A generic approach to proving NP-hardness of partition type problems

Kovalyov, Mikhail Y.; Pesch, Erwin

doi:10.1016/j.dam.2010.08.001

Cited by 8 publications

(4 citation statements)

References 8 publications

(7 reference statements)

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“…It is easy to observe that the well-known 2 − PARTITION problem can be reduced to our problem 2 | merging, tree | C max . Recall that the 2 − PARTITION problem consists in determining if a set of N given integers (a 1 , a 2 , ..., a N ) can be partitioned into two disjoint subsets S 1 and S 2 such that i∈S1 a i = i∈S2 a i (for further information, see [11]. Indeed, we can consider an input where we have a tree with one root and m = N leaves (i.e., m tiles).…”

Section: General Description Of the Subproblemmentioning

confidence: 99%

Fully polynomial time approximation scheme for the pagination problem with hierarchical structure of tiles

2023

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The pagination problem is described as follows. We have a set of symbols, a collection of subsets over these symbols (we call these subsets the tiles) and an integer capacity C. The objective is to find the minimal number of pages (a type of container) on which we can schedule all the tiles while following two fundamental rules. We cannot assign more than C symbols on each page and a tile cannot be broken into several pieces, all of its symbols must be assigned to at least one of the pages. The dierence from the Bin Packing Problem is that tiles can merge. If two tiles share a subset of symbols and if they are assigned to the same page, this subset will be assign only once to the page (and not several times). In this paper, as this problem is NP-complete, we are going to look at a particular case of the dual problem, where we have exactly two pages for which the capacity must be minimized. We are going to present a fully polynomial time approximation scheme (FPTAS) to solve it. This approximation scheme is based on the simplification of a dynamic programming algorithm and it has a strongly polynomial time complexity. The conducted numerical experiments show its practical eectiveness.

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Section: General Description Of the Subproblemmentioning

confidence: 99%

Fully polynomial time approximation scheme for the pagination problem with hierarchical structure of tiles

2023

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“…The computational problem MIN-MAX-PROB. It will be convenient to reduce to δ-OPT-CONTRACT from a computational problem we call MIN-MAX-PROB, which is a variant of MIN-MAX PRODUCT PARTITION [40] and thus NP-hard.…”

Section: Approximate Implementabilitymentioning

confidence: 99%

The Complexity of Contracts

Dütting¹,

Roughgarden²,

Talgam-Cohen³

2021

SIAM J. Comput.

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We initiate the study of computing (near-)optimal contracts in succinctly representable principal-agent settings. Here optimality means maximizing the principal's expected payoff over all incentive-compatible contracts-known in economics as "second-best" solutions. We also study a natural relaxation to approximately incentive-compatible contracts.We focus on principal-agent settings with succinctly described (and exponentially large) outcome spaces. We show that the computational complexity of computing a near-optimal contract depends fundamentally on the number of agent actions. For settings with a constant number of actions, we present a fully polynomial-time approximation scheme (FPTAS) for the separation oracle of the dual of the problem of minimizing the principal's payment to the agent, and use this subroutine to efficiently compute a δ-incentive-compatible (δ-IC) contract whose expected payoff matches or surpasses that of the optimal IC contract.With an arbitrary number of actions, we prove that the problem is hard to approximate within any constant c. This inapproximability result holds even for δ-IC contracts where δ is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear δ-IC contracts with constant δ are sufficient to achieve a constant-factor approximation of the "first-best" (full-welfareextracting) solution, and that such a contract can be computed in polynomial time.

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“…The partition problem is one of the problems that require an exponential time for their solution (Kovalyov & Pesch, 2010). Several mathematical methods were suggested to deal with the partition problem and try to reduce the time needed for its solution (Shyu & Lee, 1990;Deineko & Woeginger, 2006) but still the time is exponential.…”

Section: Related Workmentioning

confidence: 99%

A Molecular Solution to the Three-Partition Problem

Nuser

2012

Journal of Information Technology Research

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Given a set of numbers, the three-partition problem is to divide them into disjoint triplets that all have the same sum. The problem is NP-complete. This paper presents an algorithm to solve this problem using the biomolecular computing approach. The algorithm uses a distinctive encoding technique that depends on the numbers values which omits the need to an adder to find the sum. The algorithm is explained and an analysis of its complexity in terms of time, the number of strands, number of tubes, and the longest library strand used is presented. A simulation of the algorithm is implemented and tested. This algorithm further proves the ability of molecular computing in solving hard problems.

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A generic approach to proving NP-hardness of partition type problems

Cited by 8 publications

References 8 publications

Fully polynomial time approximation scheme for the pagination problem with hierarchical structure of tiles

Fully polynomial time approximation scheme for the pagination problem with hierarchical structure of tiles

The Complexity of Contracts

A Molecular Solution to the Three-Partition Problem

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