The first part of this report describes the following result that, logarithmic approximation factor for hard capacitated set cover can be achieved from Wolsey's work [9], using a simpler and more intuitive analysis. We further show in our work, that O(log n) approximation factor can be achieved for the same problem by applying analysis of general set cover to analyze Wolsey's algorithm [5]. This work is based on the key observation that we make in Lemma 3 of this report. The second part of the report describes the geometric hitting set problem, where X is a ground set of points in a plane and S is a set of axis parallel rectangles. It is shown that epsilon-nets of size O(1/epsilon log log 1/epsilon) can be computed in polynomial time. Applying Bronnimann and Goodrich result [3] gives the hitting set of size O(log log OPT) for this problem.
We present the first near optimal approximation schemes for the maximum weighted (uncapacitated or capacitated) $b$--matching problems for non-bipartite graphs that run in time (near) linear in the number of edges. For any $δ>3/\sqrt{n}$ the algorithm produces a $(1-δ)$ approximation in $O(m \poly(δ^{-1},\log n))$ time. We provide fractional solutions for the standard linear programming formulations for these problems and subsequently also provide (near) linear time approximation schemes for rounding the fractional solutions. Through these problems as a vehicle, we also present several ideas i…
We study problems that integrate depot location decisions along with the inventory routing problem of serving clients from these depots over time balancing the costs of routing vehicles from the depots with the holding costs of demand delivered before they are due. Since the inventory routing problem is already complex, we study the version that assumes that the daily vehicle routes are direct connections from the depot thus forming stars as solutions, and call this problem the Star Inventory Routing Problem with Facility Location (SIRPFL). As a stepping stone to solving SIRPFL, we first study…
In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client's location. However, when a client who has been also one of the open facilities is to be removed, then…
Given a graph $G=(V,E)$ with non-negative real edge lengths and an integer parameter $k$, the Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The objective is to minimize the maximum length among all the trees. We give the first constant factor approximation for the hard uniform capacitated version of this problem, where, an input parameter $λ$ upper bounds the number of vertices that can be covered by any of the trees. Our result extends to the rooted version of the problem, where we are given a set of $k…
In this article, we consider the following capacitated covering problem. We are given a set $P$ of $n$ points and a set $\mathcal{B}$ of balls from some metric space, and a positive integer $U$ that represents the capacity of each of the balls in $\mathcal{B}$. We would like to compute a subset $\mathcal{B}' \subseteq \mathcal{B}$ of balls and assign each point in $P$ to some ball in $\mathcal{B}$ that contains it, such that the number of points assigned to any ball is at most $U$. The objective function that we would like to minimize is the cardinality of $\mathcal{B}$. We consider this probl…
Constant Approximation Algorithm for Non-Uniform Capacitated Multi-Item Lot-Sizing via Strong Covering Inequalities
研究了非均匀容量约束下的多产品批量计划（\lotsizing）问题。在这个问题中，在$T$时间段的规划范围内有一组需求，并且所有需求都必须按时满足。我们可以在每个周期$s$开始时下订单，产生订购成本$K_s$。在时间$s$订购的所有产品的总数量不能超过给定的容量$C_s$。另一方面，不时持有存货会产生存货持有成本。该问题的目标是找到一个可行的解决方案，使订购和持有成本之和最小化。Levi等人(Levi，Lodi和Sviridenko，Mathmatics of Operations Research 33（2），2008）给出了当容量$C_s$相同时该问题的2-近似。在本文中，我们将他们的结果推广到非一致容量的情况。即对具有一般能力的多品种批量问题给出了一个常数近似算法。常数逼近是通过增加一个指数大的新的覆盖不等式的自然设施位置型线性规划松弛的问题。沿着我们的算法的方式，我们减少了\lotsizing问题的经典背包覆盖问题的两个概括。我们给LP为基础的常数近似算法的推广，通过迭代舍入技术。
Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of $(\sqrt{41}-1)/4$ by Asano et al., while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 ap…
This study considers the (soft) capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph $G=(V,E)$, which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex $v$ in the cover, the number of $v$'s incident edges covered by the copy is up to a given …
In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to the balls in $\mathcal{B}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(\log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(\log |P|)$ even with $β< 3$ factor expansion of th…
In the unsplittable capacitated vehicle routing problem, we are given a metric space with a vertex called depot and a set of vertices called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the depot such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1. Our main result is a polynomial-time $(2+ε)$-approximation algorithm for this problem in the two-dimen…
Improved Approximation Algorithms for Capacitated Network Design and Flexible Graph Connectivity
Amazon, Bellevue, WA, USA（亚马逊）；Department of Combinatorics & Optimization, University of Waterloo, Canada（滑铁卢大学组合优化系）；Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA（宾夕法尼亚大学计算机与信息科学系）
The relationship between the sparsest cut and the maximum concurrent multi-flow in graphs has been studied extensively. For general graphs with $k$ terminal pairs, the flow-cut gap is $O(\log k)$, and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flow-cut gap in planar networks is $O(1)$, while the known bounds place the gap somewhere between $2$ (Lee and Raghavendra, 2003) and $O(\sqrt{\log k})$ (Rao, 1999). A seminal result of Okamura and Seymour (1981) shows that when all the …
We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem…
Obtaining strong linear relaxations of capacitated covering problems constitute a major technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities achieves an integrality gap of 2. These inequalities have been exploited in more general environments, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a new general setting in which items can be taken fractionally to cover a given demand. The cost incurred by …
We tackle the problem of accelerating column generation (CG) approaches to set cover formulations in operations research. At each iteration of CG we generate a dual solution that approximately solves the LP over all columns consisting of a subset of columns in the nascent set. We refer to this linear program (LP) as the Family Restricted Master Problem (FRMP), which provides a tighter bound on the master problem at each iteration of CG, while preserving efficient inference. For example, in the single source capacitated facility location problem (SSCFLP) the family of a column $l$ associated wi…
