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3-partition
The 3-partition problem is a strongly NP-complete problem in computer science. The problem is to decide whether a given multiset of integers can be partitioned into triplets that all have the same sum. More precisely: * Input: a multiset ''S'' containing ''n'' positive integer elements. * Conditions: ''S'' must be partitionable into ''m'' triplets, ''S''1, ''S''2, …, ''S''''m'', where ''n = 3m''. These triplets partition ''S'' in the sense that they are disjoint and they cover ''S''. The target value ''T'' is computed by taking the sum of all elements in ''S'', then dividing by ''m''. * Output: whether or not there exists a partition of ''S'' such that, for all triplets, the sum of the elements in each triplet equals ''T''. The 3-partition problem remains strongly NP-complete under the restriction that every integer in ''S'' is strictly between ''T''/4 and ''T''/2. Example # The set S = \ can be partitioned into the four sets \, \, \ , \, each of which sums to ''T'' = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle Packing
Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap. Several variants of this problem have been studied. Packing identical rectangles in a rectangle In this variant, there are multiple instances of a single rectangle of size (''l'',''w''), and a bigger rectangle of size (''L'',''W''). The goal is to pack as many small rectangles as possible into the big rectangle without overlap between any rectangles (small or large). Common constraints of the problem include limiting small rectangle rotation to 90° multiples and requiring that each small rectangle is orthogonal to the large rectangle. This problem has some applications such as loading of boxes on pallets and, specifically, woodpulp stowage. As an example result: it is possible to pack 147 small rectangles of size (137,95) in a big rectangle of size (1600,1230). Packing identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical 3-dimensional Matching
Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers X, Y and Z, each containing k elements, and a bound b. The goal is to select a subset M of X\times Y\times Z such that every integer in X, Y and Z occurs exactly once and that for every triple (x,y,z) in the subset x+y+z=b holds. This problem is labeled as P16in.Garey, Michael R. and David S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. Example Take X=\, Y=\ and Z=\, and b=10. This instance has a solution, namely \. Note that each triple sums to b=10. The set \ is not a solution for several reasons: not every number is used (a 4\in X is missing), a number is used too often (the 3\in X) and not every triple sums to b (since 3+4+2=9\neq b=10). However, there is at least one solution to this problem, which is the property we are interested in with decision problems. If we would take b=11 for the same X, Y and Z, this problem would ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partition of a set, partitioned into two subsets ''S''1 and ''S''2 such that the sum of the numbers in ''S''1 equals the sum of the numbers in ''S''2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are Heuristic, heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization problem, optimization version of the partition problem, which is to partition the multiset ''S'' into two subsets ''S''1, ''S''2 such that the difference between the sum of elements in ''S''1 and the sum of elements in ''S''2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. The partition problem is a special case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetris
''Tetris'' () is a puzzle video game created in 1985 by Alexey Pajitnov, a Soviet software engineer. In ''Tetris'', falling tetromino shapes must be neatly sorted into a pile; once a horizontal line of the game board is filled in, it disappears, granting points and preventing the pile from overflowing. Over 200 versions of ''Tetris'' have been published by numerous companies on more than 65 platforms, often with altered game mechanics, some of which have become standard over time. To date, these versions of ''Tetris'' collectively serve as the second-best-selling video game series with over 520 million sales, mostly on mobile devices. In the 1980s, Pajitnov worked for the Computing Center of the Academy of Sciences, where he programmed ''Tetris'' on the Elektronika 60 and adapted it to the IBM PC with the help of Dmitry Pavlovsky and Vadim Gerasimov. Floppy disk copies were distributed freely throughout Moscow, before spreading to Eastern Europe. Robert Stein of Andro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiway Number Partitioning
In computer science, multiway number partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are as similar as possible. It was first presented by Ronald Graham in 1969 in the context of the identical-machines scheduling problem. The problem is parametrized by a positive integer ''k'', and called ''k''-way number partitioning. The input to the problem is a multiset ''S'' of numbers (usually integers), whose sum is ''k*T''. The associated decision problem is to decide whether ''S'' can be partitioned into ''k'' subsets such that the sum of each subset is exactly ''T''. There is also an optimization problem: find a partition of ''S'' into ''k'' subsets, such that the ''k'' sums are "as near as possible". The exact optimization objective can be defined in several ways: * Minimize the difference between the largest sum and the smallest sum. This objective is common in papers about multiway number partitioning, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong NP-completeness
