Component Order Connectivity in Directed Graphs

A digraph is semicomplete if it has no pair of non-adjacent vertices. It is complete if every pair of distinct vertices induces a 2-cycle. A digraph is semicomplete multipartite if it can be obtained from a semicomplete digraph D by choosing a collection of vertex-disjoint subsets X₁,…,X_c of V(D) and then deleting all arcs both of whose end-vertices lie inside some X_i. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets V₁,…,V_c by adding the arcs of a semicomplete digraph D_i on V_i for each partite set V_i. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph S which is obtained as above from a semicomplete multipartite digraph D and semicomplete digraphs D_i=(V_i,A_i), i∈[c] such that the path or cycle uses as few arcs of A₁∪…A_c as possible. We obtain a number of results for the case when each D_i is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the D_i’s as possible in polynomial time and obtain a number of results, both structural and algorithmic on hamiltonian cycles that use the minimum or close to the minimum number of arcs from the D_i’s. Our results imply the polynomial solvability of some special cases of the NP-complete {0,1}-TSP problem. Finally we show that two natural questions about properties of quasi-hamiltonian cycles, that is, cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

A polynomial algorithm for hamiltonian-connectedness in semicomplete digraphs

Minimum cost and list homomorphisms to semicomplete digraphs

Decomposing locally semicomplete digraphs into strong spanning subdigraphs

We propose a new approach to the Directed Feedback Vertex Set Problem (DFVSP), where the input is a directed graph and the solution is a minimum set of vertices whose removal makes the graph acyclic. Our approach, implemented in the solver DAGER, is based on two novel contributions: Firstly, we add a wide range of data reductions that are partially inspired by reductions for the similar vertex cover problem. For this, we give a theoretical basis for lifting reductions from vertex cover to DFVSP but also incorporate novel ideas into strictly more general and new DFVSP reductions. Secondly, we propose dynamically encoding DFVSP in propositional logic using cycle propagation for improved performance. Cycle propagation builds on the idea that already a limited number of the constraints in a propositional encoding is usually sufficient for finding an optimal solution. Our algorithm, therefore, starts with a small number of constraints and cycle propagation adds additional constraints when necessary. We propose an efficient integration of cycle propagation into the workflow of MaxSAT solvers, further improving the performance of our algorithm. Our extensive experimental evaluation shows that DAGER significantly outperforms the state-of-the-art solvers and that our data reductions alone directly solve many of the instances. * The full version of the paper can be accessed at http://arxiv.org/abs/2211.06109

formula omitted]-strong spanning local tournaments in locally semicomplete digraphs

The so-called Kelly conjecture1 A proof of the Kelly conjecture for large k has been announced by R. Haggkvist at several conferences and in [5] but to this date no proof has been published.states that every regular tournament on 2k+1 vertices has a decomposition into k-arc-disjoint hamiltonian cycles. In this paper we formulate a generalization of that conjecture, namely we conjecture that every k-arc-strong tournament contains k arc-disjoint spanning strong subdigraphs. We prove several results which support the conjecture:If D = (V, A) is a 2-arc-strong semicomplete digraph then it contains 2 arc-disjoint spanning strong subdigraphs except for one digraph on 4 vertices.Every tournament which has a non-trivial cut (both sides containing at least 2 vertices) with precisely k arcs in one direction contains k arc-disjoint spanning strong subdigraphs. In fact this result holds even for semicomplete digraphs with one exception on 4 vertices.Every k-arc-strong tournament with minimum in- and out-degree at least 37k contains k arc-disjoint spanning subdigraphs H1, H2, . . . , Hk such that each Hi is strongly connected.The last result implies that if T is a 74k-arc-strong tournament with speci.ed not necessarily distinct vertices u1, u2, . . . , uk, v1, v2, . . . , vk then T contains 2k arc-disjoint branchings $$F^{ - }_{{u_{1} }} ,F^{ - }_{{u_{2} }} ,...,F^{ - }_{{u_{k} }} ,F^{ + }_{{v_{1} }} ,F^{ + }_{{v_{2} }} ,...,F^{ + }_{{v_{k} }}$$ where $$F^{ - }_{{u_{i} }}$$ is an in-branching rooted at the vertex ui and $$F^{ + }_{{v_{i} }}$$ is an out-branching rooted at the vertex vi, i=1,2, . . . , k. This solves a conjecture of Bang-Jensen and Gutin [3].We also discuss related problems and conjectures.

