The Money Changing Problem Revisited: Computing the Frobenius Number in Time O(ka 1)

For a non-negative integer $p$, one of the generalized Frobenius numbers, which is called the $p$-Frobenius number, is the largest integer that is represented at most in $p$ ways as a linear combination with nonnegative integer coefficients of a given set of positive integers whose greatest common divisor is one. The famous so-called Frobenius number proposed by Frobenius is reduced to the $0$-Frobenius number when $p=0$. The explicit formula for the Frobenius number with two variables was found in the 19th century, but a formula with more than two variables is very difficult to find, and closed formulas of Frobenius numbers have been found only in special cases such as geometric, Thabit, Mersenne, and so on. The case of $p>0$ was even more difficult, and not a single formula was known. However, most recently, we have finally succeeded in giving the $p$-Frobenius numbers as closed-form expressions of the triangular number triplet ([8]), repunits ([9]), Fibonacci triplet ([12]) and Jacobsthal triplet ([11]). In this paper, we give closed-form expressions of the $p$-Frobenius number for the finite sequence $\{a b^n-c\}_n$, where $a$, $b$ and $c$ are integers with $a\ge 1$, $b\ge 2$ and $c\ne 0$. This sequence includes the cases for geometric, Thabit and Mersenne as well as their variations.

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

We present an application of a local recompression technique, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and replacement of pairs of letters appearing in the equation by a 'fresh' letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give new self-contained proofs of many known results for word equations: the presented nondeterministic algorithm runs in O(n log n) space and in time polynomial in log N and n, where N is the size of the length-minimal solution of the word equation. It can be easily generalised to a generator of all solutions of the word equation. A further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields a new proof of the exponential bound on exponent of periodicity. For O(1) variables with arbitrary many appearances it works in linear space.

Parallel algorithms are given for the exact solution of the hidden-line problem. Most of the parallel algorithms proposed for visibility problems in the literature give approximate solutions. and thus cannot yield an upper bound on the complexity of the particular problem. The first algorithm proposed here is worth mentioning not only for its simplicity. but also from a practical point of view: a speed up of a factor P is achieved by using P processors. 1≤P≤N, where N is the number of edges used to describe a polygonal scene. Additionally. the problem of aliasing inherent with approximation methods is avoided. The significance of the second algorithm, which is based on the first one, is mainly on the theoretical level: it is used to establish the parallel complexity of the hidden-line problem. The sequential complexity of this problem has recently been proved to be Θ(N2), and now we can prove that in the parallel case the problem is in the complexity class NC, i.e., it can be solved in time polynomial in log N by using a number of processors polynomial in N, assuming any reasonable model of parallel computation. More particularly, an O (log N) parallel time solution is given which cannot be further improved even if arbitrarily many processors of a concurrent read, exclusive write parallel RAM model are available.

The probably almost exact model (PAExact) can be viewed as the exact model relaxed so that: 1. The counterexamples to equivalence queries are distributionally drawn rather than adversarially chosen. 2. The output hypothesis is equal to the target with negligible error (1//spl omega/(poly) for any poly). This model allows studying (almost) exact learnability of infinite classes and is in some sense analogous to the Exact-learning model for finite classes. It is known that PAExact-learnable/spl rArr/PAC-learnable [BJT02]. In this paper we show that if a class is PAC-learnable (in polynomial time) then it is PAExact-learnable (in polynomial time). Therefore, PAExact-learnable=PAC-learnable. It follows from this result that if a class is PAC-learnable then it is learnable in the probabilistic prediction model from examples with an algorithm that runs in polynomial time for each prediction (polynomial in log(the number of trials)) and that after polynomial number of mistakes achieves a hypothesis that predicts the target with probability 1-1/2/sup poly/. We also show that if a class is PAC-learnable in parallel then it is PAExact-learnable in parallel.

Let x 1, x 2, ..., x n be positive integers. It is well-known that every sufficiently large integer can be represented as a non-negative integer linear combination of the x i if and only if \(\gcd(x_1, x_2, \ldots, x_n) = 1\). The Frobenius problem is the following: given positive integers x 1, x 2, ..., x n with \(\gcd(x_1, x_2, \ldots, x_n) = 1 \), compute the largest integer not representable as a non-negative integer linear combination of the x i . This largest integer is sometimes denoted g(x 1,..., x n ).KeywordsRegular ExpressionLarge IntegerRegular LanguageLonge WordFrobenius NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given relatively prime integers a1,…, an, the Frobenius number g(a1,…, an) is defined as the largest integer which cannot be expressed as x1a1 +⋯+ xnan with xi nonnegative integers. In this paper, we give the Frobenius number of primitive Pythagorean triples:

To be considered fast, algorithms for operations on large data structures should operate in polylog time, i.e., with the number of steps bounded by a polynomial in log(N) where N is the size of the data structure. Example: an ordered list of reasonably short strings can be searched in log2 (N) time via binary search. To measure the time and space complexity of such operations, the usual Turing machine with its serial-access input tape is replaced by a random access model. To compare such problems and define completeness, the appropriate relation is loglog reducibility: the relation generated by random-access transducers whose work tapes have length at most log(log(N)). The surprise is that instead of being a refinement of the standard log space, polynomial time, polynomial space, ... hierarchy, the complexity classes for these random-access Turing machines form a distinct parallel hierarchy, namely, polylog time, polylog space, exppolylog time, ... . Propositional truth evaluation, context-free language recognition and searching a linked list are complete for polylog space. Searching ordered lists and searching unordered lists are complete for polylog time and nondeterministic polylog time respectively. In the serial-access hierarchy, log-space reducibility is not fine enough to classify polylog-time problems and there can be no complete problems for polylog space even with polynomial-time Turing reducibility

This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2,...an such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the “covering radius”. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytopeK, the setK+ℤh={x∶x∈ℝn;x=y+z;y∈K;z∈ℤn} which is the portion of space covered by all lattice translates ofK. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovasz [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovasz and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.

We study the minimal generating systems of the irreducible numerical semigroups that can be represented as the set of solutions to a Diophantine inequality of the form ax mod b  cx with a, b and c positive integers. In particular we obtain a simple method to count all these numerical semigroups with a given Frobenius number

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect $$t^{th}$$ power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case $$t=1$$ . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of $$\mathbb {N}^t$$ , and Jensen’s inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

Gathering a group of mobile agents is a fundamental task in the field of distributed and mobile systems. This can be made drastically more difficult to achieve when some agents are subject to faults, especially the Byzantine ones that are known as being the worst faults to handle. In this paper we study, from a deterministic point of view, the task of Byzantine gathering in a network modeled as a graph. In other words, despite the presence of Byzantine agents, all the other (good) agents, starting from {possibly} different nodes and applying the same deterministic algorithm, have to meet at the same node in finite time and stop moving. An adversary chooses the initial nodes of the agents (the number of agents may be larger than the number of nodes) and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label. The agents move in synchronous rounds and can communicate with each other only when located at the same node. Within the team, f of the agents are Byzantine. A Byzantine agent acts in an unpredictable and arbitrary way. For example, it can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. Besides its label, which corresponds to a local knowledge, an agent is assigned some global knowledge denoted by GK that is common to all agents. In literature, the Byzantine gathering problem has been analyzed in arbitrary n-node graphs by considering the scenario when GK=(n,f) and the scenario when GK=f. In the first (resp. second) scenario, it has been shown that the minimum number of good agents guaranteeing deterministic gathering of all of them is f+1 (resp. f+2). However, for both these scenarios, all the existing deterministic algorithms, whether or not they are optimal in terms of required number of good agents, have the major disadvantage of having a time complexity that is exponential in n and L, where L is the value of the largest label belonging to a good agent. In this paper, we seek to design a deterministic solution for Byzantine gathering that makes a concession on the proportion of Byzantine agents within the team, but that offers a significantly lower complexity. We also seek to use a global knowledge whose the length of the binary representation (that we also call size) is small. In this respect, assuming that the agents are in a strong team i.e., a team in which the number of good agents is at least some prescribed value that is quadratic in f, we give positive and negative results. On the positive side, we show an algorithm that solves Byzantine gathering with all strong teams in all graphs of size at most n, for any integers n and f, in a time polynomial in n and the length |l_{min}| of the binary representation of the smallest label of a good agent. The algorithm works using a global knowledge of size O(log log log n), which is of optimal order of magnitude in our context to reach a time complexity that is polynomial in n and |l_{min}|. Indeed, on the negative side, we show that there is no deterministic algorithm solving Byzantine gathering with all strong teams, in all graphs of size at most n, in a time polynomial in n and |l_{min}| and using a global knowledge of size o(log log log n).

Given a d-dimensional convex polytope P and nonnegative integer k not exceeding d−1, let ${\mathcal{G}}_{k}(P)$ denote the simple graph on the node set of k-dimensional faces of P in which two such faces are adjacent if there exists a (k+1)-dimensional face of P which contains them both. The graph ${\mathcal{G}}_{k}(P)$ is isomorphic to the dual graph of the (d−k)-dimensional skeleton of the normal fan of P. For fixed values of k and d, the largest integer m such that ${\mathcal{G}}_{k}(P)$ is m-vertex-connected for all d-dimensional polytopes P is determined. This result generalizes Balinski’s theorem on the one-dimensional skeleton of a d-dimensional convex polytope.

b-Chromatic Sum of Mycielskian of Paths