Efficient Algorithms for the Inverse Spanning-Tree Problem

Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree $$\le f(x)$$ , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection $$\mathcal{A}$$ of edge-disjoint augmenting trails such that $$M\bigoplus _{A\in {\mathcal{A}}} A $$ is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs. arXiv:2010.01102 , 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time $$O(m\alpha (m,n))$$ . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time $$O(m\,\alpha (m,n))$$ using Huang and Pettie (2022), and our algorithm eliminates the factor $$\alpha (m,n)$$ . Similarly a maximum weight f-factor is found in time $$O(\sqrt{\Phi \log \Phi }\, m\,\alpha (m,n)\, \log (\Phi W))\,$$ using Huang and Pettie (2022) , ( $$\Phi =\sum _{v\in V} f(v)$$ , W the maximum edge weight) and our algorithm eliminates the $$\alpha (m,n)$$ factor, making the time within a factor $$\sqrt{\log {\Phi }}$$ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the “skew blossom”. Also they can make blossoms become “incomplete”, i.e., partially processed yet still relevant in future searches.

We consider a balance-constrained stochastic bottleneck spanning tree problem (BCSBSTP) where edge weights are independently distributed but may follow arbitrary continuous distributions. The goal is to minimize a threshold variable that may be exceeded by the maximum edge weight at certain risk, subject to the minimum edge weight being no less than a fixed threshold with a probability guarantee. We characterize these two requirements as chance constraints, which are typically used for bounding the risk of undesirable random outcomes. Given independently distributed edge weights, we reformulate BCSBSTP as a mixed-integer nonlinear program, approximated by two mixed-integer linear programs based on special ordered set of type one (SOS1) and special ordered set of type two (SOS2) variables. By relaxing the probabilistic guarantee on the minimum edge weight in BCSBSTP, we also consider a stochastic bottleneck spanning tree problem (SBSTP), of which optimal tree solutions are approximated via a bisection algorithm in pseudopolynomial time. We demonstrate computational results of our models and algorithms by testing randomly generated instances with edge weights following a diverse set of independent distributions.

We perform an experimental study of algorithms for online bipartite matching under the known i.i.d input model with integral types. In the last decade, there has been substantial effort in designing complex algorithms to improve worst-case approximation ratios. Our goal is to determine how these algorithms perform on more practical instances rather than worst-case instances. In particular, we are interested in whether the ranking of the algorithms by their worst-case performance is consistent with the ranking of the algorithms by their average-case/practical performance. We are also interested in whether preprocessing times and implementation difficulties that are introduced by these algorithms are justified in practice. To that end, we evaluate these algorithms on different random inputs as well as real-life instances obtained from publicly available repositories. We compare these algorithms against several simple greedy-style algorithms. Most of the complex algorithms in the literature are presented as being non-greedy (i.e., an algorithm can intentionally skip matching a node that has available neighbors) to simplify the analysis. Every such algorithm can be turned into a greedy one without hurting its worst-case performance. On our benchmarks, non-greedy versions of these algorithms perform much worse than their greedy versions. Greedy versions perform about as well as the simplest greedy algorithm by itself. This, together with our other findings, suggests that simplest greedy algorithms are competitive with the state-of-the-art worst-case algorithms for online bipartite matching on many average-case and practical input families. Greediness is by far the most important property of online algorithms for bipartite matching.

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in [EQUATION] time, [EQUATION] per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs [29, 18]. Here m, n, and N bound the number of edges, vertices, and magnitude of any integer edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in [EQUATION] log(nN)) time.

A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic \(+2\) and \(+4\) unweighted spanners (both all-pairs and pairwise) to \(+2W\) and \(+4W\) weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) \(+2W\) and \(+4W\) weighted spanners of size \(O(n^{3/2})\) and \(O(n^{7/5})\) (\(O(np^{1/3})\) and \(O(np^{2/7})\)) respectively. For a technical reason, the \(+6\) unweighted spanner becomes a \(+8W\) weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) \(+8W\) weighted spanners of size \(O(n^{4/3})\) (\(O(np^{1/4})\)).

We develop an FPT algorithm and a compression for the Weighted Edge Clique Partition (WECP) problem, where a graph with n vertices and integer edge weights is given together with an integer k, and the aim is to find k cliques, such that every edge appears in exactly as many cliques as its weight. The problem has been previously only studied in the unweighted version called Edge Clique Partition (ECP), where the edges need to be partitioned into k cliques. It was shown that ECP admits a kernel with k² vertices [Mujuni and Rosamond, 2008], but this kernel does not extend to WECP. The previously fastest algorithm known for ECP has a runtime of 2^𝒪(k²)n^O(1) [Issac, 2019]. For WECP we develop a compression (to a slightly more general problem) with 4^k vertices, and an algorithm with runtime 2^𝒪(k^(3/2)w^(1/2)log(k/w))n^O(1), where w is the maximum edge weight. The latter in particular improves the runtime for ECP to 2^𝒪(k^(3/2)log k)n^O(1).

