#PolynomialTime
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Polynomial-time algorithm recovers algebraic structure in sets with small doubling constants via entropic Ruzsa distance, enabling applications to quantum state tomography and quadratic Fourier analysis.

#QuantumAlgorithms #AdditiveStructure #PolynomialTime
Algorithmic Polynomial Freiman-Ruzsa via Marton's Conjecture
arxiv.org
September 18, 2026 at 4:35 AM
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Project 3 – The Knapsack Problem – Solved 3 Ways CS2223 Answered

The Knapsack Problem is a well known NP-hard problem. This means that no polynomialtime (PT) algorithm is known to solve this problem. Many computer scientists believe that a PT algorithm cannot be found to solve Knapsack, although…
Project 3 – The Knapsack Problem – Solved 3 Ways CS2223 Answered
The Knapsack Problem is a well known NP-hard problem. This means that no polynomialtime (PT) algorithm is known to solve this problem. Many computer scientists believe that a PT algorithm cannot be found to solve Knapsack, although this hypothesis has not been proven. For this project, you will explore three ways to solve one instance of the knapsack problem, and compare time and space efficiencies for them.
jarviscodinghub.com
September 9, 2026 at 10:14 AM
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I was lead down that #RabbitHole by:

en.wikipedia.org/wiki/BQP

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In #ComputationalComplexity theory, #BoundedError #QuantumPolynomialTime ( #BQP) is the class of #DecisionProblems solvable by a #QuantumComputer in #PolynomialTime, with an #ErrorProbability of at most 1/3 for all instances.
BQP - Wikipedia
en.wikipedia.org
August 29, 2026 at 4:27 AM
mathiastck.bsky.social
It is the #QuantumAnalogue to the #ComplexityClass #BPP.

