Buyback Problem - Approximate matroid intersection with cancellation costs

In the buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to some constraints on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost that is a fixed fraction of the bid value. Previous to our work, deterministic and randomized algorithms were known when the constraint is a matroid constraint. We extend this and give a deterministic algorithm for the case when the constraint is an intersection of $k$ matroid constraints. We further prove a matching lower bound on the competitive ratio for this problem and extend our results to arbitrary downward closed set systems. This problem has applications to banner advertisement, semi-streaming, routing, load balancing and other problems where preemption or cancellation of previous allocations is allowed

オンラインナップサックと関連する諸問題に対するアルゴリズム論的研究

学位の種別:課程博士University of Tokyo(東京大学

Prophet Inequalities with Cancellation Costs

Most of the literature on online algorithms and sequential decision-making focuses on settings with “irrevocable decisions” where the algorithm’s decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider “prophet inequalities with recourse” in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts

Bayesian Ranking and Selection of Fishing Boat Efficiencies

The steadily accumulating literature on technical efficiency in fisheries attests to the importance of efficiency as an indicator of fleet condition and as an object of management concern. In this paper, we extend previous work by presenting a Bayesian hierarchical approach that yields both efficiency estimates and, as a byproduct of the estimation algorithm, probabilistic rankings of the relative technical efficiencies of fishing boats. The estimation algorithm is based on recent advances in Markov Chain Monte Carlo (MCMC) methods—Gibbs sampling, in particular—which have not been widely used in fisheries economics. We apply the method to a sample of 10,865 boat trips in the US Pacific hake (or whiting) fishery during 1987–2003. We uncover systematic differences between efficiency rankings based on sample mean efficiency estimates and those that exploit the full posterior distributions of boat efficiencies to estimate the probability that a given boat has the highest true mean efficiency.Ranking and selection, hierarchical composed-error model, Markov Chain Monte Carlo, Pacific hake fishery, Resource /Energy Economics and Policy, Q2, L5, C1,

A study of closed-loop supply chain models with governmental incentives and fees

A rich mixture of government incentives and fees to encourage the collection of used products and the subsequent remanufacturing has been increasingly utilized both domestically and internationally. In this paper, toward a fuller understanding of such government participation in closed-loop supply chains (CLSC\u27s), we construct and analyze a series of game-theoretic CLSC models with remanufacturing. Specifically, we investigate a basic decentralized CLSC model, two government participation models of linear incentives and fees as well as of central coordination via alternative financial instruments, and a revenue-sharing contract model without the government participation. We also analyze the impact of competition among manufacturers in our results. A key differentiating feature in our government participation models is the incorporation of the revenue neutrality requirement from a government\u27s perspective whose financial sources for such incentives must eventually reconcile with the financial sinks for such fees. By comparing and contrasting the equilibrium solutions and the economic consequences of these models, managerial insights and economic implications relevant to academics and practitioners including decision and policy makers are obtained. For example, we show how the government participation can induce an entry or prevent an exit of a CLSC when one or more members are unprofitable

Long Term Government Bonds

We study the impact of debt maturity on optimal fiscal policy by focusing on the case where the government issues a bond of maturity N > 1: Isolating these effects helps provide insight into the construction of optimal government debt portfolios. We find long bonds may not complete the market even in the absence of uncertainty, generate an incentive to twist interest rates and induce additional tax volatility compared to short term bonds. By focusing just on the issuance of long bonds we show that as well as their well known advantage in providing fiscal insurance long bonds also have less attractive features that induce additional tax volatility. In the case of long bonds, governments induce tax volatility in order to twist interest rates at maturity. This interest rate twisting effect is what makes optimal debt management models so difficult to solve computationally as the state space rapidly becomes cumbersome due to the need to keep track of promises about future tax rates. We provide an alternative institutional setup (\independent powers\) that eliminates this problem offering a simpler solution method. Introducing maturity requires making more institutional assumptions than is the case for one period bonds. In particular assumptions have to be made whether the government does or doesn't buy back each period all outstanding debt irrespective of maturity and whether long bonds pay coupons. This is important as the literature to date makes assumptions that are diametrically opposite to what is observed in practice. We show that this is an important divide as if we model optimal policy under the empirically motivated assumption that governments do not buyback bonds until maturity then long bonds induce additional tax volatility due to the existence of N period roll over cycles. These can be reduced in magnitude by the government issuing long bonds that pay coupons although because coupons reduce the duration of a bond below its maturity this does compromise the ability of long bonds to provide fiscal insurance

Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $\Omega(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes value oracle access to $f$ and membership oracles for the matroids defining the $p$-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201

