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Maximin-share
Maximin share (MMS) is a criterion of fair item allocation. Given a set of items with different values, the ''1-out-of-n maximin-share'' is the maximum value that can be gained by partitioning the items into ''n'' parts and taking the part with the minimum value. An allocation of items among ''n'' agents with different valuations is called MMS-fair if each agent gets a bundle that is at least as good as his/her 1-out-of-''n'' maximin-share. MMS fairness was invented by Eric Budish as a relaxation of the criterion of proportionality - each agent gets a bundle that is at least as good as the equal split (1/''n'' of every resource). Proportionality can be guaranteed when the items are divisible, but not when they are indivisible, even if all agents have identical valuations. In contrast, MMS fairness can always be guaranteed to identical agents, so it is a natural alternative to proportionality even when the agents are different. Motivation and examples Identical items. Suppose fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Item Allocation
Fair item allocation is a kind of a fair division problem in which the items to divide are ''discrete'' rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios: * Several heirs want to divide the inherited property, which contains e.g. a house, a car, a piano and several paintings. * Several lecturers want to divide the courses given in their faculty. Each lecturer can teach one or more whole courses. *White elephant gift exchange parties The indivisibility of the items implies that a fair division may not be possible. As an extreme example, if there is only a single item (e.g. a house), it must be given to a single partner, but this is not fair to the other partners. This is in contrast to the fair cake-cutting problem, where the dividend is divisible and a fair division always exists. In some cases, the indivisibility pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truthful Mechanism
In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about what the others do, you fare best or at least not worse by being truthful. SP is also called truthful or dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better off, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse off. Examples Typical examples of SP mechanisms are majority voting between two alternatives, second- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Item Allocation
Envy-free (EF) item allocation is a fair item allocation problem, in which the fairness criterion is envy-freeness - each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent. Since the items are indivisible, an EF assignment may not exist. The simplest case is when there is a single item and at least two agents: if the item is assigned to one agent, the other will envy. One way to attain fairness is to use monetary transfers; see Fair allocation of items and money. When monetary transfers are not allowed or not desired, there are allocation algorithms providing various kinds of relaxations. Finding an envy-free allocation whenever it exists Preference-orderings on bundles: envy-freeness The undercut procedure finds a complete EF allocation for two agents, if-and-only-if such allocation exists. It requires the agents to rank bundles of items, but it does not require cardinal utility information. It works whenever the agents' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borda Score
The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets ''n'' − 1 points, where ''n'' is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one. The Borda count was developed independently several times, being first proposed in 1435 by Nicholas of Cusa (see History below), but is named for the 18th-century French mathematician and naval engineer Jean-Charles de Borda, who devised the system in 1770. It is currently used to elect two ethnic minority ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is . One way of addressing this is to define the efficient polynomial-time a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) which ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP-complete
In Computational complexity theory, complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P (complexity), P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the complement (complexity), complement of an NP-complete problem. There are some problems in both NP (complexity), NP and co-NP, for example all problems in P (complexity), P or integer factorization. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and NP-complete for more details. Fortune showed in 1979 that if any sparse language is co-NP-complete (or even j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identical-machines Scheduling
Identical-machines scheduling is an optimization problem in computer science and operations research. We are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' identical machines, such that a certain objective function is optimized, for example, the makespan is minimized. Identical machine scheduling is a special case of uniform machine scheduling, which is itself a special case of optimal job scheduling. In the general case, the processing time of each job may be different on different machines; in the case of identical machine scheduling, the processing time of each job is the same on each machine. Therefore, identical machine scheduling is equivalent to multiway number partitioning. A special case of identical machine scheduling is single-machine scheduling. In the standard three-field notation for optimal job scheduling problems, the identical-machines variant is denoted by P in the first field. For example, " P, , ... [...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, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picking Sequence
A picking sequence is a protocol for fair item assignment. Suppose ''m'' items have to be divided among ''n'' agents. One way to allocate the items is to let one agent select a single item, then let another agent select a single item, and so on. A picking-sequence is a sequence of ''m'' agent-names, where each name determines what agent is the next to pick an item. As an example, suppose 4 items have to be divided between Alice and Bob. Some possible picking sequences are: * AABB - Alice picks two items, then Bob picks the two remaining items. * ABAB - Alice picks one item, then Bob picks one item, then Alice again, then Bob again. This is more "fair" than AABB since it lets Bob more chance to get a better item. * ABBA - Alice picks one item, then Bob picks two items, then Alice receives the remaining item. This is intuitively even more "fair" than ABAB, since, in ABAB, Bob is always behind of Alice, while ABBA is more balanced. Advantages A picking-sequence has several merits as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multifit Algorithm
The multifit algorithm is an algorithm for multiway number partitioning, originally developed for the problem of identical-machines scheduling. It was developed by Coffman, Garey and Johnson. Its novelty comes from the fact that it uses an algorithm for another famous problem - the bin packing problem - as a subroutine. The algorithm The input to the algorithm is a set ''S'' of numbers, and a parameter ''n''. The required output is a partition of ''S'' into ''n'' subsets, such that the largest subset sum (also called the makespan) is as small as possible. The algorithm uses as a subroutine, an algorithm called '' first-fit-decreasing bin packing'' (FFD). The FFD algorithm takes as input the same set ''S'' of numbers, and a bin-capacity ''c''. It heuristically packs numbers into bins such that the sum of numbers in each bin is at most ''C'', aiming to use as few bins as possible. Multifit runs FFD multiple times, each time with a different capacity ''C'', until it finds some ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LPT Algorithm
Longest-processing-time-first (LPT) is a greedy algorithm for job scheduling. The input to the algorithm is a set of ''jobs'', each of which has a specific processing-time. There is also a number ''m'' specifying the number of ''machines'' that can process the jobs. The LPT algorithm works as follows: # Order the jobs by descending order of their processing-time, such that the job with the longest processing time is first. # Schedule each job in this sequence into a machine in which the current load (= total processing-time of scheduled jobs) is smallest. Step 2 of the algorithm is essentially the list-scheduling (LS) algorithm. The difference is that LS loops over the jobs in an arbitrary order, while LPT pre-orders them by descending processing time. LPT was first analyzed by Ronald Graham in the 1960s in the context of the identical-machines scheduling problem. Later, it was applied to many other variants of the problem. LPT can also be described in a more abstract way, as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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