Dependent randomized rounding for clustering and partition systems with knapsack constraints

Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among the cluster-centers. This, and much more general clustering problems, can be formulated with "knapsack" and "partition" constraints. We develop new randomized algorithms targeting such problems, and study two in particular: multi-knapsack median and multi-knapsack center. Our rounding algorithms give new approximation and pseudo-approximation algorithms for these problems. One key technical tool, which may be of independent interest, is a new tail bound analogous to Feige (2006) for sums of random variables with unbounded variances. Such bounds are very useful in inferring properties of large networks using few samples

Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results

APPROXIMATION ALGORITHMS FOR FACILITY LOCATION AND CLUSTERING PROBLEMS

Facility Location (FL) problems are among the most fundamental problems in combinatorial optimization. FL problems are also closely related to Clustering problems. Generally, we are given a set of facilities, a set of clients, and a symmetric distance metric on these facilities and clients. The goal is to ``open'' the ``best'' subset of facilities, subject to certain budget constraints, and connect all clients to the opened facilities so that some objective function of the connection costs is minimized. In this dissertation, we consider generalizations of classical FL problems. Since these problems are NP-hard, we aim to find good approximate solutions in polynomial time. We study the classic $k$-median problem which asks to find a subset of at most $k$ facilities such that the sum of connection costs of all clients to the closest facility is as small as possible. Our main result is a $2.675$-approximation algorithm for this problem. We also consider the Knapsack Median (KM) problem which is a generalization of the $k$-median problem. In the KM problem, each facility is assigned an opening cost. A feasible set of opened facilities should have the total opening cost at most a given budget. The main technical challenge here is that the natural LP relaxation has unbounded integrality gap. We propose a $17.46$-approximation algorithm for the KM problem. We also show that, after a preprocessing step, the integrality gap of the residual instance is bounded by a constant. The next problem is a generalization of the $k$-center problem, which is called the Knapsack Center (KC) problem and has a similar budget constraint as in the KM problem. Here we want to minimize the maximum distance from any client to its closest opened facility. The KC problem is well-known to be $3$-approximable. However, the current approximation algorithms for KC are deterministic and it is not hard to construct instances in which almost all clients have the worst-possible connection cost. Unfairness also arises in this context: certain clients may consistently get connected to distant centers. We design a randomized algorithm which guarantees that the expected connection cost of ``most'' clients will be at most $(1+2/e) \approx 1.74$ times the optimal radius and the worst-case distance remains the same. We also show a similar result for the $k$-center problem: all clients have expected approximation ratio about $1.592$ with a deterministic upper-bound of $3$ in the worst case. It is well-known that a few \emph{outliers} (very distant clients) may result in a very large optimal radius in the center-type problems. One way to deal with this issue is to cover only some $t$ out of $n$ clients in the so-called robust model. In this thesis, we give tight approximation algorithms for both robust $k$-center and robust matroid center problems. We also introduce a lottery model in which each client $j$ wants to be covered with probability at least $p_j \in [0,1]$. We then give randomized approximation algorithms for center-type problems in this model which match the worst-case bounds of the robust model and slightly violate the coverage and fairness constraints. Several of our results for FL problems in this thesis rely on novel dependent rounding schemes. We develop these rounding techniques in the general setting and show that they guarantee new correlation properties. Given the wide applicability of the standard dependent rounding, we believe that our new techniques are of independent interests

Duality in process of noncommutative deformation and topological nature of Cherepanov-Rice integral

In t his paper it is showed, that for the noncommutative deformation simultaneously there exist also loading deformation H, and unloading deformation $H^v$. The real deformation is a combination of these types of deformations. The criterion of destruction J reflects topological character of medium, i.e. it defines properties of symmetry of medium at destruction. It is possible to tell, that during destruction the energy is released not continuously but and discretely. This situation is reflected through topological number Q or number of unloading, connected to him

Scarcity effects on consumer purchase intention in the context of E-commerce

Objectives of the Study: The purpose of this study is to examine effects of different types of scarcity messages on consumer purchase intention in the context of electronic commerce. The study also investigates the moderating roles of several individual-difference variables. Academic background and methodology: Prior research has demonstrated effects of scarcity on consumer purchase intention in many aspects. Only a few studies, however, have examined scarcity effects in the context of electronic commerce, where the ease of searching for alternative online deals may change the effectiveness of scarcity messages. Thus, it is critical to gain insights into how different types of scarcity messages influence consumer purchase intention in online shopping. Specifically, the study compares effects of scarcity between two contexts of e-commerce: high versus low ease of searching for deals. Accordingly, an online-survey experiment was conducted. The participants of the survey were exposed to two contexts. In each context, they were randomly allocated into one of six conditions containing different types of scarcity messages. Their purchase intentions were measured and investigated in order to figure out variances between conditions in each searching-ease context and the differences between two contexts. Additionally, the study examined the interaction between scarcity and three potential moderators of scarcity effects: uncertainty avoidance, need for cognitive closure, and product familiarity, of which their moderating roles were demonstrated in prior research. Findings and conclusions: The study results showed that in the context of electronic commerce, scarcity messages became less effective. In the context of high searching-ease, no significant effect of scarcity was found. In the context of low searching-ease, only the scarcity message in form of intensive time limit, in association with a signal of price promotion, presented a significant effect on consumer purchase intention. Additionally, contrary to the findings of prior research, three investigated moderators showed no significant interaction with scarcity. This outcome suggested that to explain the underlying factors of scarcity effects in the context of e-commerce, other mediators should be considered. This finding is significant for managers who intend to use scarcity as a marketing tool for their online businesses. The result also contributes to the research area of scarcity effects

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博士（農学）東京農工大