A strongly polynomial algorithm for the transportation problem

For the (linear) transportation problem withm supply nodes,n demand nodes andk feasible arcs we describe an algorithm which runs in time proportional tom logm(k + n logn) (assuming w.l.o.g.mn). The algorithm uses excess scaling. The complexity bound is a slight improvement over the bound achieved by an application of a min-cost-flow algorithm of Orlin to the transportation problem.Corresponding author. Research supported in part by grant no. I-84-095.06/88 of the German—Israeli-Foundation for Scientific Research and Development.     

The MA-ordering max-flow algorithm is not strongly polynomial for directed networks

Quite recently, Fujishige (Oper. Res. Lett. 31 (2003) 176-178) developed a weakly polynomial-time algorithm for the maximum flow problem by applying the maximum adjacency (MA) ordering technique to directed networks. In this note, we show that the algorithm is not strongly polynomial by giving a real-valued instance for which the algorithm does not terminate.     

A strongly polynomial simplex method for the linear fractional assignment problem

In this paper we show that the complexity of the simplex method for the linear fractional assignment problem (LFAP) is strongly polynomial. Although LFAP can be solved in polynomial time using various algorithms such as Newton’s method or binary search, no polynomial time bound for the simplex method for LFAP is known.     

A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables

We show that the production-transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time. The algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem: monotonicity of the objective function along certain directions, small proportion of nonlinear variables and combinatorial properties implied by transportation constraints.     

The value iteration algorithm is not strongly polynomial for discounted dynamic programming

This note provides a simple example demonstrating that, if exact computations are allowed, the number of iterations required for the value iteration algorithm to find an optimal policy for discounted dynamic programming problems may grow arbitrarily quickly with the size of the problem. In particular, the number of iterations can be exponential in the number of actions. Thus, unlike policy iterations, the value iteration algorithm is not strongly polynomial for discounted dynamic programming.     

A strongly polynomial algorithm for linear systems having a binary solution

This paper proposes a strongly polynomial algorithm which either finds a solution of a linear system Ax?=?b, 0 ≤?x?≤?1, or correctly decides that the system has no 0,1-solutions. The algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming.     

A strongly polynomial FPTAS for the symmetric quadratic knapsack problem

The symmetric quadratic knapsack problem (SQKP), which has several applications in machine scheduling, is NP-hard. An approximation scheme for this problem is known to achieve an approximation ratio of (1 + ?) for any ? > 0. To ensure a polynomial time complexity, this approximation scheme needs an input of a lower bound and an upper bound on the optimal objective value, and requires the ratio of the bounds to be bounded by a polynomial in the size of the problem instance. However, such bounds are not mentioned in any previous literature. In this paper, we present the first such bounds and develop a polynomial time algorithm to compute them. The bounds are applied, so that we have obtained for problem (SQKP) a fully polynomial time approximation scheme (FPTAS) that is also strongly polynomial time, in the sense that the running time is bounded by a polynomial only in the number of integers in the problem instance.     

A minimum residual algorithm for solving linear systems

A minimum residual algorithm for solving a large linear system (I+S)x=b, with b∈ℂ ⁿ and S∈ℂ ^n×n being readily invertible, is proposed. For this purpose Krylov subspaces are generated by applying S and S ^-1 cyclically. The iteration can be executed with every linear system once the coefficient matrix has been split into the sum of two readily invertible matrices. In case S is a translation and a rotation of a Hermitian matrix, a five term recurrence is devised. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65F10     

Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming

This paper presents a “standard form” variant of Karmarkar's algorithm for linear programming. The tecniques of using duality and cutting objective are combined in this variant to maintain polynomial-time complexity and to bypass the difficulties found in Karmarkar's original algorithm. The variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value. Some computational results are also reported.     

A new strongly polynomial dual network simplex algorithm

This paper presents a new dual network simplex algorithm for the minimum cost network flow problem. The algorithm works directly on the original capacitated network and runs in O(mn(m +n logn) logn) time for the network withn nodes andm arcs. This complexity is better than the complexity of Orlin, Plotkin and Tardos’ (1993) dual network simplex algorithm by a factor ofm/n.     

Modified policy iteration algorithms are not strongly polynomial for discounted dynamic programming

This note shows that the number of arithmetic operations required by any member of a broad class of optimistic policy iteration algorithms to solve a deterministic discounted dynamic programming problem with three states and four actions may grow arbitrarily. Therefore any such algorithm is not strongly polynomial. In particular, the modified policy iteration and $\lambda$-policy iteration algorithms are not strongly polynomial.     

