Approximating Bin Packing within O(log OPT * log log OPT) bins

For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT + O(log^2 OPT) bins. We provide the first improvement in now 3 decades and show that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory. The result is constructive via algorithms of Bansal and Lovett-Meka

Prizing on Paths: A PTAS for the Highway Problem

In the highway problem, we are given an n-edge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NP-hard only recently [Elbassioni,Raman,Ray-'09]. The best-known approximation is O(\log n/\log\log n) [Gamzu,Segev-'10], which improves on the previous-best O(\log n) approximation [Balcan,Blum-'06]. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. We demonstrate the versatility of our technique by presenting PTASs for two variants of the highway problem: the tollbooth problem with a constant number of leaves and the maximum-feasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]

Polynomiality for Bin Packing with a Constant Number of Item Types

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant

Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes P⊆R[superscript n] that do not admit a compact LP formulation. More precisely we show that for every n there is a set X⊆{0,1}[superscript n] such that conv(X) must have extension complexity at least 2[superscript n/2⋅(1−o(1)] . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on NP⊈P[subscript /poly], our result rules out the existence of a compact formulation for any NP -hard optimization problem even if the formulation may contain arbitrary real numbers