Nir Halman

Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems

AbstractWe show that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LP-type problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs. To the best of our knowledge, the LP-type framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LP-type framework in other fields, and for achieving subexponential algorithms.

A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

The single-item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon. In this paper, we prove that this problem is intractable and design for it a fully polynomial-time approximation scheme.

One-way and round-trip center location problems

In the classical p-center problem there is a set V of points (customers) in some metric space, and the objective is to locate p centers (servers), minimizing the maximum distance between a customer and his respective nearest server. In this paper we consider an extension, where each customer is associated with a set of existing depots or distribution stations he can use. The service of a customer consists of the travel of a server to some permissible depot, loading of some package (e.g., a spare part) at the depot, and the delivery of the package to the customer. This model is called the customer one-way problem. In the round-trip version of the problem, the service also includes the travel from the customer to the home base of the server. In both problems the customer cost of the service is a linear function of the distance travelled by the server. The objective is to locate p servers, minimizing the maximum customer cost (weighted distance travelled by the respective server). Since the classical p-center problem is NP-hard, so are the one-way and the round-trip models we study. We present efficient constant factor approximation algorithms for these problems on general networks. Turning to special networks, we prove that the one-way problem is strongly NP-hard even on path networks. We then present polynomial time algorithms for the round-trip problem on general tree networks. We also discuss the single center case, and provide polynomial time algorithms for general networks, tree networks and planar Euclidean and rectilinear metric spaces.

The convex dimension of a graph

The convex dimension of a graph G=(V,E) is the smallest dimension d for which G admits an injective map f:V@?R^d of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this paper we study the convex and strong convex dimensions of graphs.

Fully polynomial-time approximation schemes for time-cost tradeoff problems in series-parallel project networks

We consider the deadline problem and budget problem of the nonlinear time-cost tradeoff project scheduling model in a series-parallel activity network. We develop fully polynomial-time approximation schemes for both problems using K-approximation sets and functions, together with series and parallel reductions.

On the power of discrete and of lexicographic Helly-type theorems

Hellys theorem says that if every d + 1 elements of a given finite set of convex objects in /spl Ropf//sup d/ have a common point, then there is a point common to all of the objects in the set. We define three types of Helly theorems: discrete Helly theorems - where the common point should belong to an a-priori given set, lexicographic Helly theorems - where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We show the relations between these Helly theorems and their corresponding (standard) Helly theorems. We obtain several discrete and lexicographic Helly numbers. Using these types of Helly theorems we get linear time solutions for various optimization problems. For this, we define a framework, DLP-type (discrete linear programming type), and provide algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. We show that the complexity of the DLP-type class stands somewhere between linear programming (LP) and integer programming (IP). Finally, we use our results in order to solve in randomized linear time problems such as the discrete p-center on the real line, the discrete weighted 1-center problem in /spl Ropf//sup d/ with l/sub /spl infin// norm, the standard (continuous) problem of finding a line transversal for a totally separable set of planar convex objects, a discrete version of the problem of finding a line transversal for a set of axis-parallel planar rectangles, and the (planar) lexicographic rectilinear p-center problem for p = 1,2,3. These are the first known linear time algorithms for these problems.

A linear time algorithm for the weighted lexicographic rectilinear 1-center problem in the plane

We present an optimal O(n) time algorithm for the weighted lexicographic rectilinear 1-center problem in the plane and prove that calculating the optimal value of the objective function requires Θ(n log n) time.

A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy

Given n elements with nonnegative integer weights w = ( w 1 , ź , w n ) , an integer capacity C and positive integer ranges u = ( u 1 , ź , u n ) , we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error ź in time polynomial in n, log ź U and 1 / ź , where U = max i ź u i . More precisely, our algorithm runs in O ( n 3 log 2 ź U ź log ź n log ź U ź ) time. This is an improvement of n 2 and 1 / ź (up to log terms) over the best known deterministic algorithm by Gopalan et al. (2011) 5. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.

On the Algorithmic Aspects of Discrete and Lexicographic Helly-Type Theorems and the Discrete LP-Type Model

Hellys theorem says that, if every

A Deterministic Fully Polynomial Time Approximation Scheme For Counting Integer Knapsack Solutions Made Easy

d+1