In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal lattices include lattices of Voronoi's first kind and tensor products of root lattices of type A. The combinatorial structure of zonotopal lattices can be described by regular matroids/totally unimodular matrices. We observe that a linear algebra version of the minimum mean cycle canceling method can be applied for efficiently solving the closest vector problem in a zonotopal lattice if the lattice is given as the integral kernel of a totally unimodular matrix.
We give a $2^{n+o(n)}$-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on $n$-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic $\widetilde{O}(4^{n})$-time and $\widetilde{O}(2^{n})$-space algorithm of Micciancio and Voulgaris. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian sampling over lattice shifts, $L- t$, with very low parameters. While the actual algorithm is a natural generalization of [ADRS15], t…
In this paper, we present a deterministic algorithm for the closest vector problem for all l_p-norms, 1 < p < \infty, and all polyhedral norms, especially for the l_1-norm and the l_{\infty}-norm. We achieve our results by introducing a new lattice problem, the lattice membership problem. We describe a deterministic algorithm for the lattice membership problem, which is a generalization of Lenstra's algorithm for integer programming. We also describe a polynomial time reduction from the closest vector problem to the lattice membership problem. This approach leads to a deterministic algorithm t…
In a seminal work, Micciancio & Voulgaris (2013) described a deterministic single-exponential time algorithm for the Closest Vector Problem (CVP) on lattices. It is based on the computation of the Voronoi cell of the given lattice and thus may need exponential space as well. We address the major open question whether there exists such an algorithm that requires only polynomial space. To this end, we define a lattice basis to be $c$-compact if every facet normal of the Voronoi cell is a linear combination of the basis vectors using coefficients that are bounded by $c$ in absolute value. Given s…
This study investigates whether reoptimization can help in solving the closest substring problem. We are dealing with the following reoptimization scenario. Suppose, we have an optimal l-length closest substring of a given set of sequences S. How can this information be beneficial in obtaining an (l+k)-length closest substring for S? In this study, we show that the problem is still computationally hard even with k=1. We present greedy approximation algorithms that make use of the given information and prove that it has an additive error that grows as the parameter k increases. Furthermore, we …
We give a deterministic algorithm for solving the (1+eps)-approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2^{O(n)}(1+1/eps)^n time and 2^n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the "AKS Sieve" based algorithms for (1+eps)-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a poly(n)-space and 2^{O(n)} time algorithm for exact CVP in the…
The closest string problem is an NP-hard problem, whose task is to find a string that minimizes maximum Hamming distance to a given set of strings. This can be reduced to an integer program (IP). However, to date, there exists no known polynomial-time algorithm for IP. In 2004, Meneses et al. introduced a branch-and-bound (B & B) method for solving the IP problem. Their algorithm is not always efficient and has the exponential time complexity. In the paper, we attempt to solve efficiently the IP problem by a greedy iterative rounding technique. The proposed algorithm is polynomial time and muc…
The closest pair problem (CPP) is one of the well studied and fundamental problems in computing. Given a set of points in a metric space, the problem is to identify the pair of closest points. Another closely related problem is the fixed radius nearest neighbors problem (FRNNP). Given a set of points and a radius $R$, the problem is, for every input point $p$, to identify all the other input points that are within a distance of $R$ from $p$. A naive deterministic algorithm can solve these problems in quadratic time. CPP as well as FRNNP play a vital role in computational biology, computational…
We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any $α< 1/2$, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most $α$ times the length of the shortest nonzero lattice vector, requires as preprocessing advice only $N \approx \widetilde{O}(n \exp(α^2 n /(1-2α)^2))$ vectors, and runs in ti…
Network enrichment analysis methods allow to identify active modules without being biased towards a priori defined pathways. One of mathematical formulations of such analysis is a reduction to a maximum-weight connected subgraph problem. In particular, in analysis of metabolic networks a generalized maximum-weight connected subgraph (GMWCS) problem, where both nodes and edges are scored, naturally arises. Here we present the first to our knowledge practical exact GMWCS solver. We have tested it on real-world instances and compared to similar solvers. First, the results show that on node-weight…
In this work, we exhibit a hierarchy of polynomial time algorithms solving approximate variants of the Closest Vector Problem (CVP). Our first contribution is a heuristic algorithm achieving the same distance tradeoff as HSVP algorithms, namely $\approx β^{\frac{n}{2β}}\textrm{covol}(Λ)^{\frac{1}{n}}$ for a random lattice $Λ$ of rank $n$. Compared to the so-called Kannan's embedding technique, our algorithm allows using precomputations and can be used for efficient batch CVP instances. This implies that some attacks on lattice-based signatures lead to very cheap forgeries, after a precomputati…
We revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis-Laarhoven-De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than $2^{0.076d + o(d)}$ memory, and we show how to obtain time-memory trade-offs even when using less than $2^{0.048d + o(d)}$ memory. We also settle the op…
We give a randomized $2^{n+o(n)}$-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic $\widetilde{O}(4^n)$-time and $\widetilde{O}(2^n)$-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample $2^{n/2}$ vectors from the discrete Gaussian distribution at any paramet…
The closest pair of points problem or closest pair problem (CPP) is an important problem in computational geometry where we have to find a pair of points from a set of points in metric space with the smallest distance between them. This problem arises in a number of applications, such as but not limited to clustering, graph partitioning, image processing, patterns identification, and intrusion detection. For example, in air-traffic control, we must monitor aircrafts that come too close together, since this may potentially indicate a possible collision. Numerous algorithms have been presented f…
Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular $k$-center variant which, given a set $S$ of points from some metric space and a parameter $k<|S|$, requires to identify a subset of $k$ centers in $S$ minimizing the maximum distance of any point of $S$ from its closest center. A more general formulation, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $S$ (outliers) to be disregarded when computing the maximum distance from the cent…
Let $G=(V,E)$ be a graph with unit-length edges and nonnegative costs assigned to its vertices. Being given a list of pairwise different vertices $S=(s_1,s_2,\ldots,s_p)$, the {\em prioritized Voronoi diagram} of $G$ with respect to $S$ is the partition of $G$ in $p$ subsets $V_1,V_2,\ldots,V_p$ so that, for every $i$ with $1 \leq i \leq p$, a vertex $v$ is in $V_i$ if and only if $s_i$ is a closest vertex to $v$ in $S$ and there is no closest vertex to $v$ in $S$ within the subset $\{s_1,s_2,\ldots,s_{i-1}\}$. For every $i$ with $1 \leq i \leq p$, the {\em load} of vertex $s_i$ equals the sum…
In the Manhattan Sequence Consensus problem (MSC problem) we are given $k$ integer sequences, each of length $l$, and we are to find an integer sequence $x$ of length $l$ (called a consensus sequence), such that the maximum Manhattan distance of $x$ from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequ…
In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings $A$ and $B$, compute exactly the maximum $\textsf{LCS}(a, b)$ with $(a, b) \in A \times B$) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-…
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the M-ellipsoid, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010) for lattice problems in the l_2 norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag…
In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets $A$ and $B$ of vectors, and the goal is to find $a \in A$ and $b \in B$ maximizing inner product $a \cdot b$. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact $\ell_2$-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three …
