The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. $\bullet$ A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer $4\leq q\leq \sqrt{n}$, our algorithm takes $q^{13n+o(n)}$ time and requires $poly(n)\cdot q^{16n/q^2}$ memory. This tradeoff which ranges from enumeration ($q=\sqrt{n}$) to sieving ($q$ constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter.
$\bullet$ A quantum algorithm for SVP that runs in time $2^{0.950n+o(n)}$ and requires $2^{0.5n+o(n)}$ classical memory and poly(n) qubits. In Quantum Random Access Memory (QRAM) model this algorithm takes only $2^{0.835n+o(n)}$ time and requires a QRAM of size $2^{0.293n+o(n)}$, poly(n) qubits and $2^{0.5n}$ classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [ADRS15] that has a time and space complexity $2^{n+o(n)}$.
$\bullet$ A classical algorithm for SVP that runs in time $2^{1.669n+o(n)}$ time and $2^{0.5n+o(n)}$ space. This improves over an algorithm of [CCL18] that has the same space complexity.
The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number which is $2^{0.402n}$. We conjecture that for most lattices this quantity is a $2^{o(n)}$. Assuming that this is the case, our classical algorithm runs in time $2^{1.292n+o(n)}$, our quantum algorithm runs in time $2^{0.750n+o(n)}$ and our quantum algorithm in QRAM model runs in time $2^{0.667n+o(n)}$.
We revisit the problem of privately releasing the all-pairs shortest path distances of a weighted undirected graph up to low additive error, which was first studied by Sealfon [Sea16]. In this paper, we improve significantly on Sealfon's results, both for arbitrary weighted graphs and for bounded-weight graphs on $n$ nodes. Specifically, we provide an approximate-DP algorithm that outputs all-pairs shortest path distances up to maximum additive error $\tilde{O}(\sqrt{n})$, and a pure-DP algorithm that outputs all pairs shortest path distances up to maximum additive error $\tilde{O}(n^{2/3})$ (…
We present a proof that the number of breakpoints in the arrival function between two terminals in graphs of treewidth $w$ is $n^{O(\log^2 w)}$ when the edge arrival functions are piecewise linear. This is an improvement on the bound of $n^{Θ(\log n)}$ by Foschini, Hershberger, and Suri for graphs without any bound on treewidth. We provide an algorithm for calculating this arrival function using star-mesh transformations, a generalization of the wye-delta-wye transformations.
This paper is about the problem of finding a shortest $s$-$t$ path using at most $h$ edges in edge-weighted graphs. The Bellman--Ford algorithm solves this problem in $O(hm)$ time, where $m$ is the number of edges. We show that this running time is optimal, up to subpolynomial factors, under popular fine-grained complexity assumptions. More specifically, we show that under the APSP Hypothesis the problem cannot be solved faster already in undirected graphs with non-negative edge weights. This lower bound holds even restricted to graphs of arbitrary density and for arbitrary $h \in O(\sqrt{m})$…
The shortest path (SP) and shortest paths tree (SPT) problems arise both as direct applications and as subroutines of overlay algorithms solving more complex problems such as the constrained shortest path (CSP) or the constrained minimum Steiner tree (CMST) problems. Often, such algorithms do not use the result of an SP subroutine if its total cost is greater than a given bound. For example, for delay-constrained problems, paths resulting from a least-delay SP run and whose delay is greater than the delay constraint of the original problem are not used by the overlay algorithm to construct its…
We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices. We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe t…
In this paper, we provide a comprehensive overview of a recent debate over the quantum versus classical solvability of bounded distance decoding (BDD). Specifically, we review the work of Eldar and Hallgren [EH22], [Hal21] demonstrating a quantum algorithm solving $λ_1 2^{-Ω(\sqrt{k \log q})}$-BDD in polynomial time for lattices of periodicity $q$, finite group rank $k$, and shortest lattice vector length $λ_1$. Subsequently, we prove the results of [DvW21a], [DvW21b] with far greater detail and elaboration than in the original work. Namely, we show that there exists a deterministic, classical…
We introduce a model for differentially private analysis of weighted graphs in which the graph topology $(V,E)$ is assumed to be public and the private information consists only of the edge weights $w:E\to\mathbb{R}^+$. This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters $ε$ and $δ$, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions $w,w'$ are …
The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be solved efficiently. Specifically, if for an $n$-dimensional lattice, a Gram matrix is known that can be written as the difference of a diagonal matrix and a positive semidefinite matrix of rank $k$ (for some constant $k$), we show that the SLV problem can be reduced to a $k$-dimensional optimization problem with countably many candidate points. Moreover, we s…
We combine the well known Shortest Paths (SP) problem and the Bottleneck Paths (BP) problem to introduce a new problem called the Shortest Paths for All Flows (SP-AF) problem that has relevance in real life applications. We first solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem on directed graphs with unit edge costs in $O(mn)$ worst case time bound. We then present two algorithms to solve SSSP-AF on directed graphs with integer edge costs bounded by $c$ in $O(m^2 + nc)$ and $O(m^2 + mn\log{(\frac{c}{m})})$ time bounds. Finally we extend our algorithms for the SSSP-AF pro…
We investigate the single-source-single-destination "shortest" paths problem in acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated paths vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated paths vectors from source to sink.
Blömer and Seifert showed that $\mathsf{SIVP}_2$ is NP-hard to approximate by giving a reduction from $\mathsf{CVP}_2$ to $\mathsf{SIVP}_2$ for constant approximation factors as long as the $\mathsf{CVP}$ instance has a certain property. In order to formally define this requirement on the $\mathsf{CVP}$ instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of Blömer and Seifert to show a reduction from the Gap Closest Vector Problem with Bounded Minima to $\mathsf{SIVP}$ for any $\ell_p$ norm for some constant approxima…
The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s. For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971 and 1972], and an $O(K(m+n \log n))$ algorithm for the undirected version by Katoh et al. [Networks, 1982], both using $O(Kn + m)$ space. In this work, we consider a different parameterization for this problem: instead of bounding the number of $st$-paths output, we bound their …
Due to the computational complexity of finding almost shortest simple paths, we propose that identifying a larger collection of (nonbacktracking) paths is more efficient than finding almost shortest simple paths on positively weighted real-world networks. First, we present an easy to implement $O(m\log m+kL)$ solution for finding all (nonbacktracking) paths with bounded length $D$ between two arbitrary nodes on a positively weighted graph, where $L$ is an upperbound for the number of nodes in any of the $k$ outputted paths. Subsequently, we illustrate that for undirected Chung-Lu random graphs…
$ \newcommand{\Z}{\mathbb{Z}} \newcommand{\eps}{\varepsilon} \newcommand{\cc}[1]{\mathsf{#1}} \newcommand{\NP}{\cc{NP}} \newcommand{\problem}[1]{\mathrm{#1}} \newcommand{\BDD}{\problem{BDD}} $Bounded Distance Decoding $\BDD_{p,α}$ is the problem of decoding a lattice when the target point is promised to be within an $α$ factor of the minimum distance of the lattice, in the $\ell_{p}$ norm. We prove that $\BDD_{p, α}$ is $\NP$-hard under randomized reductions where $α\to 1/2$ as $p \to \infty$ (and for $α=1/2$ when $p=\infty$), thereby showing the hardness of decoding for distances approaching …
How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? Let $G = (V,E)$ be an unweighted, connected graph of bounded degree. The edge set $E$ is initially unknown, and the graph can be accessed using a \emph{distance oracle}, which receives a pair of vertices $(u,v)$ and returns the distance between $u$ and $v$. In the \emph{verification} problem, we are given a hypothetical graph $\hat G = (V,\hat E)$ and want to check whether $G$ is equal to $\hat G$. We analyze a natural greedy algorithm and prove that it uses $n^{1+o(1)}$ distance queries.…
Fast Shortest Path in Graphs With Sparse Signed Tree Models and Applications
CNRS, ENS de Lyon, Université Claude Bernard Lyon 1（法国国家科学研究中心、里昂高等师范学院、里昂第一大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；School of Computing, KAIST（韩国科学技术院计算机学院）
We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^…
It has been shown by Alon et al. that the so-called 'all-pairs shortest-path' problem can be solved in O((MV)^2.688 * log^3(V)) for graphs with V vertices, with integer distances bounded by M. We solve the more general problem for graphs in R (assuming no negative cycles), with expected-case running time O(V^2.5 * log(V)). While our result appears to violate the Omega(V^3) requirement of "Funny Matrix Multiplication" (due to Kerr), we find that it has a sub-cubic expected time solution subject to reasonable conditions on the data distribution. The expected time solution arises when certain sub…
Given an $n$-vertex $m$-edge graph $G$ with non negative edge-weights, the girth of $G$ is the weight of a shortest cycle in $G$. For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that computes, in $\tilde{\cal O}(n^{5/3}+m)$-time, a cycle of weight at most twice the girth of $G$. Our approach combines some new insights on the previous approximation algorithms for this problem (Lingas and Lundell, IPL'09; Roditty and Tov, TALG'13) with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, J…
We introduce stronger notions for approximate single-source shortest-path distances, show how to efficiently compute them from weaker standard notions, and demonstrate the algorithmic power of these new notions and transformations. One application is the first work-efficient parallel algorithm for computing exact single-source shortest paths graphs -- resolving a major open problem in parallel computing. Given a source vertex in a directed graph with polynomially-bounded nonnegative integer lengths, the algorithm computes an exact shortest path tree in $m \log^{O(1)} n$ work and $n^{1/2+o(1)}$…
The approximate single-source shortest-path problem is as follows: given a graph with nonnegative edge weights and a designated source vertex $s$, return estimates of the distances from~$s$ to each other vertex such that the estimate falls between the true distance and $(1+ε)$ times the distance. This paper provides the first nearly work-efficient parallel algorithm with sublinear span (also called depth) for the approximate shortest-path problem on \emph{directed} graphs. Specifically, for constant $ε$ and polynomially-bounded edge weights, our algorithm has work $\tilde{O}(m)$ and span $n^{1…
Given a graph $G$ cellularly embedded on a surface $Σ$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $Σ$ along $G$ yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width.…
The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in R^N, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L in R^N, given a target vector r in R^N and a distance parameter d, output the set of all lattice points w in L that are within distance d of r. In this work we focus on combinatorial and algorithmic que…
The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in $O(n)$ amo…
We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $ \tilde Ω(\sqrt{n} + D) $ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $ n $ is the number of nodes in the network and $ D $ is its diameter. The state of the art is Elkin's ran…
In this paper we address the problem of computing a sparse subgraph of a weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a small sized subgraph that preserves distances between any pair of vertices is a well studied problem. Since in the real world any network is prone to failures, it is natural to study the fault tolerant version of the above problem. Unfortunately, it turns out that there may not always exist such a sparse subgraph even under single edge failure [Demetrescu \emph…
In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain $\mathcal{P}$ with a total of $n$ vertices. We discover many interesting observations. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of $\mathcal{P}$ that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the combinatorial size of the shortest path map equivalence decomposition of $\mathcal{P}$, which is known to be $O(n^{10})$. One key observ…
We show that the Minimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-wi…
Highly dynamic networks rarely offer end-to-end connectivity at a given time. Yet, connectivity in these networks can be established over time and space, based on temporal analogues of multi-hop paths (also called {\em journeys}). Attempting to optimize the selection of the journeys in these networks naturally leads to the study of three cases: shortest (minimum hop), fastest (minimum duration), and foremost (earliest arrival) journeys. Efficient centralized algorithms exists to compute all cases, when the full knowledge of the network evolution is given. In this paper, we study the {\em distr…
A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.
The Constraint Shortest Path (CSP) problem is as follows. An $n$-vertex graph is given, each edge/arc assigned two weights. Let us call them "cost" and "length" for definiteness. Finding a min-cost upper-bounded length path between a given pair of vertices is required. The problem is NP-hard even when the lengths of all edges are the same. Therefore, various approximation algorithms have been proposed in the literature for it. The constraint on path length can be accounted for by considering one edge weight equals to a linear combination of cost and length. By varying the multiplier value in a…
Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on…
A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this paper, we consider the problem version on curved obstacles, commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons). Each curved edge is assumed to be of O(1) complexity. Given in the plane two points s and t and a set of $h$ pairwise disjoint splinegons with a total of $n$ vertices, we com…
In large networks, using the length of shortest paths as the distance measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs and directed graphs is controversial. The second shortcoming is that a huge number of vertices may have exactly the same score. The third shortcoming is that in many applications, the distance between two vertices not only depends on the length of shortest paths, but also on the number of shortest paths. In this paper, first we develop a new distance measure between vertices of a graph that yields discriminative distance-based ce…
