Manru Zong, Yin Tat Lee, Man-Chung Yue
Short-step methods are an important class of algorithms for solving convex constrained optimization problems. In this short paper, we show that under very mild assumptions on the self-concordant barrier and the width of the $\ell_2$-neighbourhood, any short-step interior-point method is not strongly polynomial-time.
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This paper begins with a class of convex quadratic programs (QPs) with bounded variables solvable by the parametric principal pivoting algorithm with $\mathcal{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of variables of the problem. Extension of the Hessian class is also discussed. Our research is motivated by a recent reference [7] wherein the efficient solution of a quadratic program with a tridiagonal Hessian matrix in the quadratic objective is needed for the construction of a polynomial-time algorithm for solving an associated sparse variable selection problem. With t…
It is well known that the most challenging question in optimization and discrete geometry is whether there is a strongly polynomial time simplex algorithm for linear programs (LPs). This paper gives a positive answer to this question by using the parameter analysis technique presented by us (arXiv:2006.08104). We show that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem.
Recent results of Ye and Hansen, Miltersen and Zwick show that policy iteration for one or two player (perfect information) zero-sum stochastic games, restricted to instances with a fixed discount rate, is strongly polynomial. We show that policy iteration for mean-payoff zero-sum stochastic games is also strongly polynomial when restricted to instances with bounded first mean return time to a given state. The proof is based on methods of nonlinear Perron-Frobenius theory, allowing us to reduce the mean-payoff problem to a discounted problem with state dependent discount rate. Our analysis als…
We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our te…
Discrete Wasserstein barycenters correspond to optimal solutions of transportation problems for a set of probability measures with finite support. Discrete barycenters are measures with finite support themselves and exhibit two favorable properties: there always exists one with a provably sparse support, and any optimal transport to the input measures is non-mass splitting. It is open whether a discrete barycenter can be computed in polynomial time. It is possible to find an exact barycenter through linear programming, but these programs may scale exponentially. In this paper, we prove that th…
The existence of strongly polynomial-time algorithm for linear programming is a cross-century international mathematical problem, whose breakthrough will solve a major theoretical crisis for the development of artificial intelligence. In order to make it happen, this paper proposes three solving techniques based on the cone-cutting theory: 1. The principle of Highest Selection; 2. The algorithm of column elimination, which is more convenient and effective than the Ye-column elimination theorem; 3. A step-down algorithm for a feasible point horizontally shifts to the center and then falls down …
A compact implementation of a recently proposed strongly polynomial-time algorithm for the general LP problem
本文提出一种紧凑实现，通过半尺寸矩阵内的互补Gauss-Jordan旋转操作，实现通用线性规划问题的强多项式时间算法。
A numerical illustration of a recently proposed strongly polynomial-time algorithm for the general linear programming problem
本文通过数值示例展示了近期提出的通用线性规划问题的强多项式时间算法，该算法每次迭代包含两次Gauss-Jordan旋转操作，最多在2(k+n)次迭代内终止。
A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.
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.
Validation of a recently proposed strongly polynomial-time algorithm for the general linear programming problem
本文验证了一种最近提出的强多项式时间算法，该算法通过互补高斯-若尔当枢轴操作求解一般线性规划问题，并证明其迭代次数不超过2(k+n)。
We investigate the duality relation between linear programs over grids (Grid-LPs) and generalized linear complementarity problems (GLCPs) with hidden K-matrices. The two problems, moreover, share their combinatorial structure with discounted Markov decision processes (MDPs). Through proposing reduction schemes for the GLCP, we obtain a strongly polynomial reduction from Grid-LPs to linear programs over cubes (Cube-LPs). As an application, we obtain a scheme to reduce discounted MDPs to their binary counterparts. This result also suggests that Cube-LPs are the key problems with respect to solva…
We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [Végh16]. For the uncapacitated problem formulation, the complexity bound $O(mn(m+n\log n)\log (n^2/m))$ improves on the previous estimate by almost a factor $O(n^2)$. Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in …
Circuit Diameter of Polyhedra is Strongly Polynomial
该研究证明了多面体的电路直径存在强多项式界，解决了多项式Hirsch猜想，并为线性规划提供了强多项式时间算法的可能途径。
Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes
本文提出鲁棒马尔可夫决策过程的线性规划表示，通过编码鲁棒策略迭代步骤构造单一LP，并给出强多项式算法及改进的复杂度界。
A strongly polynomial-time algorithm for the general linear programming problem
本文提出了一种强多项式时间算法，通过隐含的约简过程解决一般线性规划问题，利用互补关系和非负变量构建线性方程组，每迭代步应用互补的高斯-约旦 pivoting 操作，最多需 2(k+n) 次迭代。
In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification and Mizuno's use of the simplex method, we introduce a modification of Tardos' algorithm considering only the primal problem and using simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate.
It is an open question to determine if the theory of self-concordant barriers can provide an interior point method with strongly polynomial complexity in linear programming. In the special case of the logarithmic barrier, it was shown in [Allamigeon, Benchimol, Gaubert and Joswig, SIAM J. on Applied Algebra and Geometry, 2018] that the answer is negative. In this paper, we show that none of the self-concordant barrier interior point methods is strongly polynomial. This result is obtained by establishing that, on parametric families of convex optimization problems, the log-limit of the central …
We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $Ω(2^r)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized familie…
Policy Iteration Is Not Strongly Polynomial for Deterministic Markov Decision Processes: The Price of Algorithmic Anarchy
该研究证明霍华德策略迭代在确定性折扣MDP上具有指数迭代下界，排除其强多项式性，并与单纯形法形成指数分离，揭示算法无政府状态的价格。
The theory of $n$-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an $n$-fold integer program (IP) consists of parameter $A$, dimension $n$, and numerical data of binary encoding length $L$. It was known for some time that such programs can be solved in polynomial time using $O(n^{g(A)}L)$ arithmetic operations where $g$ is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable (FPT) time using $O(f(A)n^3L)$ arithmetic operations for a single-exponential function $f$. T…
We consider integer and linear programming problems for which the linear constraints exhibit a (recursive) block-structure: The problem decomposes into independent and efficiently solvable sub-problems if a small number of constraints is deleted. A prominent example are $n$-fold integer programming problems and their generalizations which have received considerable attention in the recent literature. The previously known algorithms for these problems are based on the augmentation framework, a tailored integer programming variant of local search. In this paper we propose a different approach. O…
We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound $O(({n}^4\cdot \textrm{EO} + {…
Strongly Polynomial Frame Scaling to High Precision
本文提出一种强多项式确定性算法，以对数收敛速度解决帧缩放问题，将迭代次数从O(n^5 d^11/ε^5)降至O(n^3 log(n/ε))，并改进矩阵缩放的迭代界，同时给出解大小的新界。
Strongly-polynomial time and validation analysis of policy gradient methods
本文提出优势间隙函数作为终止准则，证明策略梯度方法能在强多项式时间内解决MDP，并提供验证方法。
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