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a pseudo-polynomial reduction to it. This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs. In special, the pseudo-polynomial reduction cannot output a numerical parameter that is not polinomially bounded by the size and value of numbers in the input ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Job Scheduling
A job scheduler is a computer application for controlling unattended background program execution of job (computing), jobs. This is commonly called batch scheduling, as execution of non-interactive jobs is often called batch processing, though traditional ''job'' and ''batch'' are distinguished and contrasted; see that page for details. Other synonyms include batch system, distributed resource management system (DRMS), distributed resource manager (DRM), and, commonly today, workload automation (WLA). The data structure of jobs to run is known as the job queue. Modern job schedulers typically provide a graphical user interface and a Device management, single point of control for definition and monitoring of background executions in a distributed network of computers. Increasingly, job schedulers are required to orchestrate the integration of real-time business activities with traditional background IT processing across different operating system platforms and business application ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David S
David (; , "beloved one") was a king of ancient Israel and Judah and the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Canaanite and Aramaic inscriptions, Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "Davidic line, House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, Historicity of the Bible, the historicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Garey
Michael Randolph Garey (born November 19, 1945) is a computer science researcher, and co-author (with David S. Johnson) of '' Computers and Intractability: A Guide to the Theory of NP-completeness''. He and Johnson received the 1979 Frederick W. Lanchester Prize from the Operations Research Society of America for the book. Garey earned his PhD in computer science in 1970 from the University of Wisconsin–Madison. He was employed by AT&T Bell Laboratories in the Mathematical Sciences Research Center from 1970 until his retirement in 1999. For his last 11 years with the organization, he served as its director. His technical specialties included discrete algorithms and computational complexity, approximation algorithms, scheduling theory, and graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-dimensional Matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph). 3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard. Definition Let ''X'', ''Y'', and ''Z'' be finite sets, and let ''T'' be a subset of ''X'' × ''Y'' × ''Z''. That is, ''T'' consists of triples (''x'', ''y'', ''z'') such that ''x'' ∈ ''X'', ''y'' ∈ ''Y'', and ''z'' ∈ ''Z''. Now ''M'' ⊆ ''T'' is a 3-dimensional matching if the following holds: for any two distinct triples (''x''1, ''y''1, ''z''1) ∈ ''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak NP-completeness
In computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms are technically exponential functions of their input size and are therefore not considered polynomial. For example, the NP-hard knapsack problem can be solved by a dynamic programming algorithm requiring a number of steps polynomial in the size of the knapsack and the number of items (assuming that all data are scaled to be integers); however, the runtime of this algorithm is exponential time since the input sizes of the objects and knapsack are logarithmic in their magnitudes. However, as Garey and Johnson (1979) observed, “A pseudo-polynomial-time algorithm … will display 'exponential behavior' only when confronted w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudopolynomial Time Number Partitioning
In computer science, pseudopolynomial time number partitioning is a pseudopolynomial time algorithm for solving the partition problem. The problem can be solved using dynamic programming when the size of the set and the size of the sum of the integers in the set are not too big to render the storage requirements infeasible. Suppose the input to the algorithm is a multiset S of cardinality N: : ''S'' = Let ''K'' be the sum of all elements in ''S''. That is: ''K'' = ''x''1 + ... + ''xN''. We will build an algorithm that determines whether there is a subset of ''S'' that sums to \lfloor K/2 \rfloor . If there is a subset, then: : if ''K'' is even, the rest of ''S'' also sums to \lfloor K/2 \rfloor : if ''K'' is odd, then the rest of ''S'' sums to \lceil K/2 \rceil . This is as good a solution as possible. e.g.1 ''S'' = , ''K'' = ''sum(S)'' = 11, ''K/2'' = 5, Find a subset from ''S'' that is closest to ''K/2'' -> = 5, 11 - 5 * 2 = 1 e.g.2 ''S'' = , ''K'' = ''sum(S)'' = 11, '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
