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Bin Packing Problem
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media and technology mapping in FPGA semiconductor chip design. Computationally, the problem is NP-hard, and the corresponding decision problem - deciding if items can fit into a specified number of bins - is NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It requires '' Θ''(''n'' log ''n'') time, where ''n' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a ''continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Continuous optimization problem The '' standard form'' of a continuous optimization problem is \begin &\underset& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_j(x) = 0, \quad j = 1, \dots,p \end where * is the objective function to be minimized over the -variable vector , * are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Online Algorithm
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. In operations research, the area in which online algorithms are developed is called online optimization. As an example, consider the sorting algorithms selection sort and insertion sort: selection sort repeatedly selects the minimum element from the unsorted remainder and places it at the front, which requires access to the entire input; it is thus an offline algorithm. On the other hand, insertion sort considers one input element per iteration and produces a partial solution without considering future elements. Thus insertion sort is an online algorithm. Note that the final result of an insertion sor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refined First-fit Bin Packing
First-fit (FF) is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The first-fit algorithm uses the following heuristic: * It keeps a list of open bins, which is initially empty. * When an item arrives, find the ''first'' bin into which the item can fit, if any. ** If such a bin is found, the new item is placed inside it. ** Otherwise, a new bin is opened and the coming item is placed inside it. Approximation ratio Denote by FF(L) the number of bins used by First-Fit, and by OPT(L) the optimal number of bins possible for the list L. The analysis of FF(L) was done in several steps. * The first upper bound of FF(L) \leq 1.7\mathrm+3 for FF was proven by Ullman in 1971. * In 1972, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Best-fit Bin Packing
Best-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The best-fit algorithm uses the following heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...: * It keeps a list of open bins, which is initially empty. * When an item arrives, it finds the bin with the ''maximum load'' into which the item can fit, if any. ** If such a bin is found, the new item is placed inside it. ** Otherwise, a new bin is opened and the coming item is placed inside it. Approximation ratio De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Next-fit Bin Packing
Next-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The next-fit algorithm uses the following heuristic: * It keeps a ''current bin'', which is initially empty. * When an item arrives, it checks whether the item fits into the current bin. ** If it fits, it is placed inside it. ** Otherwise, the current bin is closed, a new bin is opened and the coming item is placed inside this new bin. Next-Fit is a bounded space algorithm - it requires only one partially-filled bin to be open at any time. The algorithm was studied by David S. Johnson in his doctoral thesis in 1973. Run time The running time of NextFit can be bounded by \mathcal(n), where n is the number of items in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David S
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and Lyre, harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges David and Jonathan, a notably close friendship with Jonathan (1 Samuel), Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-polynomial Time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of the input (the number of bits required to represent it), which is the case for polynomial time algorithms.Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979. In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length. An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless . The strong/weak kinds of NP-hardness are defined ana ... [...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 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 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 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 two related problems: * In the subset sum problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-partition Problem
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: * The input to the problem is a multiset ''S'' of ''n'' = 3 positive integers. The sum of all integers is . * The output is whether or not there exists a partition of ''S'' into ''m'' triplets ''S''1, ''S''2, …, ''S''''m'' such that the sum of the numbers in each one is equal to ''T''. The ''S''1, ''S''2, …, ''S''''m'' must form a partition of ''S'' in the sense that they are disjoint and they cover ''S''. 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'' = 90. # The set S = \ can be partitioned into the two sets \, \ each of which sum to ''T'' = 15. # (every ... [...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 polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem. Normally numerical parameters to a problem are given in positional notation, so a probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \text && \mathbf^\mathrm \mathbf\\ & \text && A \mathbf \le \mathbf, \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
