In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.
In computational complexity theory, the complexity class #P (pronounced "sharp P" or, sometimes "number P" or "hash P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute f(x)", where f is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. The most difficult, representative problems of this class are #P-complete.
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties: a prover and a verifier. The parties interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover possesses unlimited computational resources but cannot be trusted, while the verifier has bounded computation power but is assumed to be always honest. Messages are sent between the verifier and prover until the verifier has an answer to the problem and has "convinced" itself that it is correct.
In computational complexity theory, a probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm using a bounded amount of randomness and reading a bounded number of bits of the proof. The algorithm is then required to accept correct proofs and reject incorrect proofs with very high probability. A standard proof, as used in the verifier-based definition of the complexity class NP, also satisfies these requirements, since the checking procedure deterministically reads the whole proof, always accepts correct proofs and rejects incorrect proofs. However, what makes them interesting is the existence of probabilistically checkable proofs that can be checked by reading only a few bits of the proof using randomness in an essential way.
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic space complexity. It states that for any function ,
In computational complexity theory, DSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine. It represents the total amount of memory space that a "normal" physical computer would need to solve a given computational problem with a given algorithm.
In computational complexity theory, L is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way.
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.
In computational complexity theory, the PCP theorem states that every decision problem in the NP complexity class has probabilistically checkable proofs of constant query complexity and logarithmic randomness complexity.
Vijay Virkumar Vazirani is an Indian American distinguished professor of computer science in the Donald Bren School of Information and Computer Sciences at the University of California, Irvine.
In circuit complexity, AC is a complexity class hierarchy. Each class, ACi, consists of the languages recognized by Boolean circuits with depth and a polynomial number of unlimited fan-in AND and OR gates.
Umesh Virkumar Vazirani is an Indian–American academic who is the Roger A. Strauch Professor of Electrical Engineering and Computer Science at the University of California, Berkeley, and the director of the Berkeley Quantum Computation Center. His research interests lie primarily in quantum computing. He is also a co-author of a textbook on algorithms.
In computational complexity theory, a certificate is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer "Yes" or "No".
In graph drawing, a circular layout is a style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon.
Michael Justin Kearns is an American computer scientist, professor and National Center Chair at the University of Pennsylvania, the founding director of Penn's Singh Program in Networked & Social Systems Engineering (NETS), the founding director of Warren Center for Network and Data Sciences, and also holds secondary appointments in Penn's Wharton School and department of Economics. He is a leading researcher in computational learning theory and algorithmic game theory, and interested in machine learning, artificial intelligence, computational finance, algorithmic trading, computational social science and social networks. He previously led the Advisory and Research function in Morgan Stanley's Artificial Intelligence Center of Excellence team, and is currently an Amazon Scholar within Amazon Web Services.
In the study of hierarchical clustering, Dasgupta's objective is a measure of the quality of a clustering, defined from a similarity measure on the elements to be clustered. It is named after Sanjoy Dasgupta, who formulated it in 2016. Its key property is that, when the similarity comes from an ultrametric space, the optimal clustering for this quality measure follows the underlying structure of the ultrametric space. In this sense, clustering methods that produce good clusterings for this objective can be expected to approximate the ground truth underlying the given similarity measure.
Boaz Barak is an Israeli-American professor of computer science at Harvard University.
Satish B. Rao is an American computer scientist who is a professor of computer science at the University of California, Berkeley.
In graph theory, the cutwidth of an undirected graph is the smallest integer with the following property: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at most edges. That is, if the vertices are numbered , then for every , the number of edges with and is at most .
