A polynomial time algorithm for Sylvester waves when entries are bounded

This paper deals with the presentation of polynomial time (approximation) algorithms for a variant of open‐shop scheduling, where the processing times are only machine‐dependent. This variant of scheduling is called proportionate scheduling and its applications are used in many real‐world environments. This paper develops three polynomial time algorithms for the problem. First, we present a polynomial time algorithm that solves the problem optimally if , where n and m denote the numbers of jobs and machines, respectively. If, on the other hand, , we develop a polynomial time approximation algorithm with a worst‐case performance ratio of that improves the bound existing for general open‐shops. Next, in the case of , we take into account the problem under consideration as a master problem and convert it into a simpler secondary approximation problem. Furthermore, we formulate both the master and secondary problems, and compare their complexity sizes. We finally present another polynomial time algorithm that provides optimal solution for a special case of the problem where .

P versus NP is an unsolved problem in mathematics and computational complexity. In this paper, we use the formal language theory to the computational complexity to analyze P versus NP problem from a new point of view. P versus NP problem is to determine whether some deterministic algorithm also accepts every language accepted by some nondeterministic algorithm in polynomial time in polynomial time. Then, we use the theory of formal languages to determine whether some deterministic algorithm also accepts every language accepted by some nondeterministic algorithm in polynomial time in polynomial time. We use different problems to display the question of P versus NP from Formal Languages Theory View.

The theory of NP-hardness has been very successful in identifying problems that are unlikely to be solvable in polynomial time. However, many other important problems do have polynomial time algorithms, but large exponents in their time bounds can make them run for days, weeks or more. For example, quadratic time algorithms, although practical on moderately sized inputs, can become inefficient on big data problems that involve gigabytes or more of data. Although for many problems no sub-quadratic time algorithms are known, any evidence of quadratic-time hardness has remained elusive. In this talk I will give an overview of recent research that aims to remedy this situation. In particular, I will describe hardness results for problems in string processing (e.g., edit distance computation or regular expression matching) and machine learning (e.g., Support Vector Machines or gradient computation in neural networks). All of them have polynomial time algorithms, but despite extensive amount of research, no near-linear time algorithms have been found for many variants of these problems. I will show that, under a natural complexity-theoretic conjecture, such algorithms do not exist. I will also describe how this framework has led to the development of new algorithms.

Rarely do mathematical disciplines have so direct practical relevance as combinatorial optimizations, which is why it is one of the most active areas of discrete mathematics. It became a subject in its own right only about 50 years ago, which makes it one of the youngest also.This book covers most of the important results and algorithms achieved in the field to date. Most of the problems are formulated in terms of graphs and linear programs. The book starts with reviewing basic graph theory and linear and integer programming. Next, the classical topics in the field are studied: minimum spanning trees, shortest paths, network flows, matching and matroids.Most of the problems in Chapters 6-14 have polynomial time (efficient) algorithms, while most of the problems studied in Chapters 15-21 are NP-hard, i.e. polynomial time algorithm is unlikely to exist. Although in many cases approximation algorithms are offered which at least have guaranteed performance.Some of the topics include areas which have developed very recently, and which have not appeared in a book before. Examples are algorithms for multicommodity flows, network design problems and the traveling salesman problem. The book also contains some new results and new proofs for previously known results.The authors have an unique approach in the presentation of the topics. They provided detailed proofs for almost all results, including deep classical theorems (e.g. weighted matching algorithm and Karmarkar-Karp bin-packing algorithm) whose proofs are usually sketched in previous works.

Embedding graphs on the torus is a problem with both theoretical and practical importance. It is required to embed a graph on the torus for solving many application problems such as VLSI design, graph drawing and etc. Polynomial time and exponential time algorithms for embedding graphs on the torus are known. However, the polynomial time algorithms are very complex and their implementation has been a challenge for a long time. On the other hand, the implementations of some exponential time algorithms are known but they are not efficient for large graphs in practice. To develop an efficient practical tool for embedding graphs on the torus, we propose a new exponential time algorithm for embedding graphs on the torus. Compared with a well used previous exponential time algorithm, our algorithm has a better practical running time.

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of $n$ jobs, each job having a release time $r_j$, a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial} time algorithm that has an $O(1)$ approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar settles the long standing conjecture that there is a polynomial time algorithm with $O(1)$-approximation for min-WPFT.

SIGEST

The P systems (or membrane systems) are a class of distributed parallel computing devices of a biochemical type, where membrane division is the frequently investigated way for obtaining an exponential working space in a linear time, and on this basis solving hard problems, typically NP-complete problems, in polynomial (often, linear) time. In this paper, using another way to obtain exponential working space --- membrane separation, it was shown that Satisfiability Problem and Hamiltonian Path Problem can be deterministically solved in linear or polynomial time by a uniform family of P systems with separation rules, where separation rules are not changing labels, but polarizations are used. Some related open problems are mentioned.

