Further Research on Observation Reduction in Non-Deterministic Planning

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.

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.

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.

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling problems into the original problems by polynomial-time reduction. We show that, although the labelling version of these problem seems more universal than their original counterparts, they are actually equivalent to the corresponding original problem from an algorithmic point of view. Moreover, we generalize the greedy approach for solving submodular cover problem to its labelling version, and as simple applications of our new method, we use it to solve the labelling versions of the minimum weighted spanning tree and connected vertex cover problem in a unified way.

Increasing reliability and reducing disruptions in supply networks are of increasing importance; for example, power outages in electricity distribution networks cost $35–$50 billion annually in the United States. Motivated by the operational constraints of such networks and their rapid adoption of decentralized paradigms and self-healing components, we introduce the minimum reconnection time (MRT) problem, which models reliability metrics such as the System Average Interruption Duration Index (SAIDI). MRT seeks to reduce outage time after network disruptions by programming reconnection times of different edges (i.e., switches), ensuring that the operating network is acyclic. We show that MRT is NP-hard and is a special case of the well-known (weighted) minimum sum set cover (MSSC) problem. We develop the theory of kernel-based randomized rounding approaches to give a tight polynomial-time approximation for MSSC, improving the state-of-the-art approximation factor for these instances. Further, motivated by the reliability incentive structure for utility companies and operational energy losses in distribution networks, we study minimizing energy losses and reliability metrics such as SAIDI and reconnection times simultaneously. Optimizing for any single objective at a time can create unfair duration of expected outage for industrial and residential areas. We, therefore, propose local search over spanning trees to balance these multiple objectives. We computationally validate our reconfiguration methods on the National Renewable Energy Laboratory Synthetic Models for Advanced, Realistic Testing: Distribution Systems and Scenarios Greensboro synthetic network and show that this improves equity by a factor of four across industrial and residential areas. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: C. Hettle and D. Molzahn's research was partially supported by the National Science Foundation [Grant NSF 2112533], received by the Georgia Institute of Technology. S. Gupta’s research is supported by the National Science Foundation CAREER [Grant 2239824], received by the Massachusetts Institute of Technology. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0295 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0295 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We consider several geometric special cases of the set cover problem. The first problem we consider is the decomposing coverings problem. Here, we consider a combinatorial problem: given a collection of points in the plane and a collection of objects in the plane such that each point is contained in at least k objects, partition the objects into as many sets as possible so that each set covers all of the points. We show that if the objects are translates of a convex polygon, then it is possible to partition the translates into Ω( k) covers. The second problem we consider is the planar sensor cover problem. This problem is a generalization of the decomposing coverings problem. We are given a collection of points in the plane and a collection of objects in the plane. Each of the objects can be thought of as a sensor. The sensors have a duration which can be thought of as the battery life of the sensor. The planar sensor cover problem is to schedule a start time to each of the sensors so that the points are covered by a sensor for as long as possible. We give a constant factor approximation for this problem. The key contribution to this result is a constant factor approximation to a one-dimensional version of the problem called the restricted strip cover (RSC) problem. Our result for RSC improves upon the previous best O(log log log n)-approximation, and our result for the planar sensor cover problem improves upon the previous best O (log n)-approximation. The next problem we consider is the metric clustering to minimize the sum of radii problem. Here, we are given an n-point metric (P, d), and an integer k > 0. We are interested in covering the points in P with at most k balls so that the sum of the radii of the balls is minimized. We give a randomized algorithm which solves the problem exactly in nO(log n log Δ) time, where Δ is the ratio of the maximum interpoint distance to the minimum interpoint distance. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and when the metric has constant doubling dimension. The last problem we consider is the minimum dominating set problem for disk graphs. In this problem, we are given a set of disks in the plane, and we want to choose a minimum-cardinality subset of disks such that every disk is either in the set or intersects a disk in the set. For any e>0, we show that a simple local search algorithm is a (1+e)-approximation for the problem which improves upon the previous best O(log n)-approximation algorithm.

The inter-domain role mapping is a basic method for facilitating interoperation in RBAC-based collaborating environments, where each domain employs role based access control (RBAC) to specify access control policies. Prior to concrete interoperation, one important problem is to establish role mappings. Two issues are involved in the establishing process. The first one is to generate role mappings while respecting RBAC states such as separation of duty (SoD) constraints. On the other hand, administrative works of RBAC policies are sometimes needed to generate mappings. This paper investigates these two problems, mostly from the computational perspective. In particular, we study how to find a set of roles appropriate for mappings and how to fulfill interoperation requests; it turns out that most of corresponding problems are NP-complete. Further, several useful subcases of these problems are identified. We also motivate and support partial interoperation by imposing constraints on interoperation requests. When administrative works are necessary, we examine how to minimize administrative cost; the result is that one subcase of the problem reduces to the "minimal set cover" (MSC) problem. We demonstrate that approaches to MSC can be applied to this problem, even though they are not totally equivalent. Finally, a discussion on how administrative operations made to RBAC states may influence interoperability is presented as well.

