elements while minimizing the maximum number of sets in which any element is contained. For unit squares and unit disks, we show that the problem remains NP-hard and does not admit a polynomial-time approximation algorithm with ratio smaller than 2 unless P=NP. For unit squares, we give a 5-approximation algorithm for instances where the optimum objective value is bounded by a constant.

The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. Such a partition defines a probability distribution, obtained by dividing each part size by n. The goal is to find a feasible solution minimizing the (binary) entropy of the corresponding distribution. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat =log 2 e bits $\simeq$1.4427 bits. Moreover, inspired by recent work by Feige, Lovasz and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem.

In this paper, we consider the general class of coverage and clustering problems in a dynamic environment, and propose a computationally efficient framework to address them. We define the problem of achieving instantaneous coverage as a combinatorial optimization problem in a Maximum Entropy Principle framework. We then extend the framework to a dynamic environment, thereby allowing us to address the inherent trade-off between the resolution of the clusters and the computation cost, and provides flexibility to a variety of dynamic specifications. The proposed framework addresses both the coverage as well as tracking aspects of the problem. The determination of cluster centers and their associated velocity field is cast as a control design problem ensuring that the algorithm achieves progressively better coverage with time. Simulation results presented in the paper demonstrate that the proposed algorithm achieves target coverage costs five to seven times faster than related frame-by-frame methods, with the additional ability to identify natural clusters in the dataset.

A rough set method for the minimum vertex cover problem of graphs

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

Suppose that finding the solution to a problem is P-complete. It is natural to ask if it is any easier to obtain an approximate solution. For decision problems this might mean considering the corresponding combinatorial optimization problem. That is, a problem in which we try to minimize or maximize a given quantity. As one might expect from the theory of NP-completeness, the answer is both yes (for example in the case of Bin Packing, Problem A.4.7) and no (for example in the case of the Lexicographically First Maximal Independent Set Size Problem, see Lemma 10.2.2.). There are several motivations for developing good NC approximation algorithms. First, in all likelihood P-complete problems cannot be solved fast in parallel. Therefore, it may be useful to approximate them quickly in parallel. Second, problems that are P- complete but that can be approximated well seem to be special boundary cases. Perhaps by examining these types of problems more closely we can improve our understanding of parallelism. Third, it is important to build a theoretical foundation for studying and classifying additional approximation problems. Finally, it may be possible to speed up sequential approximation algorithms, of NP-complete problems, using fast parallel approximations. Our goal in this section is to develop the basic theory of parallel approximation algorithms. We begin by showing that certain P-complete problems are not amenable to NC approximation algorithms. Later we present examples of P-complete problems that can be approximated well in parallel. We start by considering the Lexicographically First Maximal Independent Set Problem, introduced in Definition 7.1.1, and proven P-complete in Problem A.2.1. As defined, LFMIS it is not directly amenable to approximation. We can phrase the problem in terms of computing the size of the independent set. Definition 10.2.1 Lexicographically First Maximal Independent Set Size (LFMISsize) Given: An undirected graph G = (V, E) with an ordering on the vertices and an integer k. Problem: Is the size of the lexicographically first maximal independent set of G less than or equal to k ? The following lemma shows that computing just the size of the lexicographically first maximal independent set is P-complete.

Decision-making problems can be modeled as combinatorial optimization problems with Constraint Programming formalisms such as Constrained Optimization Problems. However, few Constraint Programming formalisms can deal with both optimization and uncertainty at the same time, and none of them are convenient to model problems we tackle in this paper. Here, we propose a way to deal with combinatorial optimization problems under uncertainty within the classical Constrained Optimization Problems formalism by injecting the Rank Dependent Utility from decision theory. We also propose a proof of concept of our method to show it is implementable and can solve concrete decision-making problems using a regular constraint solver, and propose a bot that won the partially observable track of the 2018 μRTS AI competition. Our result shows it is possible to handle uncertainty with regular Constraint Programming solvers, without having to define a new formalism neither to develop dedicated solvers. This brings new perspective to tackle uncertainty in Constraint Programming.

Wireless access networks are ever-changing their setup, topologies and provisions of coverage. Forthcoming deployment trends include densification of cells and convergence of the ‘native IP’ with cellular access networks with more than best-effort expectations. Greater meshing in networks due to practical deployment requirements is inevitable and needed. This paper proposes a routing protocol to address the “randomness” of topology interconnections, routing paths and sizes in emerging IP access networks. Termed as Minimum Set Cover (MSC) approach, it is a generalization of the NP and NP complete mathematical problem. The offline component of the MSC multiplies intra-domain routing installations, called Routing Planes (RPs) modelled as graphs, to suitably cover the whole routing topology prior to the traffic injections in the online component. We introduce novel offline optimization features using a dynamic cost function that plans availability of capacities and correlation of routing paths via the chosen set of RPs. Our simulations verify effective path diversity using MSC with its modest protocol overhead and a heuristic used for RPs selection. For randomly constructed access network topologies with 7, 18 and 33 routers and various meshing levels, our results show convincing suitability and key performances gains compared with rival routing solutions.