A station-oriented enumerative algorithm for two-sided assembly line balancing

In this paper, a station-oriented enumerative algorithm for two-sided assembly lines balancing (TALB) is proposed. First, the time transfer function is defined and combined with the precedence relation to compute the earliest and the latest start time of tasks. With the direction and cycle time constraints, a station-oriented procedure based on the start time is designed to assign tasks, starting from the left station to the rightstation of the position. Some unsuitable position assignments would be finally removed by checking the precedence constraints among the assigned tasks. The proposed algorithm is integrated with the Hoffmann heuristic to develop a system for solving TALB problems. The test is performed on the well-known benchmark set of problem instances. Experimental results demonstrate that the proposed procedure is efficient.     

An improvement over the GVW algorithm for inhomogeneous polynomial systems

The GVW algorithm provides a new framework for computing Gröbner bases efficiently. If the input system is not homogeneous, some J-pairs with larger signatures but lower degrees may be rejected by GVW's criteria, and instead, GVW has to compute some J-pairs with smaller signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher, and hence, the total computations become more expensive. This phenomenon happens more frequently when the coefficient field is a finite field and the field polynomials are involved in the computations. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed. The concept of mutant pairs is introduced to overcome the inconveniences brought by inhomogeneous inputs. In aspects of implementations, to obtain efficient implementations of GVW/M-GVW over boolean polynomial rings, we take advantages of the famous library M4RI. We propose a new rotating swap method of adapting efficient routines in M4RI to deal with the one-direction reductions in GVW/M-GVW. Our implementations are tested with many examples from Boolean polynomial rings, and the timings show M-GVW usually performs much better than the original GVW algorithm if mutant pairs are found.     

A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems

We propose a new and more stable variant of the CGS method [27] for solving nonsymmetric linear systems. The method is based on squaring the Composite Step BCG method, introduced recently by Bank and Chan [1,2], which itself is a stabilized variant of BCG in that it skips over steps for which the BCG iterate is not defined and causes one kind of breakdown in BCG. By doing this, we obtain a method (Composite Step CGS or CSCGS) which not only handles the breakdowns described above, but does so with the advantages of CGS, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new method does produce improved performance over CGS on practical problems.Partially supported by the Office of Naval Research grant N00014-92-J-1890, the National Science Foundation grant ASC92-01266, and the Army Research Office grant DAAL03-91-G-150.     

A parallel algorithm for solving Toeplitz linear systems

Numerical methods of solution are considered for systems which are Toeplitz and symmetric. In our case, the coefficient matrix is essentially tridiagonal and sparse. There are two distinct approaches to be considered each of which is efficient in its own way. Here we will combine the two approaches which will allow application of the cyclic reduction method to coefficient matrices of more general forms. The convergence of the approximations to the exact solution will also be examined. Solving linear systems by the adapted cyclic reduction method can be parallel processed.     

A successful algorithm for solving directed Hamiltonian path problems

In this paper we present a graph-theoretic polynomial algorithm which has positive probability of finding a Hamiltonian path in a given graph, if there is one; if the algorithm fails, it can be rerun with a randomly chosen starting solution, and there is again a positive probability it will find an answer. If there is no Hamiltonian path, the algorithm will always terminate with failure. We call this a Successful Algorithm because it has high (close to 1) empirical probability of success and it works in polynomial time. Some basic theoretical results concerning spanning arborescences of a graph are given. The concept of a ramification index is defined and it is shown that ramification index of a Hamiltonian path is zero. The algorithm starts with finding any spanning arborescence and by suitable pivots it endeavors to reduce the ramification index to zero. Probabilistic properties of the algorithm are discussed. Computational experience with graphs up to 30 000 nodes is included.     

On strongly polynomial dual simplex algorithms for the maximum flow problem

Several pivot rules for the dual network simplex algorithm that enable it to solve a maximum flow problem on ann-node,m-arc network in at most 2nm pivots and O(n ² m) time are presented. These rules are based on the concept of apreflow and depend upon the use of node labels which are either the lengths of a shortestpseudoaugmenting path from those nodes to the sink node orvalid underestimates of those lengths. Extended versions of our algorithms are shown to solve an important class of parametric maximum flow problems with no increase in the worst-case pivot and time bounds of these algorithms. This research was supported in part by NSF Grants DMS 91-06195, DMS 94-14438, and CDR 84-21402 and DOE Grant DE-FG02-92ER25126.     

A genetic algorithm for solving linear fractional bilevel problems

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. In this paper a genetic algorithm is proposed for the class of bilevel problems in which both level objective functions are linear fractional and the common constraint region is a bounded polyhedron. The algorithm associates chromosomes with extreme points of the polyhedron and searches for a feasible solution close to the optimal solution by proposing efficient crossover and mutation procedures. The computational study shows a good performance of the algorithm, both in terms of solution quality and computational time.     

An extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex

We present an extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex. It is shown that in at most O(nL) steps, the algorithm produces a feasible point or proves that the problem has no solution. The complexity is O(n ² m ² L) arithmetic operations. The algorithm is endowed with two new powerful stopping criteria.