A #DecisionProblem is a member of #BQP if there exists a #QuantumAlgorithm (an #algorithm that runs on a #QuantumComputer) that solves the decision problem with #HighProbability and is guaranteed to run in #PolynomialTime.
August 29, 2026 at 4:28 AM
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Parts 1-4/4 — EFTA02674310.jpg
#epsteinweb #efta02674310
https://epsteinweb.org
Available in the iOS app store now!
https://apps.apple.com/us/app/epstein-web/id6758880661
EFTA02674309 | EFTA02674310 The class PTIME (denoted as P) consists of problems wham solutions can be computed by an algorithm that trues polynomial time. This means the running time of the algorithm grows as a polynomial function of the size of the input. In computer science, PTIME represents the doe: of problems which can he solved efficiently. The class NP (non-deterministic polynomial time) consists of problems, for which solutions exist that are of polynomial length, and given a candidate for a solution of polynomial length, whether the candidate is indeed a solution can be checked in polynomial time. Therefore, an NI' algorithm can verify a solution in polynomial time. In order to proceed further. we need the notion of 'reduction' between classes of problems. A reduction, from a given problem Pt to n problem P2. is a translation such that a solution for P3 can provide a solution for Pr . More precisely, if there is a polynomial-time reduction from Pr to P2, then a polynomial-time algorithm for Pz implies a polynomial-time algorithm for Pi. A given problem is NP-hard if for every problem in NP there is a polynomial reduction to the given problem. A problem is NP-complete, if it is both NP-hard, as well as there is an NP algorithm for the problem. For example. consider a Boolean formula over variables. and the question whether there exists an assignment to the variables such that the formula is true. A polynomial candidate solution is an assignment of truth values to variables, and given a candidate assignment the formula can be evaluated in polynomial time. This is the famous satisliability, SAT, problem in computer science. The SM' problem is NP-complete. The clam P is contained in NP. and a major. long-standing open question in computer science is whether P=NP? A polynomial-time algorithm for an NP-complete (or an NPhard) problem would imply that P=NP, resolving the longstanding open problem. The class #P (sharpP) intuitively corresponds to counting the number of solutions. A problem is in #1' if it counts the number of distinct solutions such that (i) every peesible candidate for a solution is of polynomial length, and (ii) given a candidate for a solution, it can be checked in polynomial time whether the candidate is a solution. For example. given a Boolean formula, the problem whether there are at least k distinct satisfying assignments to the formula is a #l'-problem. A given problem is #P-hard, if for every #P-problem there is a polynomial-time reduction to the given problem. A #Pcomplete problem is a problem that is both #P-hard, and there is a #P-solution. For example, counting the number of solutions in SAT is #P-complete. The class NI' is contained in #1' because given the enumeration of solutions for #P, it is easy to check if there exists at least one solution. Intuitively, an NI' problem asks whether there is at least one solution, whereas #P is the counting version which asks if there are least k distinct solutions (and the special case of k = I gives NI'). Again a major open question is whether NP=#1)? Note that a polynomial-time algorithm for a #P-complete problem would be an even bigger result as it would imply both P=NP and P=#1'. The clam PSPACE, consists of problems which can he solved with polynomial space. Note that n polynomial space algorithm can reuse space and can in general require exponential time. Every #P problem can be solved in PSPACE by simply enumerating each candidate for a solution and checking if it is a solution. Since we can reuse space to enumerate the candidates for solutions, the enumeration can be achieved in polynomial space. Moreover, every polynomial-time algorithm uses at most polynomial space. I it follows that #1' is contained in PSPACE. The notion of PSPACF.2 I woo reascosicsilscom 1073/ress 0709640104 hardness and PSPACE-completeness is similar to the notion of NP-hardness and NP-completenets, but with respect to the problems in PSPACE. Again a long-standing open question in computer science is whether #P=PSPACE, and a polynomialtime algorithm for a PSPACE-complete (or PSPACE-hard) problem would imply P=NP=#P=PSPACE. We have mentioned that the major questions about the equality of the complexity dames are open problems, but the widely believed conjecture is that P is strictly contained in NP. NP is strictly contained in #1'. and #1' is strictly contained in PSPACE. In other words. it is widely believed that MP-complete problems cannot he solved in polynomial time, #P-complete problems are harder than NP-complete problems, and PSPACE-complete problems are harder than #Pcomplete problems. A pictorial illustration of the complexity classes is shown in Figure I. The first problem is motivated by ecological dynamics. There is an ecosystem occupied by resident species. The spatial structure of the ecosystem is given by a graph. An invading species is introduced (see Figure 2 for an illustration). We assume the invading species has a competitive advantage in the sense that once a position is occupied by the invading species the resident cannot get it back. The invading species, however, has a density constraint: if the number of invader,. around a focal invader is above a threshold, h, then the Invader in the focal node can not colonize another node. We are interested in the probability that the invader starting from a random initial position will take over the entire ecosystem (and therefore drive the resident to extinction). There are two types of questions. The 'qualitative question' is whether the take over probability is greater than zero. The 'quantitative question' is concerned with computing the take over probability subject to a small error. Figure 2 gives a pictorial illustration. We prove the following results. The qualitative question is NP-complete (SI Theorem 1). The quantitive question is #P-complete (SI Theorem 8). The second problem is concerned with evolutionary game; in structured populations. There are two types, A and B. whose reproductive rates depend on local interactions. We consider the setting of games on graphs. Each vertex is occupied by one individual. which is either A or B. Interactions occur pairwise with all neighbors. The payoff matrix is given by A B A ( o b) (I) B c d The entries of the payoff matrix can he positive or negative (or zero). Each individual interacts with all of its neighbors on the graph to derive a payoff sum. The payoff sum is translated into reproductive success as follows. If the payoff sum is positive. then the fecundity equals the payoff sum. If the payoff sum is negative. then the fecundity is zero. We refer to this translation as linear fitness. In any one time step, a random individual is chosen for reproduction proportional to its fecundity. The offspring. which is of the same type as the parent, is placed into an adjacent position on the graph (see Figure 3 and Figure 4 for an illustration). We are interested in the probability that a single A individual starting in a random position on the graph generous a lineage which will take over the entire population; this probability is generally called fixation probability. As before, there are two type; of questions. The 'qualitative question' is whether the fixation probability is positive. The 'quantitative question' is concerned with computing the fixation probability subject to a small error. We prove the following results. The qualitative question is NP-hard and in PSPACE. The quantiFootline Author EFTA_R1_01954779 EFTA02674310
April 17, 2026 at 8:03 PM