Essays in Retail Operations and Humanitarian Logistics

This dissertation introduces and analyzes research problems related to Retail Operations and Humanitarian Logistics. In Retail Operations, the inventory that ends up as unsaleable at primary markets can be significant (up to 20% of the retail product). Thus retailers look for strategies like selling in secondary markets at a discounted price. In such a setting, the decisions of how much to order for a product of limited shelf life and when (if at all) to start selling the product in the secondary market become critical because these decisions not only affect the retailer's cost of procurement and sales revenues obtained from the product but also affect utilization of shelf space, product rollover and assortment decisions of the retailer. Apart from using secondary markets, retailers that sell seasonal products or products with sales horizons shorter than the typical production/procurement lead time also enter into contractual agreements with suppliers. These contracts are in place to share risks associated with unknown or uncertain demand for the product. Presence of such contracts does affect a retailer's order quantity as well as the time to start selling in the secondary market. In our two essays on retail operations, we analyze a retailer's optimal order quantity and when he/she starts selling in the secondary market. We refer to the former as the 'ordering decision' and the latter as the 'timing decision.' These two decisions are studied first without risk sharing contracts in Essay 1, and then in the presence of contracts in Essay 2. In Essay 1, we build a two-stage model with demand uncertainty. The ordering decision is made in the first stage considering cost of procurement and expected sales revenue. The timing decision is made in the second stage and is conditional on the order quantity determined in the first stage. We introduce a new class of aggregate demand model for this model. We study the structural properties of the retailer's timing and ordering problem and identify optimality conditions for the timing decision. Finally, we complement our analytical results with computational experiments and show how retailer's optimal decisions change when problem parameters are varied. In Essay 2, we extend the work in first essay to include the contracts between the retailer and a supplier. In this essay, we introduce a time-based Poisson demand model. We define three di®erent types of contracts and investigate the effect of each of these contracts on the retailer's ordering and timing decisions. We investigate how the analytical structure of the retailer's decision changes in the presence of these contracts. For a given order quantity, we show that the timing decision depends on the type of contract. Our analytical results on the timing decision are complemented with computational experiments where we investigate the impact of contract type on the optimal order quantity of the retailer. In Humanitarian Logistics, non-profit organizations receive several-billion-dollars-worth of donations every year but lack a sophisticated system to handle their complex logistics operations; the absence of expertly-designed systems is one of the significant reasons why there has been a weak link in the distribution of relief aid. The distribution of relief aid is a complex problem as the goal is humanitarian yet at the same time, due to limited resources, the operations have to be efficient. In the two essays on humanitarian logistics, we study the distribution of aid using homogeneous fleet, with and without capacity restrictions. In Essay 3, we discuss routing for relief operations using one vehicle without capacity restrictions. Contrary to the existing vehicle routing models, the key property of our routing models is that the nodes have priorities along with humanitarian needs. We formulate this model with d-Relaxed Priority rule that captures distance and response time. We formulate routing models with strict and relaxed forms of priority restrictions as Mixed Integer Programs (MIP). We derive bounds for this problem and show that this bound is attained in limiting condition for a worst-case example. Finally, we evaluate the optimal solutions on test problems for response time and distance and show that our vehicle routing model with priorities captures the trade-off between distance and response time unlike existing Vehicle Routing Problem (VRP) models without priorities. In Essay 4, we extend the problem dealt in third essay to consider fleet consisting of multiple vehicles (homogeneous) with capacity and route length restrictions. First, we show that the humanitarian aspect imposes additional challenges and develop routing models that capture performance metrics like fill rate, distance traversed, response time and number of victims satisfied. Proposed routing models are formulated as Mixed Integer Programs and are solved to optimality for small test problems. We conduct computational experiment and show that our models perform well on these performance metrics

Equity Issuance and Divident Policy under Commitment

This paper studies a model of corporate finance in which firms use stock issuance to finance investment. Since the firm recognizes the relationship between future dividends and stock prices, future variables enter in the constraints and optimal policy is in general time inconsistent. We discuss the nature of time inconsistency and show that it arises because managers promise to incorporate value maximization gradually into their objective function. This shows how one could change managers’ incentives in order to enforce the optimal contract under full commitment. We then characterize several cases where time consistency arises and we study different examples where policy is time inconsistent. This allows us to address some outstanding issues in the literature about dividend policy and equity issuance. In particular, our results suggest that growing firms that can credibly commit will pay lower dividends at the beginning and promise higher dividends in the future, consistent with empirical evidence. Our results also suggests that compensation that is tied to stock options creates incentives to inflate prices and pay lower dividends. This is consistent with the empirical evidence of increased stock option compensation and payout through repurchases instead to dividends during the last decades.Stock Issuance; time inconsistency; dividend policy

Small Space Stream Summary for Matroid Center

In the matroid center problem, which generalizes the k-center problem, we need to pick a set of centers that is an independent set of a matroid with rank r. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than Delta-approximation for partition-matroid center must use Omega(r^2) bits of space, where Delta is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and k-center, for which the Doubling algorithm [Charikar et al., 1997] gives an 8-approximation using O(k)-space and one pass. To complement this, we give a one-pass algorithm for matroid center that stores at most O(r^2 log(1/epsilon)/epsilon) points (viz., stream summary) among which a (7+epsilon)-approximate solution exists, which can be found by brute force, or a (17+epsilon)-approximation can be found with an efficient algorithm. If we are allowed a second pass, we can compute a (3+epsilon)-approximation efficiently. We also consider the problem of matroid center with z outliers and give a one-pass algorithm that outputs a set of O((r^2+rz)log(1/epsilon)/epsilon) points that contains a (15+epsilon)-approximate solution. Our techniques extend to knapsack center and knapsack center with z outliers in a straightforward way, and we get algorithms that use space linear in the size of a largest feasible set (as opposed to quadratic space for matroid center)