Speeding up Karmarkar's algorithm for multicommodity flows

In recent years, several approaches have been investigated to obtain multiplier-free realizations of digital filters. One approach makes use of periodically time-varying (PTV) structures. The idea is to distribute the computation in time and in space. Distribution in time provides reuse of the same hardware by means of PTV coefficients. Distribution in space increases the number of coefficients but simplifies the values of the coefficients. Computation distribution is based on radix-r number representation, and it can be carried out to the extent that each computation involves a simple coefficient that can be realized using only addition and shift (no hardware multiplier). Previous PTV realizations could not exploit the coefficient symmetry of finite impulse response (FIR) filters to reduce the number of coefficients. This paper proposes design and realizations of FIR filters with punctured radix-8 coefficients belonging to the septuple set {0, ±1, ±2, ±4), which can be implemented using only a shift operation without requiring any hardware multiplier. The realizations exploit the coefficient symmetry to reduce the hardware by about one-half. Due to a non-uniform grid of representation, we apply a modified Karmarkar's linear programming algorithm to find the optimum set of discrete coefficients that minimizes the weighted peak ripple error. Comparison with a conventional FIR filter with sum-of-powers-of-two (SOPOT) coefficients shows that the proposed filter is faster and uses less hardware than one with SOPOT coefficients. However, the punctured radix-8 system has a limit on the achievable ripple.

We present a new parallel algorithm for computing a least squares solution to a sparse overdetermined system of linear equations Ax=b such that m×n matrix A is sparse and the graph, G=(V,E), of the matrix \(H = \left[ {\begin{array}{*{20}c}I & {A^T } \\A & O \\\end{array} } \right]\) has an s(m+n)-separator family, that is, either |V| <n0 for a fixed constant n0 or, by deleting a separator subset S of vertices of the size ≤ s(m+n), G can be partitioned into two disconnected subgraphs having vertex sets V1,V2 of the sizes ≤ 2/3 (m+n) and each of the two resulting subgraphs induced by the vertex sets S ∪ Vi, i=1,2, can be recursively s(|S ∪ Vi|)-separated in a similar way. Our algorithm uses O(log (m+n) log2s(m+n)) steps and ≤ s3(m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems and has several immediate applications of interest, in particular to mathematical programming, to sparse nonsymmetric systems of linear equations, and to the path algebra computations. We most closely examine the impact on the linear programming problem, l.p.p., which requires to maximize cTy subject to ATy ≤ b, y ≥ 0 where A is an m×n matrix. Hereafter it is assumed that m ≥ n. The recent algorithm by N. Karmarkar gives the best known upper estimate (O(m3.5 L) arithmetic operations where L is the input size) for the cost of the solution of this problem in the worst case. We prove an asymptotic improvement of that result in the case where the graph of the associated matrix H has an s(m+n)-separator family; then our algorithm can be implemented using O(m L log m log2s(m+n)) parallel arithmetic steps, s3(m+n) processors and a total of O(m L s3(m+n) log m log2s(m+n)) arithmetic operations. In many cases of practical importance this is a considerable improvement of the known estimates: for example, s(m+n) = √8 (m+n) if G is planar (as occurs in many operations research applications, for instance, in the problem of computing the maximum multicommodity flow with a bounded number of commodities in a network having an s(m+n)-separator family), so that the processor bound is only 8 √8 (m+n)1.5 and the total number of arithmetic steps is O(m2.5L) in that case. Similarly Karmarkar's algorithm and the known algorithms for the solution of overdetermined linear systems are accelerated in the case of dense input matrices via our recent parallel algorithms for the inversion of dense k×k matrices using O(log2k) steps, k3 processors. Combined with a modification of Karmarkar's algorithm, this implies solution of l.p.p. using O(Lm log2m) steps, m2.5 processors. The stated results promise some important practical applications. Theoretically the above processor bounds can be reduced for dense matrix inversion to o(k2.5) and for l.p.p. to o(m2.165) in the dense case and to o(s2.5(m+n)) in the sparse case (preserving the same number of parallel steps); this also decreases the sequential time bound for the l.p.p. by a factor of m0.335, that is, to O(Lm3.165).

We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $$\sqrt {m + n} $$ .

