Polynomial Time Method for Solving Nash Equilibria of Zero-Sum Games

There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.

Solving the Binary Linear Programming Model in Polynomial Time

The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.

Optimizing Polynomial-Time Solutions to a Network Weighted Vertex Cover Game

Weighted vertex cover(WVC)is one of the most important combinatorial optimization problems.In this paper,we provide a new game optimization to achieve efficiency and time of solutions for the WVC problem of weighted networks.We first model the WVC problem as a general game on weighted networks.Under the framework of a game,we newly define several cover states to describe the WVC problem.Moreover,we reveal the relationship among these cover states of the weighted network and the strict Nash equilibriums(SNEs)of the game.Then,we propose a game-based asynchronous algorithm(GAA),which can theoretically guarantee that all cover states of vertices converging in an SNE with polynomial time.Subsequently,we improve the GAA by adding 2-hop and 3-hop adjustment mechanisms,termed the improved game-based asynchronous algorithm(IGAA),in which we prove that it can obtain a better solution to the WVC problem than using a the GAA.Finally,numerical simulations demonstrate that the proposed IGAA can obtain a better approximate solution in promising computation time compared with the existing representative algorithms.

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS （PART I：EQUAL TIMES AND COSTS）

Abstract Most papers in scheduling research have treated individual job processing times as fixed parameters. However, in many practical situations, a manager may control processing time by reallocating resources. In this paper, authors consider a machine scheduling problem with controllable processing times. In the first part of this paper, a special case where the processing times and compression costs are uniform among jobs is discussed. Theoretical results are derived that aid in developing an O(n 2) algorithm to slove the problem optimally. In the second part of this paper, authors generalize the discussion to general case. An effective heuristic to the general problem will be presented.

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS ： PROOF OF THEOREMS

Abstract This report is virtually the appendix part of the author's previous paper which includes the proofs for the theorems and lemmas.

Network Decomposition and Maximum Independent Set Part Ⅱ: Application Research

According to the researches on theoretic basis in part Ⅰ of the paper, the spanning tree algorithms solving the maximum independent set both in even network and in odd network have been developed in this part, part Ⅱ of the paper. The algorithms transform first the general network into the pair sets network, and then decompose the pair sets network into a series of pair subsets by use of the characteristic of maximum flow passing through the pair sets network. As for the even network, the algorithm requires only one time of transformation and decomposition, the maximum independent set can be gained without any iteration processes, and the time complexity of the algorithm is within the bound of O(V3). However, as for the odd network, the algorithm consists of two stages. In the first stage, the general odd network is transformed and decomposed into the pseudo-negative envelope graphs and generalized reverse pseudo-negative envelope graphs alternately distributed at first; then the algorithm turns to the second stage, searching for the negative envelope graphs within the pseudo-negative envelope graphs only. Each time as a negative envelope graph has been found, renew the pair sets network by iteration at once, and then turn back to the first stage. So both stages form a circulation process up to the optimum. Two available methods, the adjusting search and the picking-off search are specially developed to deal with the problems resulted from the odd network. Both of them link up with each other harmoniously and are embedded together in the algorithm. Analysis and study indicate that the time complexity of this algorithm is within the bound of O(V5).

Network Decomposition and Maximum Independent Set Part Ⅰ:Theoretic Basis

The structure and characteristics of a connected network are analyzed, and a special kind of sub-network, which can optimize the iteration processes, is discovered. Then, the sufficient and necessary conditions for obtaining the maximum independent set are deduced. It is found that the neighborhood of this sub-network possesses the similar characters, but both can never be allowed incorporated together. Particularly, it is identified that the network can be divided into two parts by a certain style, and then both of them can be transformed into a pair sets network, where the special sub-networks and their neighborhoods appear alternately distributed throughout the entire pair sets network. By use of this characteristic, the network decomposed enough without losing any solutions is obtained. All of these above will be able to make well ready for developing a much better algorithm with polynomial time bound for an odd network in the the application research part of this subject.

UKF-based multi-sensor passive tracking with active assistance

A new synergy tracking method of infrared and radar is presented. To improve tracking accuracy, the unscented Kalman filter （UKF）, which has better nonlinear approximation ability, is adopted. In addition, to reduce the possibility of radar being locked-on by adverse electronic support measure （ESM）, radar is under the intermittent-working state. After radar is turned off, the possible target position is estimated by a set of time polynomials, which is constructed based on the sufficient observations done before radar is turned off, the estimated values from time polynomials are compared with the current observation values from infrared to determine the time when radar is turned on. Simulation results show the method has a good tracking accuracy and effectively reduces the possibility of radar being locked-on by adverse ESM.

The Polynomially Exponential Time Restrained Analytical Hierarchy

A polynomially exponential time restrained analytical hierarchy is introduced with the basic proper ties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level. And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.

