Tight Approximation Algorithms for Maximum Separable Assignment Problems

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, f[subscript ij], for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)[superscript 1] and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give i. A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm. ii. A simple polynomial-time local search (β/(β + 1) − ε)-approximation algorithm, for any ε > 0. Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 − 1/e − ε)-approximation and a local search algorithm with (½ - ε)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a ½-approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1/e unless NP ⊆ DTIME(n[superscript O(log log n)]), even if there exists a polynomial-time exact algorithm for the single-bin problem. We extend the (1 − 1/e)-approximation algorithm to a constant-factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a ½ - ε approximation algorithm for the maximum k-median problem with hard capacities.National Science Foundation (U.S.) (Contract CCF-0728869)National Science Foundation (U.S.) (Contract CCF-0829878)United States. Office of Naval Research (Grant N00014-11-1-0053

Length-constrained Path-matchings in Graphs

The path-matching problem is to nd a set of vertex or edge disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has application in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of dierent cases of this problem. In each case, we either provide a polynomial time algorithm or prove that the problem is NP-complete

A fast vision system for middle size robots in robocup

Abstract. A mobile robot should be able to analyze what it is seeing in real time rate and decide accordingly.Fast and reliable analysis of image data is one of the key points in soccer robot performance.In this paper we suggest a very fast method for object finding which uses the concept of perspective view.In our method, we introduce a set of jump points in perspective on which we search for objects.An object is estimated by a rectangle surrounding it.A vector based calculation is introduced to find the distance and angle of a robot from objects in the field.In addition we present a new color model which takes its components from different color models.The proposed method can detect all objects in each frame and their distance and angle in one scan on the 1 jump points in that frame.This process takes about of a second. 50 Our vision system uses a commercially available frame grabber and is implemented only in software.It has shown a very good performance in RoboCup competitions.