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.

Optimal Sensor Placement for Contamination Detection and Identification in Water Distribution Networks

One of the most fundamental problems in web search is how to re-rank result web pages based on user logs. Most traditional models for re-ranking assume each query has a single intent. That is, they assume all users formulating the same query have similar preferences over the result web pages. It is clear that this is not true for a large portion of queries as different users may have different preferences over the result web pages. Accordingly, a more accurate model should assume that queries have multiple intents. In this paper, we introduce the multiple intents re-ranking problem. This problem captures scenarios in which some user makes a query, and there is no information about its real search intent. In such cases, one would like to re-rank the search results in a way that minimizes the efforts of all users in finding their relevant web pages. More formally, the setting of this problem consists of various types of users, each of which interested in some subset of the search results. Moreover, each user type has a non-negative profile vector. Consider some ordering of the search results. This order sets a position for each search result, and induces a position vector of the results relevant to each user type. The overhead of a user type is the dot product of its profile vector and its induced position vector. The goal is to order the search results as to minimize the average overhead of the users. Our main result is an O(log r)-approximation algorithm for the problem, where r is the maximum number of search results that are relevant to any user type. The algorithm is based on a new technique, which we call harmonic interpolation. In addition, we consider two important special cases. The first case is when the profile vector of each user type is non-increasing. This case is a generalization of the well-known min-sum set cover problem. We extend the techniques of Feige, Lovasz and Tetali (Algorithmica '04), and present an algorithm achieving 4-approximation. The second case is when the profile vector of each user type is non-decreasing. This case generalizes the minimum latency set cover problem, introduced by Hassin and Levin (ESA '05). We devise an LP-based algorithm that attains 2-approximation for it.

Interest in role-based access control has generated considerable research activity in recent years. A number of interesting problems related to the well known set cover problem have come to light as a result of this activity. However, the computational complexity of some of these problems is still not known. In this paper, we explore some variations on the set cover problem and use these variations to establish the computational complexity of these problems. Most significantly, we introduce the minimal cover problem - a generalization of the set cover problem - which we use to determine the complexity of the inter-domain role mapping problem.

On the approximation ability of evolutionary optimization with application to minimum set cover

The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm.Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2O(log* n)-approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant.

The problems dealt with in this paper are generalizations of the set cover problem, min{cx | Ax ≥ b, x ∈{0,1} n }, where c ∈ Q+ n , A ∈ {0,1}m × n, b ∈ 1. The covering 0-1 integer program is the one, in this formulation, with arbitrary nonnegative entries of A and b, while the partial set cover problem requires only m–K constrains (or more) in Ax ≥ b to be satisfied when integer K is additionall specified. While many approximation algorithms have been recently developed for these problems and their special cases, using computationally rather expensive (albeit polynomial) LP-rounding (or SDP-rounding), we present a more efficient purely combinatorial algorithm and investigate its approximation capability for them. It will be shown that, when compared with the best performance known today and obtained by rounding methods, although its performance comes short in some special cases, it is at least equally good in general, extends for partial vertex cover, and improves for weighted multicover, partial set cover, and further generalizations.

The availability of effective test suites is critical for the development and maintenance of reliable software systems. To increase test effectiveness, software developers tend to employ larger and larger test suites. The recent availability of software tools for automatic test generation makes building large test suites affordable, therefore contributing to accelerating this trend. However, large test suites, though more effective, are resources and time consuming and therefore cannot be executed frequently. Reducing them without decreasing code coverage is a needed compromise between efficiency and effectiveness of the test, hence enabling a more regular check of the software under development. We propose a novel approach, namely REDUNET, to reduce a test suite while keeping the same code coverage. We integrate this approach in a complete framework for the automatic generation of efficient and effective test suites, which includes test suite generation, code coverage analysis, and test suite reduction. Our approach formulates the test suite reduction as a set cover problem and applies integer linear programming and a network-based optimisation, which takes advantage of the properties of the control flow graph. We find the optimal set of test cases that keeps the same code coverage in fractions of seconds on real software projects and test suites generated automatically by Randoop. The results on ten real software systems show that the proposed approach finds the optimal minimisation and achieves up to 90% reduction and more than 50% reduction on all systems under analysis. On the largest project our reduction algorithm performs more than three times faster than both integer linear programming alone and the state-of-the-art heuristic Harrold Gupta Soffa.