Exponential Stabilization for C_0-Semigroups Under Compact Perturbation

It is proved that a system under compact perturbation cannot be uniformly exponentially stable for an isometric C0-semigroup or a C0-group with polynomial growth for negative time in a Banach space. The results extend and improve the corresponding results of previous literature.

Improved Approximation Schemes for Early Work Scheduling on Identical Parallel Machines with a Common Due Date

We study the early work scheduling problem on identical parallel machines in order to maximize the total early work,i.e.,the parts of non-preemptive jobs that are executed before a common due date.By preprocessing and constructing an auxiliary instance which has several good properties,for any desired accuracy,we propose an efficient polynomial time approximation scheme with running time O(f(1/ε)n),where n is the number of jobs and f(1/ε)is exponential in 1/ε,and a fully polynomial time approximation scheme with running time O(1/ε^(2m+1)+n)when the number of machines is fixed.

Uniform Parallel-Machine Scheduling with Time Dependent Processing Times

We consider several uniform parallel-machine scheduling problems in which the processing time of a job is a linear increasing function of its starting time.The objectives are to minimize the total completion time of all jobs and the total load on all machines.We show that the problems are polynomially solvable when the increasing rates are identical for all jobs;we propose a fully polynomial-time approximation scheme for the standard linear deteriorating function,where the objective function is to minimize the total load on all machines.We also consider the problem in which the processing time of a job is a simple linear increasing function of its starting time and each job has a delivery time.The objective is to find a schedule which minimizes the time by which all jobs are delivered,and we propose a fully polynomial-time approximation scheme to solve this problem.

Human Machine Collaborative Support Scheduling System of Intelligence Information from Multiple Unmanned Aerial Vehicles Based on Eye Tracker

Many human-machine collaborative support scheduling systems are used to aid human decision making by providing several optimal scheduling algorithms that do not take operator's attention into consideration.However, the current systems should take advantage of the operator's attention to obtain the optimal solution.In this paper, we innovatively propose a human-machine collaborative support scheduling system of intelligence information from multi-UAVs based on eye-tracker. Firstly, the target recognition algorithm is applied to the images from the multiple unmanned aerial vehicles(multi-UAVs) to recognize the targets in the images. Then,the support system utilizes the eye tracker to gain the eye-gaze points which are intended to obtain the focused targets in the images. Finally, the heuristic scheduling algorithms take both the attributes of targets and the operator's attention into consideration to obtain the sequence of the images. As the processing time of the images collected by the multi-UAVs is uncertain, however the upper bounds and lower bounds of the processing time are known before. So the processing time of the images is modeled by the interval processing time. The objective of the scheduling problem is to minimize mean weighted completion time. This paper proposes some new polynomial time heuristic scheduling algorithms which firstly schedule the images including the focused targets. We conduct the scheduling experiments under six different distributions. The results indicate that the proposed algorithm is not sensitive to the different distributions of the processing time and has a negligible computational time. The absolute error of the best performing heuristic solution is only about 1%. Then, we incorporate the best performing heuristic algorithm into the human-machine collaborative support systems to verify the performance of the system.

The Complexity of Checking Consistency of Pedigree Information and Related Problems

Consistency checking is a fundamental computational problem in genetics.Given a pedigree and information on the genotypes (of some) of the individuals in it, the aim ofconsistency checking is to determine whether these data are consistent with the classic Mendelianlaws of inheritance. This problem arose originally from the geneticists'' need to filter their inputdata from erroneous information, and is well motivated from both a biological and a sociologicalviewpoint. This paper shows that consistency checking is NP-complete, even with focus on a singlegene and in the presence of three alleles. Several other results on the computational complexity ofproblems from genetics that are related to consistency checking are also offered. In particular, itis shown that checking the consistency of pedigrees over two alleles, and of pedigrees withoutloops, can be done in polynomial time.

Notes on Liveness and Boundedness of Extended Strong Asymmetric Choice Nets Ⅱ

In this paper, the Extended Strong Asymmetric Choice Nets Ⅱ (ESACN Ⅱ), a subclass of Asymmetric Choice Nets (ACN) including Extended Free Choice Nets (EFCN) and Strong Asymmetric Choke Nets Ⅱ (SACN Ⅱ, is presented. A necessary and sufficient condition for liveness of ESACN Ⅱ is proposed. Moreover, a criterion is introduced, which is necessary and sufficient for judgement of liveness and boundedness of ESACN Ⅱ. Meanwhile a polynomial time algorithm is given to decide liveness and boundedness for ESACN Ⅱ.

On Scheduling a Deteriorating Rate-Modifying Activity to Minimize the Number of Tardy Jobs

We investigate a single-machine scheduling problem,where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of jobs.The objective is to minimize the number of tardy jobs.Under the assumption that the duration of the rate-modifying activity is a nonnegative and nondecreasing function on its starting time,we propose an optimal polynomial time algorithm running in O(n^(3))time.