In modern manufacturing industry, in order to adapt to changes in the general environment, the manufacturing industry must improve production efficiency. To this end, this article introduces an improved genetic algorithm based on rule selection to tackle the nondeterministic polynomial hard problem stemming from inventory fibre resources and fibre selection principles in optical cable production. The algorithm aims to maximize inventory score and minimize fibre segmentation rate. It employs a permutation encoding approach to link the genetic algorithm with fibre allocation solutions and applies a self-attention mechanism to determine subset solution weight within each solution. To boost the recombination of favourable gene segments from different chromosomes, a rule optimization strategy is integrated into the crossover operation based on the weights. This operations enhance the algorithm's global search capability and convergence speed. A feasibility repair strategy is then used to inspect and rectify chromosomes, preventing the generation of illegal solutions. The legitimate mutation operation, founded on weight optimization rules, effectively reduces the algorithm's running time by avoiding illegal solutions. By leveraging actual production data from an optical cable manufacturer for simulation, the experimental results confirm the effectiveness of the improved genetic algorithm in addressing the fibre allocation problem. Comparative simulations with the unimproved genetic algorithm and a stepwise greedy algorithm underscore the superiority of the improved genetic algorithm in resolving the fibre allocation problem.

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis.

Map matching (MM), pins the drifting position data to the correct road link on which a vehicle is travelling, is a crucial step needed by many industrial or research ITS projects which rely on post-hoc analysis of trajectories. To address the unprecedented challenge of massive GPS data processing in urban transportation data center nowadays, this paper proposed an improved parallel topological map-matching algorithm that aims to achieve highest efficiency as well as guaranteed accuracy. The main contributions of this work include: I) a weighting scheme based on cost-effectiveness ratio to reduce candidate path set in low time cost, II) a novel leapfrog method to omit the redundant GPS points that are not needed in path determination, III) parallelized processing using Map Reduce paradigm. Experiment show that these improvements greatly reduced algorithm's running time when compare to the state of the art.

It is important processes that phase-space diagram and computation of geometrical eigenvalues are reconstituted in nonlinear dynamical analysis. It's difficult to analyze nonlinear system such as EEG real-time because the algorithms of phase-space diagram reconstitution and geometrical eigenvalue computation are complex on both time and space. The algorithms were optimized to reduce their complexity, after that the algorithms were parallelized, at last the integrated algorithm's running time is 1/30 of the running time before optimization and parallelization. It was found that the value of correlation dimension can reflect sleep stages after analyzing the sleep EEG, final sleep stages were also forecasted simply.

When the density parameter is close to unity, the universe has a large curvature radius independent of its being hyperbolic or spherical, or in the limiting case of an infinite curvature radius, flat. Whatever the curvature, the universe may have either a simply connected or a multiply connected topology. In the flat case, the topology scale is arbitrary, and there is no a priori reason for this scale to be of the same order as the size of the observable universe. In the hyperbolic case, any nontrivial topology would almost surely be on a length scale too large to detect. In the spherical case, in contrast, the topology could easily occur on a detectable scale. The present paper shows how, in the spherical case, the assumption of a nearly flat universe simplifies the algorithms for detecting a multiply connected topology, but also reduces the amount of topology that can be seen. This is of primary importance for the upcoming cosmic microwave background data analysis.This paper shows that for spherical spaces one may restrict the search to diametrically opposite pairs of circles in the circles-in-the-sky method and still detect the cyclic factor in the standard factorization of the holonomy group. This vastly decreases the algorithm's run time. If the search is widened to include pairs of candidate circles whose centres are almost opposite and whose relative twist varies slightly, then the cyclic factor along with a cyclic subgroup of the general factor may also be detected. Unfortunately, the full holonomy group is, in general, unobservable in a nearly flat spherical universe, and so a full six-parameter search is unnecessary. Crystallographic methods could also potentially detect the cyclic factor and a cyclic subgroup of the general factor, but nothing else.

Near-linear-time deterministic plane Steiner spanners for well-spaced point sets

Complex networks describe a wide range of systems in nature and society. To understand complex networks, it is crucial to investigate their community structure. In this paper, we develop an online community detection algorithm with linear time complexity for large complex networks. Our algorithm processes a network edge by edge in the order that the network is fed to the algorithm. If a new edge is added, it just updates the existing community structure in constant time, and does not need to re-compute the whole network. Therefore, it can efficiently process large networks in real time. Our algorithm optimizes expected modularity instead of modularity at each step to avoid poor performance. The experiments are carried out using 11 public data sets, and are measured by two criteria, modularity and NMI (Normalized Mutual Information). The results show that our algorithm's running time is less than the commonly used Louvain algorithm while it gives competitive performance.

Bandit online optimization over the permutahedron

Fair division is a topic that has significant social and industrial value. In this work, we study allocations that simultaneously satisfy definitions of fairness and efficiency: EFx and PO. First, we prove that the problem of finding such allocations is NP-hard for two agents. Then, we propose a concept for an ILP-based solving algorithm, the running time of which depends on the number of EFx allocations. We generate input data and analyze algorithm's running time based on the results obtained.