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.

Elementary Linear Programming with Applications

In this paper, a new algorithm for convex quadratic programming (QP) is presented. Firstly, the surrogate problem for QP is developed, and the Karush-Kuhn-Tucker conditions of the surrogate problem hold if the unconstrained minimum of the objective function does not satisfy any constraints. Then, Karmarkar's algorithm for linear programming (LP) is introduced to solve the surrogate dual problem. In addition, the case of general constraints is also discussed, and some examples of optimum truss sizing problems show that the proposed algorithm is robust and efficient.

This paper studies energy efficient routing for data aggregation in wireless sensor networks. Our goal is to maximize the lifetime of the network, given the energy constraint on each sensor node. Using linear programming (LP) formulation, we model this problem as a multicommodity flow problem, where a commodity represents the data generated from a sensor node and delivered to a base station. A fast approximate algorithm is presented, which is able to compute (1-e)-approximation to the optimal lifetime for any e > 0. Then along this baseline, we further study several advanced topics. First, we design an algorithm, which utilizes the unique characteristic of data aggregation, and is proved to reduce the running time of the fastest existing algorithm by a factor of K, K being the number of commodities. Second, we extend our algorithm to accommodate the same problem in the setting of multiple base stations, and study its impact on network lifetime improvement. All algorithms are evaluated through both solid theoretical analysis and extensive simulation results.

This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected O(d 2 n) + (log n)O(d) d/2+O(1) + O(d 4 √nlog n) arithmetic operations, as n→∞ . The constant factors do not depend on d. Also, an algorithm is given for integer linear programming. Let φ bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then, the algorithm requires expected O(2 d dn + 8 d d√n ln ln n) + d O(d) φ ln n operations on numbers with d O(1) φ bits, as n→∞ , where the constant factors do not depend on d or φ to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in E d has the same time bound.

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a 1-ε approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find 1-ε approximate solutions to the maximum concurrent flow problem and maximum weighted multicommodity flow problem in time O(m4/3poly(k,ε-1)).

We simplify and strengthen the analysis of the improvement obtained in one step of Karmarkar's algorithm.

The use of KORBX, a linear problem solver based on Karmarkar's algorithm, to solve military airlift problems is described. Such problems involve so many variables and constraints that using a simplex-based system is not feasible. Three specific applications are examined: optimal channel routing of cargoes, cargo distribution, and wartime patient evacuation. >

This paper considers the optimal design of high order digital differentiators in the L/sub 1/ sense. Conventionally, using L/sub 1/ error criterion for this design problem results in a nonlinear optimization problem since the corresponding objective function contains an absolute error function. We first reformulate the design problem as a linear programming problem in the frequency domain. To avoid the requirement of huge computation load and storage space when using linear programming based algorithms, we present a method based on a modification of Karmarkar's algorithm to solve the design problem so that an analytical weighted least-squares (WLS) solution formula can be obtained. This leads to a very efficient procedure for the considered design problem. Computer simulations show that the designed differentiators can achieve more accurate wideband differentiation than those designed by using L/sub 2/ and Chebyshev (minimax) error criteria.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameterα (so that we must have 0 <α ≤ 1) the best possible guaranteeδn(α) inn dimensional space is essentially ln 2 ≃ 0.69; and to achieve this we must takeα about 1. Indeed we show the precise result thatδn(α) equals ln(1 +α)-ln(1 −α/(n − 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to aboutδ* ≃ 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.

Multicommodity flows in certain planar directed networks

The minimum-cost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost. Multicommodity flow problems can be expressed as linear programs, and most theoretical and practical algorithms use linear-programming algorithms specialized for the problems'' structures. Combinatorial approximation algorithms yield flows with costs slightly larger than the minimum cost and use capacities slightly larger than the given capacities. Theoretically, the running times of these algorithms are much less than that of linear-programming-based algorithms. We combine and modify the theoretical ideas in these approximation algorithms to yield a fast, practical implementation solving the minimum-cost multicommodity flow problem. Experimentally, the algorithm solved our problem instances (to 1% accuracy) two to three orders of magnitude faster than the linear-programming package CPLEX and the linear-programming based multicommodity flow program PPRN.