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Register a new userImproved Approximation Algorithms for Inventory Problems
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Added by
Neil Olver
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of $N$ items and a discrete time horizon of $T$ days in which given demands for the items must be satisfied. Ordering a set of items incurs a cost according to a set function, with properties depending on the problem under consideration. Demand for an item at time $t$ can be satisfied by an order on any day prior to $t$, but a holding cost is charged for storing the items during the intermediate period; the goal is to minimize the sum of the ordering and holding cost. Our approximation factor for both problems is $O(log log min(N,T))$; this improves exponentially on the previous best results.
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In the relay placement problem the input is a set of sensors and a number $r ge 1$, the communication range of a relay. In the one-tier version of the problem the objective is to place a minimum number of relays so that between every pair of sensors there is a path through sensors and/or relays such that the consecutive vertices of the path are within distance $r$ if both vertices are relays and within distance 1 otherwise. The two-tier version adds the restrictions that the path must go through relays, and not through sensors. We present a 3.11-approximation algorithm for the one-tier version and a PTAS for the two-tier version. We also show that the one-tier version admits no PTAS, assuming P $ e$ NP.
The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any non-trivial factor, and little progress has been made despite its importance in modeling activity allocation under uncertainty. We consider a special case that we call Feedback MAB, where the reward obtained by playing each of n independent arms varies according to an underlying on/off Markov process whose exact state is only revealed when the arm is played. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is also an instance of a Partially Observable Markov Decision Process (POMDP), and is widely studied in wireless scheduling and unmanned aerial vehicle (UAV) routing. Unlike the stochastic MAB problem, the Feedback MAB problem does not admit to greedy index-based optimal policies. We develop a novel and general duality-based algorithmic technique that yields a surprisingly simple and intuitive 2+epsilon-approximate greedy policy to this problem. We then define a general sub-class of restless bandit problems that we term Monotone bandits, for which our policy is a 2-approximation. Our technique is robust enough to handle generalizations of these problems to incorporate various side-constraints such as blocking plays and switching costs. This technique is also of independent interest for other restless bandit problems. By presenting the first (and efficient) O(1) approximations for non-trivial instances of restless bandits as well as of POMDPs, our work initiates the study of approximation algorithms in both these contexts.
We consider the $k$-clustering problem with $ell_p$-norm cost, which includes $k$-median, $k$-means and $k$-center cost functions, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points $P$ of size $n$, a set of $k$ centers induces a fair clustering if for every point $vin P$, $v$ can find a center among its $n/k$ closest neighbors. Recently, Mahabadi and Vakilian [2020] showed how to get a $(p^{O(p)},7)$-bicriteria approximation for the problem of fair $k$-clustering with $ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $ell_p$-norm cost of the solution is at most $p^{O(p)}$ times the cost of an optimal fair solution. In this work, for any $varepsilon>0$, we present an improved $(16^p +varepsilon,3)$-bicriteria approximation for the fair $k$-clustering with $ell_p$-norm cost. To achieve our guarantees, we extend the framework of [Charikar et al., 2002, Swamy, 2016] and devise a $16^p$-approximation algorithm for the facility location with $ell_p$-norm cost under matroid constraint which might be of an independent interest. Besides, our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by Kleindessner et al. [2019], which is essentially the median matroid problem [Krishnaswamy et al., 2011].
In the Priority Steiner Tree (PST) problem, we are given an undirected graph $G=(V,E)$ with a source $s in V$ and terminals $T subseteq V setminus {s}$, where each terminal $v in T$ requires a nonnegative priority $P(v)$. The goal is to compute a minimum weight Steiner tree containing edges of varying rates such that the path from $s$ to each terminal $v$ consists of edges of rate greater than or equal to $P(v)$. The PST problem with $k$ priorities admits a $min{2 ln |T| + 2, krho}$-approximation [Charikar et al., 2004], and is hard to approximate with ratio $c log log n$ for some constant $c$ [Chuzhoy et al., 2008]. In this paper, we first strengthen the analysis provided by [Charikar et al., 2004] for the $(2 ln |T| + 2)$-approximation to show an approximation ratio of $lceil log_2 |T| rceil + 1 le 1.443 ln |T| + 2$, then provide a very simple, parallelizable algorithm which achieves the same approximation ratio. We then consider a more difficult node-weighted version of the PST problem, and provide a $(2 ln |T|+2)$-approximation using extensions of the spider decomposition by [Klein & Ravi, 1995]. This is the first result for the PST problem in node-weighted graphs. Moreover, the approximation ratios for all above algorithms are tight.
In this paper, we consider several finite-horizon Bayesian multi-armed bandit problems with side constraints which are computationally intractable (NP-Hard) and for which no optimal (or near optimal) algorithms are known to exist with sub-exponential running time. All of these problems violate the standard exchange property, which assumes that the reward from the play of an arm is not contingent upon when the arm is played. Not only are index policies suboptimal in these contexts, there has been little analysis of such policies in these problem settings. We show that if we consider near-optimal policies, in the sense of approximation algorithms, then there exists (near) index policies. Conceptually, if we can find policies that satisfy an approximate version of the exchange property, namely, that the reward from the play of an arm depends on when the arm is played to within a constant factor, then we have an avenue towards solving these problems. However such an approximate version of the idling bandit property does not hold on a per-play basis and are shown to hold in a global sense. Clearly, such a property is not necessarily true of arbitrary single arm policies and finding such single arm policies is nontrivial. We show that by restricting the state spaces of arms we can find single arm policies and that these single arm policies can be combined into global (near) index policies where the approximate version of the exchange property is true in expectation. The number of different bandit problems that can be addressed by this technique already demonstrate its wide applicability.
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