Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K_1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general.The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K_i,j-free graphs, which exclude, for some fixed i,j∈N, the complete bipartite graph K_i,j as a subgraph. For i,j≥2, the class of claw-free graphs and any class of K_i,j-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable.We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t≥4, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K_1,t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes.Finally, we show that for any t∈N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.     

Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability

Data reduction is a key technique in the study of fixed parameter algorithms. In the AI literature, pruning techniques based on simple and efficient-to-implement reduction rules also play a crucial role in the success of many industrial-strength solvers. Understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly-generated instances of NP-complete problems.In this paper, we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem, the weighted d-CNF satisfiability problem. We propose a non-trivial random model for the problem and study the probabilistic behavior of the random instances from the model. We design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed-parameter polynomial time O(dknm) where n is the number of variables, m is the number of clauses, and k is the fixed parameter. We establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space, unsatisfiable random instances of the problem have parametric resolution proof of fixed-parameter polynomial size. Also discussed is a more general random model and the generalization of the results to the model.     

Total completion time minimization in a 2-stage differentiation flowshop with fixed sequences per job type

This paper addresses the total completion time minimization in a two-stage differentiation flowshop where the sequences of jobs per type are predetermined. The two-stage differentiation flowshop consists of a stage-1 common machine and m stage-2 parallel dedicated machines. The goal is to determine an optimal interleaved processing sequence of all jobs at the first stage. We propose an dynamic programming algorithm, where nk is the number of type-k jobs. The running time is polynomial when m is constant.     

Fast algorithms for the dominating set problem on permutation graphs

AnO(n log logn) (resp.O(n² log² n)) algorithm is presented to solve the minimum cardinality (resp. weight) dominating set problem on permutation graphs, assuming the input is a permutation. The best-known previous algorithm was given by FÄrber and Keil, where they use dynamic programming to get anO(n² (resp.O(n³)) algorithm. Our improvement is based on the following three factors: (1) an observation on the order among the intermediate terms in the dynamic programming, (2) a new construction formula for the intermediate terms, and (3) efficient data structures for manipulating these terms.This research was supported in part by the National Science Foundation under Grant CCR-8905415 to Northwestern University.     

Cluster editing problem for points on the real line: A polynomial time algorithm

In this paper we consider the cluster editing problem for a special type of graphs, where the vertices represent points on the real line and there is an edge between each two vertices for which the distance between their corresponding points on the line is less than a given constant. We give a polynomial time cluster editing algorithm for this class of graphs.     

Solution time reduction techniques of a stochastic dynamic programming approach for hazardous material route selection problem

We use a stochastic dynamic programming (SDP) approach to solve the problem of determining the optimal routing policies in a stochastic dynamic network. Due to its long time for solving SDP, we propose three techniques for pruning stochastic dynamic networks to expedite the process of obtaining optimal routing policies. The techniques include: (1) use of static upper/lower bounds, (2) pre-processing the stochastic dynamic networks by using the start time and origin location of the vehicle, and (3) a mix of pre-processing and upper/lower bounds. Our experiments show that while finding optimal routing policies in stochastic dynamic networks, the last two of the three strategies have a significant computational advantage over conventional SDP. Our main observation from these experiments was that the computational advantage of the pruning strategies that depend on the start time of the vehicle varies according to the time input to the problem. We present the results of this variation in the experiments section. We recommend that while comparing the computational performances of time-dependent techniques, it is very important to test the performance of such strategies at various time inputs.     

A branch, bound, and remember algorithm for the 1|r i |∑t i scheduling problem

This paper presents a modified Branch and Bound (B&B) algorithm called, the Branch, Bound, and Remember (BB&R) algorithm, which uses the Distributed Best First Search (DBFS) exploration strategy for solving the 1|r _i |∑t _i scheduling problem, a single machine scheduling problem where the objective is to find a schedule with the minimum total tardiness. Memory-based dominance strategies are incorporated into the BB&R algorithm. In addition, a modified memory-based dynamic programming algorithm is also introduced to efficiently compute lower bounds for the 1|r _i |∑t _i scheduling problem. Computational results are reported, which shows that the BB&R algorithm with the DBFS exploration strategy outperforms the best known algorithms reported in the literature.     

