Efficient Algorithm for Solving Hyperbolic Programs

Hyperbolic polynomials is a class of real-roots polynomials that has wide range of applications in theoretical computer science. Each hyperbolic polynomial also induces a hyperbolic cone that is of particular interest in optimization due to its generality, as by choosing the polynomial properly, one can easily recover the classic optimization problems such as linear programming and semidefinite programming. In this work, we develop efficient algorithms for hyperbolic programming, the problem in each one wants to minimize a linear objective, under a system of linear constraints and the solution must be in the hyperbolic cone induced by the hyperbolic polynomial. Our algorithm is an instance of interior point method (IPM) that, instead of following the central path, it follows the central Swath, which is a generalization of central path. To implement the IPM efficiently, we utilize a relaxation of the hyperbolic program to a quadratic program, coupled with the first four moments of the hyperbolic eigenvalues that are crucial to update the optimization direction. We further show that, given an evaluation oracle of the polynomial, our algorithm only requires $O(n^2d^{2.5})$ oracle calls, where $n$ is the number of variables and $d$ is the degree of the polynomial, with extra $O((n+m)^3 d^{0.5})$ arithmetic operations, where $m$ is the number of constraints

Shared Integer Dichotomy

The Integer Dichotomy Diagram IDD(n) represents a natural number n ∈ N by a Directed Acyclic Graph in which equal nodes are shared to reduce the size s(n). That IDD also represents some finite set of integers by a Digital Search DAG where equal subsets are shared. The same IDD also represents representing Boolean Functions, IDDs are equivalent to (Zero-suppressed) ZDD or to (Binary Moment) BMD Decision Diagrams. The IDD data-structure and algorithms combines three standard software packages into one: arithmetics, sets and Boolean functions. Unlike the binary length l(n), the IDD size s(n) < l(n) is not monotone in n. Most integers are dense, and s(n) is near l(n). Yet, the IDD size of sparse integers can be arbitrarily smaller. We show that a single IDD software package combines many features from the best known specialized packages for operating on integers, sets, Boolean functions, and more. Over dense structures, the time/space complexity of IDD operations is proportional to that of its specialized competitors. Yet equality testing is performed in unit time with IDDs, and the complexity of some integer operations (e.g. n < m, n ± 2 m , 2 2 n ,. . .) is exponentially lower than with bit-arrays. In general, the IDD is best in class over sparse structures, where both the space and time complexities can be arbitrarily lower than those of un-shared representations. We show that sparseness is preserved by most integer operations, including arithmetic and logic operations, but excluding multiplication and division. Keywords: computer arithmetic, integer dichotomy & trichotomy, sparse & dense structures , dictionary package, digital search tree, minimal acyclic automata, binary Trie, boolean function, decision diagram, store/compute/code once.

Faster Approximate String Matching for Short Patterns

We study the classical approximate string matching problem, that is, given strings $P$ and $Q$ and an error threshold $k$, find all ending positions of substrings of $Q$ whose edit distance to $P$ is at most $k$. Let $P$ and $Q$ have lengths $m$ and $n$, respectively. On a standard unit-cost word RAM with word size $w \geq \log n$ we present an algorithm using time $$O(nk \cdot \min(\frac{\log^2 m}{\log n},\frac{\log^2 m\log w}{w}) + n)$$ When $P$ is short, namely, $m = 2^{o(\sqrt{\log n})}$ or $m = 2^{o(\sqrt{w/\log w})}$ this improves the previously best known time bounds for the problem. The result is achieved using a novel implementation of the Landau-Vishkin algorithm based on tabulation and word-level parallelism.Comment: To appear in Theory of Computing System

Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings

Evaluating geometric queries using few arithmetic operations

Let \cp:=(P_1,...,P_s) be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family \cp is in a suitable sense \emph{generic}. We construct a database $\cal D$, supported by an algebraic computation tree, such that for each $x\in [0,1]^n$ the query for the signs of $P_1(x),...,P_s(x)$ can be answered using h d^{\cO(n^2)} comparisons and $nL$ arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of $\cal D$ are then employed to exhibit example classes of systems of $n$ polynomial equations in $n$ unknowns whose consistency may be checked using only few arithmetic operations, admitting however an exponential number of comparisons

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.Comment: 28 pages, 10 figure