Purpose of work is the development of a new method for estimating the quantum resilience of modern blockchain platforms based on the effective solution of cryptanalysis problems for asymmetric encryption schemes (RSA, El-Gamal) and digital signature (DSA, ECDSA or RSA-PSS), based on computationally difficult problems of factorization and discrete logarithm. Research method is the use of quantum algorithms providing exponential gain (eg Shor’s algorithm) and quadratic gain (eg Grover’s algorithm). Due to the fact that the class of problems solved by quantum algorithms in polynomial time cannot yet be significantly expanded, more attention is paid to cryptanalysis based on the quantum Shor algorithm and other polynomial algorithms. Results of the study include a classification of well-known algorithms and software packages for cryptanalysis of asymmetric encryption schemes (RSA, El-Gamal) and digital signature (DSA, ECDSA or RSA-PSS) based on computationally difficult problems of factorization and discrete logarithm has been built. A promising method for solving problems of cryptanalysis of asymmetric encryption schemes (RSA, ElGamal) and digital signature (DSA, ECDSA or RSA-PSS) of known blockchain platforms in polynomial time in a quantum computing model is proposed. Algorithms for solving problems of quantum cryptanalysis of two-key cryptography schemes of known blockchain platforms in polynomial time are developed, taking into account the security of the discrete algorithm (DLP) and the discrete elliptic curve algorithm (ECDLP). A structural and functional diagram of the software package for quantum cryptanalysis of modern blockchain platforms “Kvant-K”, adapted to work in a hybrid computing environment of the IBM Q quantum computer (20 and 100 qubits) and the IBM BladeCenter (2022) supercomputer, has been designed. A methodology has been developed for using the “Kvant-K” software package to assess the quantum stability of blockchain platforms: InnoChain (Innopolis University), Waves Enterprise (Waves, Vostok), Hyperledger Fabric (Linux, IBM), Corda Enterprise, Bitfury Exonum, Blockchain Industrial Alliance, Exonum (Bitfury CIS), NodesPlus (b41), Masterchain (Sberbank), Microsoft Azure Blockchain, Enterprise Ethereum Alliance, etc. Practical relevance: The developed new solution for computationally difficult problems of factorization and discrete logarithm, given over finite commutative (and non-commutative) associative algebras, in a quantum model of computing in polynomial time. It is essential that the obtained scientific results formed the basis for the development of the corresponding software and hardware complex “Kvant-K”, which was tested in a hybrid computing environment (quantum computer IBM Q (20 and 100 qubits) and/or 5th generation supercomputer: IBM BladeCenter (2022), RCS based on FPGA Virtex UltraScale (2020), RFNC-VNIIEF (2022) and SKIF P-0.5 (2021). An appropriate method for estimating the quantum stability of these blockchain platforms based on the author’s models, methods and algorithms of quantum cryptanalysis has been developed and tested. Keywords: blockchain and distributed ledger technologies (DLT), SMART contracts, blockchain security threat model, quantum security threat, cryptographic attacks, quantum cryptanalysis, quantum and post-quantum cryptography, quantum algorithms Shor, Grover and Simon algorithms, quantum Fourier transform, factorization and discrete logarithm problem, post-quantum cryptography, quantum resilience of blockchain platforms.

Computer science and physics have been closely linked since the birth of modern computing. This book is about that link. John von Neumann’s original design for digital computing in the 1940s was motivated by applications in ballistics and hydrodynamics, and his model still underlies today’s hardware architectures. Within several years of the invention of the first digital computers, the Monte Carlo method was developed, putting these devices to work simulating natural processes using the principles of statistical physics. It is difficult to imagine how computing might have evolved without the physical insights that nurtured it. It is impossible to imagine how physics would have evolved without computation. While digital computers quickly became indispensable, a true theoretical understanding of the efficiency of the computation process did not occur until twenty years later. In 1965, Hartmanis and Stearns [227] as well as Edmonds [139, 140] articulated the notion of computational complexity, categorizing algorithms according to how rapidly their time and space requirements grow with input size. The qualitative distinctions that computational complexity draws between algorithms form the foundation of theoretical computer science. Chief among these distinctions is that of polynomial versus exponential time. A combinatorial problem belongs in the complexity class P (polynomial time) if there exists an algorithm guaranteeing a solution in a computation time, or number of elementary steps of the algorithm, that grows at most polynomially with input size. Loosely speaking, such problems are considered computationally feasible. An example might be sorting a list of n numbers: even a particularly naive and inefficient algorithm for this will run in a number of steps that grows as O(n2), and so sorting is in the class P. A problem belongs in the complexity class NP (non-deterministic polynomial time) if it is merely possible to test, in polynomial time, whether a specific presumed solution is correct. Of course, P ⊆ NP: for any problem whose solution can be found in polynomial time, one can surely verify the validity of a presumed solution in polynomial time.