The traditional algorithm of L. Stockmeyer (1983) for area minimization of slicing floorplans has time (and space) complexity O(n/sup 2/) in the worst case, or O(n log n) for balanced slicing. For more than a decade, it is considered the best possible. In this paper, we present a new algorithm of worst-case time (and space) complexity O(n log n), where n is the total number of realizations for the basic blocks, regardless whether the slicing is balanced or not. We also prove /spl Omega/(n log n) is the lower bound and the time complexity of any area minimization algorithm. Therefore, the new algorithm not only finds the optimal realization, but also has an optimal running time.

We evaluate the performance of FPT algorithms for the directed feedback vertex set problem (DFVS). We propose several new data reduction rules for DFVS. which can significantly reduce the search space. We also propose various heuristics to accelerate the FPT search. Finally, we demonstrate that DFVS is not more helpful for deadlock recovery (with mutex locks) than simple cycle detection.

Big data is technology that has the ability to manage very large amounts of data, in very fast time to allow real-time analysis and reactions. Several clustering methods which are used to group data are Fuzzy C-Means (FCM) and K-Means Clustering. K-Means Clustering algorithm is a method of partitioning existing data into two or more group. This research goal was to compare the performance of K-Means and Fuzzy C-Means algorithms in clustering data using big data technology. In this research, Hadoop and Hive were chosen the big data technology. The knowledge of Shia history on student and lecturer of Syarif Hidayatullah State Islamic University Jakarta were the data which used in this research, The testing was done by constructing application K-Means Fuzzy C-Means using Java language, Hadoop and Hive and then test the performance of K-Means and Fuzzy C-Means algorithms in data clustering. It compares both algorithms in terms of accuracy, execution time, and time complexity of the algorithms. In the application K-Means Fuzzy C-Means, evaluation were performed with data filter and the average accuracy difference result of K-Means and Fuzzy C-Means is 8.03% with the better accuracy owned by K-Means. The average execution time difference is 718.58 ms, which K-Means was faster than is Fuzzy C-Means. The time complexities of both algorithms have the same value O(n2) and the Big O equation resulted in an average difference of 93,568 with the smallest value on K-Means. Thus, K-Means algorithm is better than the Fuzzy C-Means in terms of accuracy, execution time, and the time complexity.

Planar graphs are known to allow subexponential algorithms running in time \(2^{O(\sqrt{n})}\) or \(2^{O(\sqrt{n} \log n)}\) for most of the paradigmatic problems, while the brute-force time \(2^{\varTheta (n)}\) is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in \(2^{O(n^{2/3}\log n)}\) by Fox and Pach [SODA’11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time \(2^{O(n^{2/3}\log ^{O(1)}n)}\) on string graphs while an algorithm running in time \(2^{o(n)}\) for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a \(2^{o(n^{2/3})}\) algorithm (under the ETH). The construction exploits the celebrated Erdős-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.

