On the Minimum Satisfiability Problem

In this paper we study the minimum rainbow subgraph problem, motivated by applications in bioinformatics. The input of the problem consists of an undirected graph where each edge is coloured with one of the p possible colors. The goal is to find a subgraph of minimum order (i.e. minimum number of vertices) which has precisely one edge from each color class.In this paper we show a \(\max(\sqrt{2p}, \min_q(q + \frac{\Delta}{e^{p q^2/\Delta n}}))\)-approximation algorithm using LP rounding, where Δ is the maximum degree in the input graph. In particular, this is a \(\max(\sqrt{2n}, \sqrt{2\Delta\ln{\Delta}})\)-approximation algorithm. On the other hand we prove that there exists a constant c such that the minimum rainbow subgraph problem does not have a cln Δ-approximation, unless NP ⊆ TIME(n O(loglogn)).

We study an approximation algorithm with a performance guarantee to solve a new $$\mathcal {NP}$$ -hard optimization problem on planar graphs. The problem, which is referred to as the minimum hub cover problem, has recently been introduced to the literature to improve query processing over large graph databases. Planar graphs also arise in various graph query processing applications, such as; biometric identification, image classification, object recognition, and so on. Our algorithm is based on a well-known graph decomposition technique that partitions the graph into a set of outerplanar graphs and provides an approximate solution with a proven performance ratio. We conduct a comprehensive computational experiment to investigate the empirical performance of the algorithm. Computational results demonstrate that the empirical performance of the algorithm surpasses its guaranteed performance. We also apply the same decomposition approach to develop a decomposition-based heuristic, which is much more efficient than the approximation algorithm in terms of computation time. Computational results also indicate that the efficacy of the decomposition-based heuristic in terms of solution quality is comparable to that of the approximation algorithm.

In this paper, we study the minimum power partial cover problem (MinPPC). Suppose X is a set of points and $${\mathcal {S}}$$ is a set of sensors on the plane, each sensor can adjust its power and the covering range of a sensor s with power p(s) is a disk of radius r(s) satisfying $$p(s)=c\cdot r(s)^\alpha $$. Given an integer $$k\le |X|$$, the MinPPC problem is to determine the power assignment on every sensor such that at least k points are covered and the total power consumption is the minimum. We present a primal-dual algorithm for MinPPC with approximation ratio at most $$3^{\alpha }$$. This ratio coincides with the best known ratio for the minimum power full cover problem, and improves previous ratio $$(12+\varepsilon )$$ for MinPPC which was obtained only for $$\alpha =2$$.

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy and in the absence of it. This model also contains a control function responsible for the transition from the stage of such therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the problem of minimizing the cancer cells load both during the entire period of melanoma treatment and at its final moment is stated. Such a minimization problem has a nonconvex control set, which can lead to the absence of an optimal solution to the stated minimization problem in the classes of admissible modes traditional for applications. To avoid this problem, the control set is imposed to be convex. As a result, a relaxed minimization problem arises, in which the optimal solution exists. An analytical study of this minimization problem is carried out using the Pontryagin maximum principle. The corresponding optimal solution is found in the form of synthesis and may contain a singular arc. It shows that there are values of the parameters of the bilinear control model, its initial conditions, and the time interval for which the original minimization problem does not have an optimal solution, because it has a sliding mode. Then for such values it is possible to find an approximate optimal solution to the original minimization problem in the class of piecewise constant controls with a predetermined number of switchings. This research presents the results of the analysis of the connection between such an approximate solution of the original minimization problem and the optimal solution of the relaxed minimization problem based on numerical calculations performed in the Maple environment for the specific values of the parameters of the bilinear control model, its initial conditions, and the time interval.

Affine transformational HDMR and linearised rational least squares approximation

Article Approximability and nonapproximability results for minimizing total flow time on a single machine Share on Authors: Hans Kellerer Institut für Statistik, Ökonometrie und Operations Research, Universität Graz, A-8010 Graz, Austria Institut für Statistik, Ökonometrie und Operations Research, Universität Graz, A-8010 Graz, AustriaView Profile , Thomas Tautenhahn Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, D-39016 Magdeburg, Germany Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, D-39016 Magdeburg, GermanyView Profile , Gerhard J. Woeginger Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, NL-5600 MB Eindhoven, The NetherlandsView Profile Authors Info & Claims STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of ComputingJuly 1996 Pages 418–426https://doi.org/10.1145/237814.237989Online:01 July 1996Publication History 54citation413DownloadsMetricsTotal Citations54Total Downloads413Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

The generalized vertex cover problem and some variations

We consider the problem of makespan minimization: i.e., scheduling jobs on machines to minimize the maximum load. For the deterministic case, good approximations are known even when the machines are unrelated. However, the problem is not well-understood when there is uncertainty in the job sizes. In our setting the job sizes are stochastic, i.e., the size of a job j on machine i is a random variable Xij, whose distribution is known. (Sizes of different jobs are independent of each other.) The goal is to find a fixed assignment of jobs to machines, to minimize the expected makespan---i.e., the expected value of the maximum load over the m machines. For the identical machines special case when the size of a job is the same across all machines, a constant-factor approximation algorithm has long been known. However, the problem has remained open even for the next-harder related machines case. Our main result is a constant-factor approximation for the most general case of unrelated machines. The main technical challenge we overcome is obtaining an efficiently computable lower bound for the optimal solution. We give an exponential-sized LP that we argue gives a strong lower bound. Then we show how to round any fractional solution to satisfy only a small subset of the constraints, which are enough to bound the expected makespan of our solution. We then consider two generalizations. The first is the budgeted makespan minimization problem, where the goal is to minimize the makespan subject to scheduling any subset of jobs whose reward is at least some target reward R. We extend our above result to a constant-factor approximation here using polyhedral properties of the bipartite matching polytope. The second problem is the q-norm minimization problem, where we want to minimize the expected eq-norm of the load vectors. Here we give an O(q/ log q)-approximation algorithm using a reduction to the deterministic q-norm problem with side constraints.

A rounding algorithm for approximating minimum Manhattan networks

It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs. Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.

Surrogate Modeling for Superstructure Optimization with Generalized Disjunctive Programming

Makespan scheduling on identical machines is one of the most basic and fundamental packing problem studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NPhard, and thus we do not expect a (1 + epsilon)-approximation algorithm with a running time that depends polynomially on 1/epsilon. Furthermore, Chen et al. [Chen/JansenZhang, SODA'13] recently showed that a running time of 2^{1/epsilon}^{1-delta} + poly(n) for any delta > 0 would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on 1/epsilon, the better of which achieves a running time of 2^{~O(1/epsilon^{2})} + O(n*log(n)) [Jansen, SIAM J. Disc. Math. 2010]. In this paper we obtain an algorithm with a running time of 2^{~O(1/epsilon)} + O(n*log(n)), which is tight under ETH up to logarithmic factors on the exponent. Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time 2^{~O(1/epsilon)} + poly(n).

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P \(\neq\) NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs.