An instance of the NP-hard Quadratic Shortest Path Problem (QSPP) is called linearizable iff it is equivalent to an instance of the classic Shortest Path Problem (SPP) on the same input digraph. The linearization problem for the QSPP (LinQSPP) decides whether a given QSPP instance is linearizable and determines the corresponding SPP instance in the positive case. We provide a novel linear time algorithm for the LinQSPP on acyclic digraphs which runs considerably faster than the previously best algorithm. The algorithm is based on a new insight revealing that the linearizability of the QSPP for acyclic digraphs can be seen as a local property. Our approach extends to the more general higher-order shortest path problem.
In this paper, we present a new approach to linearizing zero-one quadratic minimization problem which has many applications in computer science and communications. Our algorithm is based on the observation that the quadratic term of zero-one variables has two equivalent piece-wise formulations, convex and concave cases. The convex piece-wise objective function and/or constraints play a great role in deducing small linearization. Further tight strategies are also discussed.
The quadratic shortest path problem (QSPP) is \textcolor{black}{the problem of finding a path with prespecified start vertex $s$ and end vertex $t$ in a digraph} such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot be approximated unless P=NP. Here, we give a simple proof for the same result. We also show that if the quadratic cost matrix is a symmetric weak sum matrix \textcolor{black}{ and…
Given an undirected, unweighted planar graph $G$ with $n$ vertices, we present a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of $G$ in constant time. For any $\varepsilon > 0$, our distance oracle takes up $O(n^{5/3+\varepsilon})$ space and is capable of answering shortest-path distance queries exactly for any pair of vertices of $G$ in worst-case time $O(\log (1/\varepsilon))$. Previously no truly sub-quadratic size distance oracles with constant query time for answering exact all-pairs shortest paths distance queries existe…
Blomer and Naewe[BN09] modified the randomized sieving algorithm of Ajtai, Kumar and Sivakumar[AKS01] to solve the shortest vector problem (SVP). The algorithm starts with $N = 2^{O(n)}$ randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice. The running time of the sieving procedure is quadratic in $N$. We study this problem for the special but important case of the $\ell_\infty$ norm. We give a new sieving procedure that runs in time linear in $N$, thereby significantly improving the running time of the algorithm for SVP in…
Let $\mathcal{P}$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $\mathcal{P}$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ "gateways" for $s$ and a set of $n_t$ "gateways" for $t$ such that there exist a shortest $s$-$t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s$-$t$ path using these gateways. Previous algorithms all take quadratic $O(n_s\cdot n_t)$ time to s…
In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm ($1\leq p\leq\infty$). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all $\ell_p$ norm ($1\leq p\leq\infty$). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of $2^{2.751n+o(n)}$, which is much less than the $2^{3.849n+o(n)}$ complexity of the previous best a…
We design an $n^{2+o(1)}$-time algorithm that constructs a cut-equivalent (Gomory-Hu) tree of a simple graph on $n$ nodes. This bound is almost-optimal in terms of $n$, and it improves on the recent $\tilde{O}(n^{2.5})$ bound by the authors (STOC 2021), which was the first to break the cubic barrier. Consequently, the All-Pairs Maximum-Flow (APMF) problem has time complexity $n^{2+o(1)}$, and for the first time in history, this problem can be solved faster than All-Pairs Shortest Paths (APSP). We further observe that an almost-linear time algorithm (in terms of the number of edges $m$) is not …
We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires $Ω(n^2/\log^2{n})$ rounds in the worst case in the CONGEST model, as well as any algorithm for $χ$-coloring a graph, where $χ$ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of round…
In this work we consider the problem of computing the $(\min, +)$-convolution of two sequences $a$ and $b$ of lengths $n$ and $m$, respectively, where $n \geq m$. We assume that $a$ is arbitrary, but $b_i = f(i)$, where $f(x) \colon [0,m) \to \mathbb{R}$ is a function with one of the following properties: 1. the linear case, when $f(x) =β+ α\cdot x$; 2. the monotone case, when $f(i+1) \geq f(i)$, for any $i$; 3. the convex case, when $f(i+1) - f(i) \geq f(i) - f(i-1)$, for any $i$; 4. the concave case, when $f(i+1) - f(i) \leq f(i) - f(i-1)$, for any $i$; 5. the piece-wise linear case, when $f…
Many popular algorithms for searching the space of leaf-labelled trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are given by pairs of trees connected by one rearrangement operation (sometimes called a move). Most popular are the classical nearest neighbour interchange, subtree prune and regraft, and tree bisection and reconnection moves. The problem of computing distances, however, is NP-hard in each of these graphs, making tree inference and comparison algorithms challengin…
In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $n\choose 2$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^3)$ when $m = Θ(n^2)$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $\tilde{O}(n^{2})$-time a…
Let $G$ be an unweighted, undirected graph. An additive $k$-spanner of $G$ is a subgraph $H$ that approximates all distances between pairs of nodes up to an additive error of $+k$, that is, it satisfies $d_H(u,v) \le d_G(u,v)+k$ for all nodes $u,v$, where $d$ is the shortest path distance. We give a deterministic algorithm that constructs an additive $O\!\left(1\right)$-spanner with $O\!\left(n^{4/3}\right)$ edges in $O\!\left(n^2\right)$ time. This should be compared with the randomized Monte Carlo algorithm by Woodruff [ICALP 2010] giving an additive $6$-spanner with $O\!\left(n^{4/3}\log^3 …
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）
Given a set of $n$ points in the Euclidean plane, such that just $k$ points are strictly inside the convex hull of the whole set, we want to find the shortest tour visiting every point. The fastest known algorithm for the version when $k$ is significantly smaller than $n$, i.e., when there are just few inner points, works in $O(k^{11\sqrt{k}} k^{1.5} n^{3})$ time [Knauer and Spillner, WG 2006], but also requires space of order $k^{c\sqrt{k}}n^{2}$. The best linear space algorithm takes $O(k! k n)$ time [Deineko, Hoffmann, Okamoto, Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a line…
This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can b…
We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bit…
A seed in a word is a relaxed version of a period in which the occurrences of the repeating subword may overlap. We show a linear-time algorithm computing a linear-size representation of all the seeds of a word (the number of seeds might be quadratic). In particular, one can easily derive the shortest seed and the number of seeds from our representation. Thus, we solve an open problem stated in the survey by Smyth (2000) and improve upon a previous O(n log n) algorithm by Iliopoulos, Moore, and Park (1996). Our approach is based on combinatorial relations between seeds and subword complexity (…
Complexity Gaps between Point and Interval Temporal Graphs for some Reachability Problems
IRIF, CNRS & Université Paris Cité（IRIF, 法国国家科学研究中心与巴黎西岱大学）；European Commission –- Joint Research Centre (JRC)（欧盟委员会联合研究中心）；Kent State University（肯特州立大学）；University of Bucharest, Faculty of Mathematics and Computer Science, and National Institute for Research and Development in Informatics, Romania（布加勒斯特大学数学与计算机科学学院，以及罗马尼亚信息学研究与开发国家研究所）；Inria, DI ENS, Paris（法国国家信息与自动化研究所，巴黎高等师范学院）
