Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant $0<\lambda<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p^{\lfloor (1-\lambda)\frac{n}{3p}\rfloor}$ in expected time less than $c^n$ for a constant $c<2$ that depends only on $p$ and $\lambda$. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in $O^*(3^{n-\alpha(G)})$ time and polynomial space, where $\alpha(G)$ is the size of the maximum independent set in $G$. In particular, this yields an $O^*(3^{n/2})$ time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an $O^*(2^k)$-time randomized algorithm for detecting out-branchings with at least $k$ internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed $k$-Leaf problem, based on a non-standard monomial detection problem

An Enumerative Perspective on Connectivity

Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be the maximum number of edge-disjoint paths from $s$ to $t$ in $G$. Much research has gone into designing fast algorithms for computing connectivities in graphs. Previous work showed that it is possible to compute connectivities for all pairs of vertices in directed graphs with $m$ edges in $\tilde{O}(m^\omega)$ time [Chueng, Lau, and Leung, FOCS 2011], where $\omega \in [2,2.3716)$ is the exponent of matrix multiplication. For the related problem of computing "small connectivities," it was recently shown that for any positive integer $k$, we can compute $\min(k,\lambda(s,t))$ for all pairs of vertices $(s,t)$ in a directed graph with $n$ nodes in $\tilde{O}((kn)^\omega)$ time [Akmal and Jin, ICALP 2023]. In this paper, we present an alternate exposition of these $\tilde{O}(m^\omega)$ and $\tilde{O}((kn)^\omega)$ time algorithms, with simpler proofs of correctness. Earlier proofs were somewhat indirect, introducing an elegant but ad hoc "flow vector framework" for showing correctness of these algorithms. In contrast, we observe that these algorithms for computing exact and small connectivity values can be interpreted as testing whether certain generating functions enumerating families of edge-disjoint paths are nonzero. This new perspective yields more transparent proofs, and ties the approach for these problems more closely to the literature surrounding algebraic graph algorithms

A Novel Method for Solving Multi-objective Shortest Path Problem in Respect of Probability Theory

Transportation process or activity can be considered as a multi-objective problem reasonably. However, it is difficult to obtain an absolute shortest path with optimizing the multiple objectives at the same time by means of Pareto approach. In this paper, a novel method for solving multi-objective shortest path problem in respect of probability theory is developed, which aims to get the rational solution of multi-objective shortest path problem. Analogically, each objective of the shortest path problem is taken as an individual event, thus the concurrent optimization of many objectives equals to the joint event of simultaneous occurrence of the multiple events, and therefore the simultaneous optimization of multiple objectives can be solved on basis of probability theory rationally. The partial favorable probability of each objective of every scheme (routine) is evaluated according to the actual preference degree of the utility indicator of the objective. Moreover, the product of all partial favorable probabilities of the utility of objective of each scheme (routine) casts the total favorable probability of the corresponding scheme (routine), which results in the decisively unique indicator of the scheme (routine) in the multi-objective shortest path problem in the point of view of system theory. Thus, the optimum solution of the multi-objective shortest path problem is the scheme (routine) with highest total favorable probability. Finally, an application example is given to illuminate the approach

Privaatsust säilitavad paralleelarvutused graafiülesannete jaoks

Turvalisel mitmeosalisel arvutusel põhinevate reaalsete privaatsusrakenduste loomine on SMC-protokolli arvutusosaliste ümmarguse keerukuse tõttu keeruline. Privaatsust säilitavate tehnoloogiate uudsuse ja nende probleemidega kaasnevate suurte arvutuskulude tõttu ei ole paralleelseid privaatsust säilitavaid graafikualgoritme veel uuritud. Graafikalgoritmid on paljude arvutiteaduse rakenduste selgroog, nagu navigatsioonisüsteemid, kogukonna tuvastamine, tarneahela võrk, hüperspektraalne kujutis ja hõredad lineaarsed lahendajad. Graafikalgoritmide suurte privaatsete andmekogumite töötlemise kiirendamiseks ja kõrgetasemeliste arvutusnõuete täitmiseks on vaja privaatsust säilitavaid paralleelseid algoritme. Seetõttu esitleb käesolev lõputöö tipptasemel protokolle privaatsuse säilitamise paralleelarvutustes erinevate graafikuprobleemide jaoks, ühe allika lühima tee, kõigi paaride lühima tee, minimaalse ulatuva puu ja metsa ning algebralise tee arvutamise. Need uued protokollid on üles ehitatud kombinatoorsete ja algebraliste graafikualgoritmide põhjal lisaks SMC protokollidele. Nende protokollide koostamiseks kasutatakse ka ühe käsuga mitut andmeoperatsiooni, et vooru keerukust tõhusalt vähendada. Oleme väljapakutud protokollid juurutanud Sharemind SMC platvormil, kasutades erinevaid graafikuid ja võrgukeskkondi. Selles lõputöös kirjeldatakse uudseid paralleelprotokolle koos nendega seotud algoritmide, tulemuste, kiirendamise, hindamiste ja ulatusliku võrdlusuuringuga. Privaatsust säilitavate ühe allika lühimate teede ja minimaalse ulatusega puuprotokollide tegelike juurutuste tulemused näitavad tõhusat meetodit, mis vähendas tööaega võrreldes varasemate töödega sadu kordi. Lisaks ei ole privaatsust säilitavate kõigi paaride lühima tee protokollide hindamine ja ulatuslik võrdlusuuringud sarnased ühegi varasema tööga. Lisaks pole kunagi varem käsitletud privaatsust säilitavaid metsa ja algebralise tee arvutamise protokolle.Constructing real-world privacy applications based on secure multiparty computation is challenging due to the round complexity of the computation parties of SMC protocol. Due to the novelty of privacy-preserving technologies and the high computational costs associated with these problems, parallel privacy-preserving graph algorithms have not yet been studied. Graph algorithms are the backbone of many applications in computer science, such as navigation systems, community detection, supply chain network, hyperspectral image, and sparse linear solvers. In order to expedite the processing of large private data sets for graphs algorithms and meet high-end computational demands, privacy-preserving parallel algorithms are needed. Therefore, this Thesis presents the state-of-the-art protocols in privacy-preserving parallel computations for different graphs problems, single-source shortest path (SSSP), All-pairs shortest path (APSP), minimum spanning tree (MST) and forest (MSF), and algebraic path computation. These new protocols have been constructed based on combinatorial and algebraic graph algorithms on top of the SMC protocols. Single-instruction-multiple-data (SIMD) operations are also used to build those protocols to reduce the round complexities efficiently. We have implemented the proposed protocols on the Sharemind SMC platform using various graphs and network environments. This Thesis outlines novel parallel protocols with their related algorithms, the results, speed-up, evaluations, and extensive benchmarking. The results of the real implementations of the privacy-preserving single-source shortest paths and minimum spanning tree protocols show an efficient method that reduced the running time hundreds of times compared with previous works. Furthermore, the evaluation and extensive benchmarking of privacy-preserving All-pairs shortest path protocols are not similar to any previous work. Moreover, the privacy-preserving minimum spanning forest and algebraic path computation protocols have never been addressed before.https://www.ester.ee/record=b555865

