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Concept
Unique games conjecture
Summary
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines.
The conjecture is unusual in that the academic world seems about evenly divided on whether it is true or not.
The unique games conjecture can be stated in a number of equivalent ways.
The following formulation of the unique games conjecture is often used in hardness of approximation. The conjecture postulates the NP-hardness of the following promise problem known as label cover with unique constraints. For each edge, the colors on the two vertices are restricted to some particular ordered pairs. Unique constraints means that for each edge none of the ordered pairs have the same color for the same node.
This means that an instance of label cover with unique constraints over an alphabet of size k can be represented as a directed graph together with a collection of permutations πe: [k] → [k], one for each edge e of the graph. An assignment to a label cover instance gives to each vertex of G a value in the set [k] = {1, 2, ... k}, often called “colours.”
File:Unique label cover yes-instance.svg|An instance of unique label cover. The 4 vertices may be assigned the colors red, blue, and green while satisfying the constraints at each edge.
File:Unique label cover yes-instance with assignment.svg|A solution to the unique label cover instance.
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Maximum cut
For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible. Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset S of the vertex set such that the number of edges between S and the complementary subset is as large as possible.
Unique games conjecture
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation.
Semidefinite programming
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems.