A General Framework for Approximating Min Sum Ordering Problems

We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of min sum set cover, several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the min sum ordering problem (MSOP), which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time α-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time [Formula: see text]-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results. Summary of Contribution: This paper provides a general framework for min sum ordering problems. Within the realm of theoretical computer science, these problems include min sum set cover and its generalizations, as well as problems in Boolean function evaluation. On the operations research side, they include problems in search theory and scheduling. We present and analyze a very general algorithm for these problems, unifying several previous results on various min sum ordering problems and resulting in new constant factor guarantees for others.

Categorical properties of intuitionistic fuzzy groups

The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a unifying notion. In this paper, we study the relationship between the category of groups and the category of intuitionistic fuzzy groups. We prove that the category of groups is a subcategory of category of intuitionistic fuzzy groups and that it is not an Abelian category. We establish a function β : Hom(A, B) → [0; 1] × [0; 1] on the set of all intuitionistic fuzzy homomorphisms between intuitionistic fuzzy groups A and B of groups G and H, respectively. We prove that β is a covariant functor from the category of groups to the category of intuitionistic fuzzy groups. Further, we show that the category of intuitionistic fuzzy groups is a top category by establishing a contravariant functor from the category of intuitionistic fuzzy groups to the lattices of all intuitionistic fuzzy groups.

Sufficient conditions for the optimality of the greedy algorithm in greedoids

Greedy algorithms are among the most elementary ones in theoretical computer science and understanding the conditions under which they yield an optimum solution is a widely studied problem. Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. One of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Indeed, many well-known algorithms of a greedy nature that cannot be interpreted in a matroid-theoretical context are special cases of the greedy algorithm on greedoids. Although this algorithm turns out to be optimal in surprisingly many cases, no general theorem is known that explains this phenomenon in all these cases. Furthermore, certain claims regarding this question that were made in the original works of Korte and Lovász turned out to be false only most recently. The aim of this paper is to revisit and straighten out this question: we summarize recent progress and we also prove new results in this field. In particular, we generalize a result of Korte and Lovász and thus we obtain a sufficient condition for the optimality of the greedy algorithm that covers a much wider range of known applications than the original one.

Using Transition Systems to Formalise Ideas from Vedanta

Vedanta is one of the oldest philosophical systems. While there are many detailed commentaries on Vedanta, there are very few mathematical descriptions of the different concepts developed there. This article shows how ideas from theoretical computer science can be used to explain Vedanta. The standard idea of transition systems and modal logic are used to develop a formal description for the different ideas in Vedanta. The generality of the formalism is illustrated via a number of examples including \samsara, \Patanjali's yoga sutras, karma, the three avasthas from the Mandukya Upanishad and the key difference between advaita and dvaita in relation to moksha.

How to Construct Quantum Random Functions

Pseudorandom functions ( PRFs ) are one of the foundational concepts in theoretical computer science, with numerous applications in complexity theory and cryptography. In this work, we study the security of PRFs when evaluated on quantum superpositions of inputs. The classical techniques for arguing the security of PRFs do not carry over to this setting, even if the underlying building blocks are quantum resistant. We therefore develop a new proof technique to show that many of the classical PRF constructions remain secure when evaluated on superpositions.

Review of Communication Complexity and Applications by Anup Rao and Amir Yehudayoff

At its core, communication complexity is the study of the amount of information two parties need to exchange in order to compute a function. For instance, Alice receives a string of characters, Bob receives another, and they should decide whether these strings are the same with as few rounds of communication as possible. Multiple settings are conceivable, for instance with multiple parties or with randomness. Upper and lower bounds for communication problems rely on a wealth of mathematical tools, from probability theory to Ramsey theory, making this a complete and exciting topic. Further, communication complexity finds applications in different aspects of theoretical computer science, including circuit complexity and data structures. This usually requires to take a "communication" view of a problem, which in itself can be an eye-opening vantage point.