Counting connected subgraphs with maximum-degree-aware sieving

Peer reviewe

Extensor-coding

We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic $2^{k}\cdot\operatorname{poly}(n)$ time algorithm to find a $k$-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect $k$-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.Comment: To appear at STOC 2018: Symposium on Theory of Computing, June 23-27, 2018, Los Angeles, CA, US

Virtual screening of DrugBank database for hERG blockers using topological Laplacian-assisted AI models

The human {\it ether-a-go-go} (hERG) potassium channel (K$_\text{v}11.1$) plays a critical role in mediating cardiac action potential. The blockade of this ion channel can potentially lead fatal disorder and/or long QT syndrome. Many drugs have been withdrawn because of their serious hERG-cardiotoxicity. It is crucial to assess the hERG blockade activity in the early stage of drug discovery. We are particularly interested in the hERG-cardiotoxicity of compounds collected in the DrugBank database considering that many DrugBank compounds have been approved for therapeutic treatments or have high potential to become drugs. Machine learning-based in silico tools offer a rapid and economical platform to virtually screen DrugBank compounds. We design accurate and robust classifiers for blockers/non-blockers and then build regressors to quantitatively analyze the binding potency of the DrugBank compounds on the hERG channel. Molecular sequences are embedded with two natural language processing (NPL) methods, namely, autoencoder and transformer. Complementary three-dimensional (3D) molecular structures are embedded with two advanced mathematical approaches, i.e., topological Laplacians and algebraic graphs. With our state-of-the-art tools, we reveal that 227 out of the 8641 DrugBank compounds are potential hERG blockers, suggesting serious drug safety problems. Our predictions provide guidance for the further experimental interrogation of DrugBank compounds' hERG-cardiotoxicity

Fully Dynamic Shortest Path Reporting Against an Adaptive Adversary

Algebraic data structures are the main subroutine for maintaining distances in fully dynamic graphs in subquadratic time. However, these dynamic algebraic algorithms generally cannot maintain the shortest paths, especially against adaptive adversaries. We present the first fully dynamic algorithm that maintains the shortest paths against an adaptive adversary in subquadratic update time. This is obtained via a combinatorial reduction that allows reconstructing the shortest paths with only a few distance estimates. Using this reduction, we obtain the following: On weighted directed graphs with real edge weights in $[1,W]$, we can maintain $(1+\epsilon)$ approximate shortest paths in $\tilde{O}(n^{1.816}\epsilon^{-2} \log W)$ update and $\tilde{O}(n^{1.741} \epsilon^{-2} \log W)$ query time. This improves upon the approximate distance data structures from [v.d.Brand, Nanongkai, FOCS'19], which only returned a distance estimate, by matching their complexity and returning an approximate shortest path. On unweighted directed graphs, we can maintain exact shortest paths in $\tilde{O}(n^{1.823})$ update and $\tilde{O}(n^{1.747})$ query time. This improves upon [Bergamaschi, Henzinger, P.Gutenberg, V.Williams, Wein, SODA'21] who could report the path only against oblivious adversaries. We improve both their update and query time while also handling adaptive adversaries. On unweighted undirected graphs, our reduction holds not just against adaptive adversaries but is also deterministic. We maintain a $(1+\epsilon)$-approximate $st$-shortest path in $O(n^{1.529} / \epsilon^2)$ time per update, and $(1+\epsilon)$-approximate single source shortest paths in $O(n^{1.764} / \epsilon^2)$ time per update. Previous deterministic results by [v.d.Brand, Nazari, Forster, FOCS'22] could only maintain distance estimates but no paths