Improving a Lagrangian decomposition for the unconstrained binary quadratic programming problem

This paper presents a new alternative of Lagrangian decomposition based on column generation technique to solve the unconstrained binary quadratic programming problem. We use a mixed binary linear version of the original quadratic problem with constraints represented by a graph. This graph is partitioned into clusters of vertices forming subproblems whose solutions use the dual variables obtained by a coordinator problem. Computational experiments consider a set of difficult instances and the results are compared against other methods reported recently in the literature.     

Exact and parallel metaheuristic algorithms for the single processor total weighted completion time scheduling problem with the sum-of-processing-time based models

In this paper, the single processor scheduling problem to minimize the total weighted completion times is analysed, where the processing times of jobs are described by functions dependent on the sum of the normal processing times of previously processed jobs, which can model learning or aging (deteriorating) effects. We construct the exact pseudopolynomial time algorithm based on the dynamic programming, which solves the problem, where the processing time of each job is described by an arbitrary stepwise function. Moreover, the parallel metaheuristic algorithms are provided for the general version of the problem with arbitrary sum-of-processing time based models. The efficiency of the proposed algorithms is evaluated during numerical analysis.     

Performance bound analysis of a heuristic for the total weighted flowtime problem with fixed delivery dates

We consider single-machine scheduling with fixed delivery dates, which are given or determined before the jobs are processed. A job is delivered on the earliest fixed delivery date that is no earlier than its completion time. The flowtime of a job is defined as its delivery date. The objective is to minimize the total weighted flowtime of the jobs. The largest ratio first (LRF) rule is a heuristic that sequences jobs in nonincreasing order of w_j/p_j, where p_j and w_j are the processing time and weight of job J_j, respectively. We investigate the performance bounds of the LRF heuristic under different scenarios of the problem. We conducted computational experiments to test the performance of the heuristic. The results show that the LRF heuristic is able to produce near-optimal and optimal solutions.     

A global shooting algorithm for the facility location and capacity acquisition problem on a line with dense demand

This paper describes the development of an exact allocation-based solution algorithm for the facility location and capacity acquisition problem (LCAP) on a line with dense demand data. Initially, the n-facility problem on a line is studied and formulated as a dynamic programming model in the allocation decision space. Next, we cast this dynamic programming formulation as a two-point boundary value problem and provide conditions for the existence and uniqueness of solutions. We derive sufficient conditions for non-empty service regions and necessary conditions for interior facility locations. We develop an efficient exact shooting algorithm to solve the problem as an initial value problem and illustrate on an example. A computational study is conducted to study the effect of demand density and other problem parameters on the solutions.     

Improving the efficiency of a mixed integer linear programming based approach for multi-class classification problem

Data classification is one of the fundamental issues in data mining and machine learning. A great deal of effort has been done for reducing the time required to learn a classification model. In this research, a new model and algorithm is proposed to improve the work of Xu and Papageorgiou (2009). Computational comparisons on real and simulated patterns with different characteristics (including dimension, high overlap or heterogeneity in the attributes) confirm that, the improved method considerably reduces the training time in comparison to the primary model, whereas it generally maintains the accuracy. Particularly, this speed-increase is significant in the case of high overlap. In addition, the rate of increase in training time of the proposed model is much less than that of the primary model, as the set-size or the number of overlapping samples is increased.     

High-order scheme for determination of a control parameter in an inverse problem from the over-specified data

The problem of finding the solution of partial differential equations with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, thermo-elasticity, chemical diffusion and control theory. In this paper we present a high order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximate the spatial derivative with a fourth order compact scheme and reduce the problem to a system of ordinary differential equations (ODEs). Then we apply a fourth order boundary value method for the solution of resulting system of ODEs. So the proposed method has fourth order accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. Several numerical examples and also some comparisons with other methods in the literature will be investigated to confirm the efficiency of the new procedure.