An improved cuckoo search algorithm with Stud crossover for Chinese TSP problem#

The travelling salesman problem (TSP) is an NP-hard problem in combinatorial optimization. It has assumed significance in operations research and theoretical computer science. The problem was first formulated in 1930 and since then, has been one of the most extensively studied problems in optimization. In fact, it is used as a benchmark for many optimization methods. This paper represents a new method to addressing TSP using an improved version of cuckoo search (CS) with Stud (SCS) crossover operator. In SCS method, similar to genetic operators used in various metaheuristic algorithms, a Stud crossover operator that is originated from classical Stud genetic algorithm, is introduced into the CS with the aim of improving its effectiveness and reliability while dealing with TSP. Various test functions had been used to test this approach, and used subsequently to find the shortest path for Chinese TSP (CTSP). Experimental results presented clearly demonstrates SCS as a viable and attractive addition to the portfolio of swarm intelligence techniques.

Algorithm NextFit for the Bin Packing ProblemThis work was supported by JSPS KAKENHI Grant Numbers JP20K11689, JP20K11676, JP16K00033, JP17K00013, JP20K11808, and JP17K00183.

Summary. The bin packing problem is a fundamental and important optimization problem in theoretical computer science [4], [6]. An instance is a sequence of items, each being of positive size at most one. The task is to place all the items into bins so that the total size of items in each bin is at most one and the number of bins that contain at least one item is minimum. Approximation algorithms have been intensively studied. Algorithm NextFit would be the simplest one. The algorithm repeatedly does the following: If the first unprocessed item in the sequence can be placed, in terms of size, additionally to the bin into which the algorithm has placed an item the last time, place the item into that bin; otherwise place the item into an empty bin. Johnson [5] proved that the number of the resulting bins by algorithm NextFit is less than twice the number of the fewest bins that are needed to contain all items. In this article, we formalize in Mizar [1], [2] the bin packing problem as follows: An instance is a sequence of positive real numbers that are each at most one. The task is to find a function that maps the indices of the sequence to positive integers such that the sum of the subsequence for each of the inverse images is at most one and the size of the image is minimum. We then formalize algorithm NextFit, its feasibility, its approximation guarantee, and the tightness of the approximation guarantee.

An Efficient Label-Correcting Algorithm for the Multiobjective Shortest Path Problem

This paper proposes an exact algorithm to solve the one-to-one multiobjective shortest path problem. The solution involves determining a set of nondominated paths between two given nodes in a graph that minimizes several objective functions. This study is motivated by the application of this solution method to determine cycling itineraries. The proposed algorithm improves upon a label-correcting algorithm to rapidly solve the problem on large graphs (i.e., up to millions of nodes and edges). To verify the performance of the proposed algorithm, we use computational experiments to compare it with the best-known methods in the literature. The numerical results confirm the efficiency of the proposed algorithm. Summary of Contribution: The paper deals with a classic operations research problem (the one-to-one multiobjective shortest path problem) and is motivated by a real application for cycling itineraries. An efficient method is proposed and is based on a label-correcting algorithm into which several additional improvement techniques are integrated. Computational experiments compare this algorithm with the best-known methods in the literature to validate the performance on large-size graphs (Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) instances from the ninth DIMACS challenge). New instances from the context of cycling itineraries are also proposed.

Uncontrolled mutation of super-intelligent machines may lead to the destruction of humanity

<p>If in future, the highly intelligent machines control the world, then what would be its advantages and disadvantages? Will, those artificial intelligence powered superintelligent machines become an anathema for humanity or will they ease out the human works by guiding humans in complicated tasks, thereby extending a helping hand to the human works making them comfortable. Recent studies in theoretical computer science especially artificial intelligence predicted something called ‘technological singularity’ or the ‘intelligent explosion’ and if this happens then there can be a further stage as transfused machinery intelligence and actual intelligence where the machines being immensely powerful with a cognitive capacity more than that of humans for solving ‘immensely complicated tasks’ can takeover the humans and even the machines by more intelligent machines of superhuman intelligence. Therefore, it is troublesome and worry-full to think that ‘if in case the machines turned out against humans for their optimal domination in this planet’. Can humans have any chances to avoid them by bypassing the inevitable ‘hard singularity’ through a set of ‘soft singularity’. This paper discusses all the facts in details along with significant calculations showing humanity, how to avoid the hard singularity when the progress of intelligence is inevitable. </p>