Constant-Time Approximation Algorithms for the Knapsack Problem

Ito, Hiro; Kiyoshima, Susumu; Yoshida, Yuichi

doi:10.1007/978-3-642-29952-0_17

Cited by 8 publications

(3 citation statements)

References 15 publications

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“…Batu, Berenbring, and Sohler [6] gave a parametertesting algorithm for the bin packing problem with query complexity Õ( √ n • poly(ϵ −1 )). Ito, Kiyoshima, and Yoshida [23] showed a parameter-testing algorithm for the knapsack problem with query complexity Õ(ϵ −4 ). For EXPTIMEcomplete problems, the generalized chess, Shogi (Japanese chess), and Xiangqi (Chinese chess) were proven to be all testable by Ito, Nagao, and Park [24].…”

Section: Other Resultsmentioning

confidence: 99%

Sublinear Computation Paradigm: Constant-Time Algorithms and Sublinear Progressive Algorithms

Chiba

Ito

2022

IEICE Trans. Fundamentals

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The challenges posed by big data in the 21st Century are complex: Under the previous common sense, we considered that polynomial-time algorithms are practical; however, when we handle big data, even a linear-time algorithm may be too slow. Thus, sublinear-and constant-time algorithms are required. The academic research project, "Foundations of Innovative Algorithms for Big Data," which was started in 2014 and will finish in September 2021, aimed at developing various techniques and frameworks to design algorithms for big data. In this project, we introduce a "Sublinear Computation Paradigm." Toward this purpose, we first provide a survey of constant-time algorithms, which are the most investigated framework of this area, and then present our recent results on sublinear progressive algorithms. A sublinear progressive algorithm first outputs a temporary approximate solution in constant time, and then suggests better solutions gradually in sublinear-time, finally finds the exact solution. We present Sublinear Progressive Algorithm Theory (SPA Theory, for short), which enables to make a sublinear progressive algorithm for any property if it has a constant-time algorithm and an exact algorithm (an exponential-time one is allowed) without losing any computation time in the big-O sense.

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Section: Other Resultsmentioning

confidence: 99%

Sublinear Computation Paradigm: Constant-Time Algorithms and Sublinear Progressive Algorithms

Chiba

Ito

2022

IEICE Trans. Fundamentals

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“…A property is said to be testable if there is a tester. In the last decade, many properties are found to be testable or non-testable [2,3,5,6,7,8,10,11,12,13,15]. In this paper, we study a testability of the generalized shogi and chess.…”

Section: Introductionmentioning

confidence: 95%

Generalized shogi and chess are constant-time testable

Ito¹,

Nagao²,

Park³

2015

12th International Symposium on Operations Research and Its Applications in Engineering, Technology and Management (ISORA 2015)

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We present constant-time testing algorithms for the generalized shogi (Japanese chess) and chess. These problems are known to be EXPTIME-complete. A testing algorithm (or a tester) for a property accepts an input if it has the property and rejects it if it's far from having the property in high probability (e.g., at least 2/3) by reading only a constant part of the input. A property is said to be testable if there is a tester. The generalized shogi (and chess) problem is, given any position on √ n × √ n board with O(n) pieces, for testing the property "the player who moves first has a winning strategy." We present that this property is testable for both shogi and chess. The shogi tester is one-sidederror and the chess tester is surprisingly no-error! In the last decade, many problems have been found to be testable. However, almost all of such problems are in class NP. This is the first result on constant-time testability of EXPTIME-complete problems.

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“…Recently, it has been learned that many problems can be approximated in sublinear time [1,6,7,8,9,10,11,12,13,16]. To solve a problem in sublinear time, it is necessary to introduce some approximations by using a parameter 0 < ǫ ≤ 1.…”

Section: Introduction 1what Is a Sublinear-time Algorithm For The Cak...mentioning

confidence: 99%

How to solve the cake-cutting problem in sublinear time

Ito,

Ueda

2015

Preprint

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The cake-cutting problem refers to the issue of dividing a cake into pieces and distributing them to players who have different value measures related to the cake, and who feel that their portions should be "fair." The fairness criterion specifies that in situations where n is the number of players, each player should receive his/her portion with at least 1/n of the cake value in his/her measure. In this paper, we show algorithms for solving the cake-cutting problem in sublinear-time. More specifically, we preassign fair portions to o(n) players in o(n)-time, and minimize the damage to the rest of the players. All currently known algorithms require Ω(n)-time, even when assigning a portion to just one player, and it is nontrivial to revise these algorithms to run in o(n)-time since many of the remaining players, who have not been asked any queries, may not be satisfied with the remaining cake.To challenge this problem, we begin by providing a framework for solving the cake-cutting problem in sublinear-time. Generally speaking, solving a problem in sublinear-time requires the use of approximations. However, in our framework, we introduce the concept of "ǫnvictims," which means that ǫn players (victims) may not get fair portions, where 0 < ǫ ≤ 1 is an arbitrary constant. In our framework, an algorithm consists of the following two parts: In the first (Preassigning) part, it distributes fair portions to r < n players in o(n)-time. In the second (Completion) part, it distributes fair portions to the remaining n − r players except for the ǫn victims in poly(n)-time. There are two variations on the r players in the first part. Specifically, whether they can or cannot be designated. We will then present algorithms in this framework. In particular, an O(r/ǫ)-time algorithm for r ≤ ǫn/127 undesignated players with ǫn-victims, and an Õ(r 2 /ǫ)-time algorithm for r ≤ ǫe √ ln n/7 designated players and ǫ ≤ 1/e with ǫn-victims are presented.

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Constant-Time Approximation Algorithms for the Knapsack Problem

Cited by 8 publications

References 15 publications

Sublinear Computation Paradigm: Constant-Time Algorithms and Sublinear Progressive Algorithms

Sublinear Computation Paradigm: Constant-Time Algorithms and Sublinear Progressive Algorithms

Generalized shogi and chess are constant-time testable

How to solve the cake-cutting problem in sublinear time

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