In this talk we describe a new type of probabilistic algorithm which we call Algorithms: a randomized algorithm which is guaranteed to run in expected polynomial time, and to produce a correct and unique solution with high probability. These algorithms are pseudo-deterministic: they can not be distinguished from deterministic algorithms in polynomial time by a probabilistic polynomial time observer with black box access to the algorithm. We show a necessary and sufficient condition for the existence of a Bellagio Algorithm for an NP relation R: R has a Bellagio algorithm if and only if it is deterministically reducible to some decision problem in BPP. Several examples of Bellagio algorithms, for well known problems in algebra and graph theory which improve on deterministic solutions, follow. The notion of pseudo-deterministic algorithms (or more generally computations) is interesting beyond just sequential algorithms. In particular, it has long been known that it is impossible to solve deterministically tasks such as consensus in a faulty distributed systems, whereas randomized protocols can achieve consensus in expected constant time. We thus explore the notion of pseudo-deterministic fault tolerant distributed protocols: randomized protocols which are polynomial time indistinguishable from deterministic protocols in presence of faults.

This paper considers multi-cell decode-and-forward (DF) relay-aided orthogonal frequency division multi-access (OFDMA) downlink systems, in which all sources and relays are coordinated by a central controller for resource allocation (RA). The improved subcarrier pair-based opportunistic DF relaying protocol proposed and studied in the IEEE International Conference on Communications, Beijing, 3795–3800, 2008 and IEEE Trans. Signal Process. 61:2512–2524, 2013 is applied. This protocol has a high spectrum efficiency (HSE) in the sense that all unpaired subcarriers are utilized for data transmission during the second time slot (TS). In particular, the sum (over all cells and all destinations) rate maximized problem with a total power constraint in each cell+ is formulated. To solve this problem, an iterative RA algorithm is proposed to optimize mode selection (decision whether the relay should help or not), subcarrier assignment and pairing (MSSAP) and power allocation (PA) in an alternate way. As for the MSSAP stage of each iteration, the formulated problem is decoupled into subproblems with the tentative PA results. Each subproblem can be easily solved by using the optimal results of a linear assignment problem (LAP), which is then solved by the Hungarian Algorithm in polynomial time. As for the PA stage of each iteration, an algorithm based on single-condensation and geometric programming (SCGP) is proposed to optimize PA in polynomial time with the tentative MSSAP results. The proposed algorithm is coordinate ascent (CA)-based and therefore can reach a local optimum in polynomial time. Finally, the convergence and effectiveness of the proposed algorithm, the impact of relay position and total power on the system performance, and the benefits of using subcarrier pairing (SP) and the HSE protocol are illustrated through numerical experiments.

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the gridlike minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter; the only other result with a similar flavor that is known to us is due to Demaine and Hajiaghayi and obtains a doubly-exponential bound on the parameter (albeit, for a more general class of parameterized problems). The second part of our work deals with providing a lower bound to Courcelle's famous theorem from almost two decades ago, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Whereas much work has been done on designing, improving, and applying algorithms on graphs of bounded treewidth, not much is known on the side of lower bounds: what bound on the treewidth of a class of graphs “forbids” polynomial-time parameterized algorithms to decide MSO-sentences? This question has only recently received attention with the first systematic study appearing in [Kreutzer 2009]. Using our results from the first part of our work we can improve on it significantly and establish a strong lower bound for Courcelle's theorem on classes of colored graphs.

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter; the only other result with a similar flavor that is known to us is due to Demaine and Hajiaghayi and obtains a doubly-exponential bound on the parameter (albeit, for a more general class of parameterized problems).The second part of our work deals with providing a lower bound to Courcelle's famous theorem from almost two decades ago, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Whereas much work has been done on designing, improving, and applying algorithms on graphs of bounded treewidth, not much is known on the side of lower bounds: what bound on the treewidth of a class of graphs forbids polynomial-time parameterized algorithms to decide MSO-sentences? This question has only recently received attention with the first systematic study appearing in [Kreutzer 2009]. Using our results from the first part of our work we can improve on it significantly and establish a strong lower bound for Courcelle's theorem on classes of colored graphs.