A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max cx, Ax = b, x ≥ 0, A g m × n, Vavasis and Ye developed a primal-dual interior point method using a g€layered least squares' (LLS) step, and showed that O(n3.5 log(χA+n)) iterations suffice to solve (LP) exactly, where χA is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure χA∗, defined as the minimum χAD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an O(m2 n2 + n3) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying χAD ≤ n(χ∗)3. This algorithm also allows us to approximate the value of χA up to a factor n (χ∗)2. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NP-hardness for approximating χA to within 2poly(rank(A)). The key insight for our algorithm is to work with ratios gi/gj of circuits of A - i.e., minimal linear dependencies Ag=0 - which allow us to approximate the value of χA∗ by a maximum geometric mean cycle computation in what we call the g€circuit ratio digraph' of A. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n2.5 lognlog(χA∗+n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/logn improvement on the iteration complexity bound of the original Vavasis-Ye algorithm

An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms

This paper presents a new exponential lower bound for the two most popular deterministic variants of the strategy improvement algorithms for solving parity, mean payoff, discounted payoff and simple stochastic games. The first variant improves every node in each step maximizing the current valuation locally, whereas the second variant computes the globally optimal improvement in each step. We outline families of games on which both variants require exponentially many strategy iterations

Study on simplified model for estimating evaporation from reservoirs

In this study, the Linacre evaporation model was tested for its accuracy using daily, weekly and monthly records. The records were collected from class A evaporation pan installed at Algardabiya Reservoir, Sirt, Libya. The records for three years were used to calibrate and validate the model. Statistical tests show that the model gives a reasonable accuracy. The errors in the model prediction are 5.8%, 8% and 8.5% for weekly, monthly and daily prediction respectively. Thus, the Linacre model can be used when the available meteorological data is limited (air temperature only) and for all types of record such as daily, weekly and monthly

Polyhedra Circuits and Their Applications

To better compute the volume and count the lattice points in geometric objects, we propose polyhedral circuits. Each polyhedral circuit characterizes a geometric region in Rd . They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedron. They can be also used to approximate a large class of d-dimensional manifolds in Rd . Barvinok [3] developed polynomial time algorithms to compute the volume of a rational polyhedron, and to count the number of lattice points in a rational polyhedron in Rd with a fixed dimensional number d. Let d be a fixed dimensional number, TV(d,n) be polynomial time in n to compute the volume of a rational polyhedron, TL(d,n) be polynomial time in n to count the number of lattice points in a rational polyhedron, where n is the total number of linear inequalities from input polyhedra, and TI(d,n) be polynomial time in n to solve integer linear programming problem with n be the total number of input linear inequalities. We develop algorithms to count the number of lattice points in geometric region determined by a polyhedral circuit in O(nd⋅rd(n)⋅TV(d,n)) time and to compute the volume of geometric region determined by a polyhedral circuit in O(n⋅rd(n)⋅TI(d,n)+rd(n)TL(d,n)) time, where rd(n) is the maximum number of atomic regions that n hyperplanes partition Rd . The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra (1−β) -lattice set cover problem, and (1−β) -continuous polyhedra set cover problem are discussed. We also show the NP-hardness of the geometric version of maximum coverage problem and set cover problem when each set is represented as union of polyhedra

On the Complexity of Semidefinite Programs

We show that the feasibility of a system of m linear inequalities over the cone of symmetric positive semidefinite matrices of order n can be tested in mn O(minfm;n 2 g) arithmetic operations with ln O(minfm;n 2 g) -bit numbers, where l is the maximum binary size of the input coefficients. We also show that any feasible system of dimension (m; n) has a solution X such that log kXk ln O(minfm;n 2 g) . 1 Introduction This paper is concerned with the general semidefinite feasibility problem (F) : Given integral n \Theta n symmetric matrices A 1 ; : : : ; Am and integers b 1 ; : : : ; b m , determine whether there exists a real n \Theta n symmetric matrix X such that A i ffl X b i ; i = 1; : : : ; m; X 0; (1) where A ffl X = tr(AX) denotes the standard inner product on the space of real symmetric matrices and the notation (\Delta) 0 indicates that (\Delta) is a symmetric positive semidefinite matrix. We also consider the following (polynomially equivalent) problem ..

Computing Integral Points in Convex Semi-algebraic Sets

Let Y be a convex set in IR k defined by polynomial inequalities and equations of degree at most d 2 with integer coefficients of binary length l. We show that if Y " ZZ k 6= ;, then Y contains an integral point of binary length ld O(k 4 ) . For fixed k, our bound implies a polynomial-time algorithm for computing an integral point y 2 Y . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming

Computing Integral Points in Convex Semi-algebraic Sets

Let Y be a convex set in IR k defined by polynomial inequalities and equations of degree at most d 2 with integer coefficients of binary length l. We show that if Y " ZZ k 6= ;, then Y contains an integral point of binary length ld O(k 4 ) . For fixed k, our bound implies a polynomial-time algorithm for computing an integral point y 2 Y . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming

Linear scheduling is close to optimality

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Linear scheduling is nearly optimal

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