On the Partition of Numbers

Mathews, G. B.

doi:10.1112/plms/s1-28.1.486

Cited by 108 publications

(45 citation statements)

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“…Theorem 2.1 states that we can "aggregate" these integer programming problems efficiently. Earliest results on aggregation are probably due to Matthews [18]. He shows that given a system of two simultaneous Diophantine equations whose variables are restricted to be nonnegative and whose coefficients are nonnegative, there is an (easily obtainable) single Diophantine equation whose set of nonnegative solutions is identical to that of the given system.…”

Section: Polynomial-time Aggregation Of Integer Programming Problems mentioning

confidence: 98%

Polynomial-Time Aggregation of Integer Programming Problems

Kannan

1983

J. ACM

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It ts shown that a set of linear Dtophantme equations m nonnegative variables with nonnegative coefficients can be reduced to a single equation with the same solution set in polynomial time. A weaker verston of the above statement ~s shown to be true when the coefficients are allowed to be negative Besides being polynomial-trine bounded, the present aggregation scheme differs from existing ones in that the final equation is m variables that are not exphcltly bounded Three applications of this aggregation technique are presented: (i) ~t Is proved that a certain type of knapsack problem cannot have a polynomialtune approxtmatlon algorithm unless NP = P; 00 an analog of Farkas' lemma for integer programming is proved; and 0ii) ~t is shown that a decision problem revolving integer variables is NP-complete.

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Section: Polynomial-time Aggregation Of Integer Programming Problems mentioning

confidence: 98%

Polynomial-Time Aggregation of Integer Programming Problems

Kannan

1983

J. ACM

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“…The knapsack problem has been studied for more than a century, with early works dating as far back as 1897 [8]. The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography and applied mathematics.…”

Section: Knapsack Problemmentioning

confidence: 99%

Adaptive variable density sampling based on Knapsack problem for fast MRI

Rajgopal

2015

2015 IEEE International Symposium on Signal Processing and Information Technology (ISSPIT)

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This paper presents a novel heuristic approach to generate a variable density sampling (VDS) pattern for fast MRI data acquisition, based on the principle of Knapsack problem. MR images are known to exhibit weak sparsity in Fourier domain. This sparsity has been exploited to devise faster k-space sampling schemes by acquiring lesser samples while using Compressed Sensing reconstruction techniques to reconstruct high quality MR images. The entire range of distribution of the magnitude of k-space is divided into fixed/variable bin widths and draw samples from these bins according to a cost criterion satisfying the undersampling factor. This will facilitate sampling the kspace coefficients by preserving the energy content for the desired undersampling factor and do away with the deterministic central region sampling resulting in a VDS method with a correlation to the magnitude spectrum of the reference image scan. Knapsack principle is used to select the relevant bins. The method is also devoid of parameter tuning, yielding significant good results at both the lower and higher under sampling ratios.

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“…This simple structure can lead to find properties that make the problem easier to solve. This is also a very general problem, because any integer program can be solved as a {0, 1}-knapsack problem (Pisinger 1995, Mathews 1897. Moreover, it can appear as a subproblem of more general combinatorial optimization problems.…”

Section: The Inverse Knapsack Problemmentioning

confidence: 99%