Let be a class of probability distributions over the discrete domain [n] = {1, …, n}. We show that if satisfies a rather general condition – essentially, that each distribution in can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknown distributions from . We analyze several natural types of distributions over [n], including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems as described below. More precisely, Log-concave distributions: We learn any mixture of k log-concave distributions over [n] using k · Õ(1/ε4) samples (independent of n) and running in time Õ(k log(n)/ε4) bit-operations (note that reading a single sample from [n] takes Θ(log n) bit operations). For the special case k = 1 we give an efficient algorithm using Õ(1/ε3) samples; this generalizes the main result of [DDS12b] from the class of Poisson Binomial distributions to the much broader class of all log-concave distributions. Our upper bounds are not far from optimal since any algorithm for this learning problem requires Ω(k/ε5/2) samples. Monotone hazard rate (MHR) distributions: We learn any mixture of k MHR distributions over [n] using O(k log(n/ε)/ε4) samples and running in time Õ(k log (n)/ε4) bit-operations. Any algorithm for this learning problem must use Ω(k log(n)/ε3) samples. Unimodal distributions: We give an algorithm that learns any mixture of k unimodal distributions over [n] using O(k log(n)/ε4) samples and running in time Õ(k log2(n)/ε4) bit-operations. Any algorithm for this problem must use Ω(k log(n)/ε3) samples.

Transportation problems are one of the particular forms that often appear in linear programs, one of which is the distribution of goods. A transportation method is needed to determine the optimal result, namely, the minimum cost from source to destination with all demand and supply fulfilled. There are several methods, one of which is the Lowest Supply Lowest Cost Method (LSLC) and the Exponential Approach Method (PE). Both methods are made in a MATLAB program, generating a script that calculates the algorithm's time complexity. Using the function notation, the Big-O Algorithm complexity of the Lowest Supply Lowest Cost method is more efficient than the Exponential Approach Method algorithm. At the same time, the optimal result for the minimum cost between the two methods is obtained by using the Exponential Approach Method.

Matching and packing problems have formed an important class of NP-hard problems. There have been a number of recently developed techniques for parameterized algorithms for these problems, including greedy localization, color-coding plus dynamic programming, and randomized divide-and-conquer. In this paper, we provide further theoretical study on the structures of these problems, and develop improved algorithmic methods that combine existing and new techniques to obtain improved algorithms for matching and packing problems. For the 3-set packing problem, we present a deterministic algorithm of time O *(4.613k), which significantly improves the previous best deterministic algorithm of time O *(12.83k). For the 3-d matching problem, we develop a new randomized algorithm of running time O *(2.323k) and a new deterministic algorithm of running time O *(2.773k). Our randomized algorithm improves the previous best randomized algorithm of running time O *(2.523k), and our deterministic algorithm significantly improves the previous best deterministic algorithm of running time O *(12.83k). Our results also imply improved algorithms for various triangle packing problems in graphs.KeywordsTime ComplexityDynamic ProgrammingMatch ProblemPacking ProblemDynamic Programming AlgorithmThese 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.

Point location problems for n points in d-dimensional Euclidean space (and ep spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than dpoly(d) · n logO(d) n time, or (Barely-Subquadratic) f(d) · n2−1/Θ(d) time, for various f. For small d and large n, running times are generally feasible, while the times are generally infeasible, requiring essentially quadratic time. For example, in the Euclidean metric, finding a Closest Pair among n points in Rd is nearly-linear, solvable in 2o(d) · n logO(1) n time, while the known algorithms for finding a Furthest Pair (the diameter of the point set) are only barely-subquadratic, requiring Ω(n2--1/Θ(d)) time. Why do these proximity problems have such different time complexities? Is there a barrier to obtaining nearly-linear algorithms for problems which are currently only barely-subquadratic? We give a novel exact and deterministic self-reduction for the Orthogonal Vectors problem on n vectors in {0, 1}d to n vectors in Zω(log d) that runs in 2o(d) time. As a consequence, barely-subquadratic problems such as Euclidean diameter, Euclidean bichromatic closest pair, and incidence detection do not have O(n2−ϵ) time algorithms (in Turing models of computation) for dimensionality d = ω(log log n)2, unless the popular Orthogonal Vectors Conjecture and the Strong Exponential Time Hypothesis are false. That is, while the poly-log-log-dimensional case of Closest Pair is solvable in n1+o(1) time, the poly-log-log-dimensional case of Furthest Pair can encode difficult large-dimensional problems conjectured to require n2−o(1) time. We also show that the All-Nearest Neighbors problem in ω(log n) dimensions requires n2−o(1) time to solve, assuming either of the above conjectures.