We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of $n$-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio.
A highlight of our conditional lower bounds results is that for any integer $k\ge 1$, under the combinatorial $4k$-clique hypothesis, there is no combinatorial algorithm for unweighted undirected $n$-PSP with approximation ratio better than $1+1/k$ that runs in $O(m^{2-2/(k+1)}n^{1/(k+1)-ε})$ time. This nearly matches an upper bound implied by the result of Agarwal (2014).
A highlight of our algorithmic results is that one can solve both $n$-PSP and ANSC in $\tilde O(m+ n^{3/2+ε})$ time with approximation factor $2+ε$ (and additive error that is function of $ε$), for any constant $ε>0$. For $n$-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for $n$-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-time algorithms on $n$-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prov…
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph $G=(V,E)$ subject to edge insertions and deletions and a source vertex $s\in V$, and the goal is to maintain the distance $d(s,t)$ for all $t\in V$. Fine-grained complexity has provided strong lower bounds for exact partially dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15]. Thus much focus has been directed towards finding efficient partially dynamic $(1+ε)$-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14, FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20].…
Given an $n$-vertex non-negatively real-weighted graph $G$, whose vertices are partitioned into a set of $k$ clusters, a \emph{clustered network design problem} on $G$ consists of solving a given network design optimization problem on $G$, subject to some additional constraint on its clusters. In particular, we focus on the classic problem of designing a \emph{single-source shortest-path tree}, and we analyze its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the \emph{unweighted} case, and prove that the problem is \np-hard. Howev…
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providin…
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in pol…
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…
In the Shortest Superstring problem, we are given a set of strings and we are asking for a common superstring, which has the minimum number of characters. The Shortest Superstring problem is NP-hard and several constant-factor approximation algorithms are known for it. Of particular interest is the GREEDY algorithm, which repeatedly merges two strings of maximum overlap until a single string remains. The GREEDY algorithm, being simpler than other well-performing approximation algorithms for this problem, has attracted attention since the 1980s and is commonly used in practical applications. Ta…
We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost $k$-cycle free graphs, for any constant $k\geq 4$. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many $4$- or $5$-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with $m^{1+o(1)}$ preprocessing time and …
The well-known $k$-disjoint path problem ($k$-DPP) asks for pairwise vertex-disjoint paths between $k$ specified pairs of vertices $(s_i, t_i)$ in a given graph, if they exist. The decision version of the shortest $k$-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest $k$-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequen…
Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse skeleton of the given graph that roughly preserves the shortest paths information. Spanners and emulators serve this purpose. This paper develops fast dynamic algorithms for sparse spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. Under the popular OMv conjecture, we show that there can be no de…
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…
Given a directed graph $G$ with arbitrary real-valued weights, the single source shortest-path problem (SSSP) asks for, given a source $s$ in $G$, finding a shortest path from $s$ to each vertex $v$ in $G$. A classical SSSP algorithm detects a negative cycle of $G$ or constructs a shortest-path tree (SPT) rooted at $s$ in $O(mn)$ time, where $m,n$ are the numbers of edges and vertices in $G$ respectively. In many practical applications, new constraints come from time to time and we need to update the SPT frequently. Given an SPT $T$ of $G$, suppose the weight on a certain edge is modified. We …
We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely …
In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's alg…
In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $α$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $α= 1$ by Ding et al. For any constant $α> 1$, we give an $O(n^{1-\frac…
In this paper, I present an algorithm called Raffica algorithm for Single-Source Shortest Path(SSSP). On random graph, this algorithm has linear time complexity(in expect). More precisely, the random graph uses configuration model, and the weights are distributed mostly positively. It is also linear for random grid graphs. Despite I made an assumption on the weights of the random graph, this algorithm is able to solve SSSP with arbitrary weights; when a negative cycle exists, this algorithm can find it out once traversed. The algorithm has a lot of appliances.
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…
We consider the problem of inferring an edge-labeled graph from the sequence of edge labels seen in a walk of that graph. It has been known that this problem is solvable in $O(n \log n)$ time when the targets are path or cycle graphs. This paper presents an online algorithm for the problem of this restricted case that runs in $O(n)$ time, based on Manacher's algorithm for computing all the maximal palindromes in a string.
The Floyd-Warshall algorithm is the most popular algorithm for determining the shortest paths between all pairs in a graph. It is very a simple and an elegant algorithm. However, if the graph does not contain any negative weighted edge, using Dijkstra's shortest path algorithm for every vertex as a source vertex to produce all pairs shortest paths of the graph works much better than the Floyd-Warshall algorithm for sparse graphs. Also, for the graphs with negative weighted edges, with no negative cycle, Johnson's algorithm still performs significantly better than the Floyd-Warshall algorithm f…
A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which is to find the shortest non-zero vector in a lattice, is one of the well-known problems that are believed to be hard to solve, even with a quantum computer. In this paper we propose space-efficient classical and quantum algorithms for solving SVP. Currently the best time-efficient algorithm for solving SVP takes $2^{n+o(n)}$ time and $2^{n+o(n)}$ space. Our clas…
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures …
We show conditional hardness of Approximate Nearest Neighbor Search (ANN) under the $\ell_\infty$ norm with two simple reductions. Our first reduction shows that hardness of a special case of the Shortest Vector Problem (SVP), which captures many provably hard instances of SVP, implies a lower bound for ANN with polynomial preprocessing time under the same norm. Combined with a recent quantitative hardness result on SVP under $\ell_\infty$ (Bennett et al., FOCS 2017), our reduction implies that finding a $(1+\varepsilon)$-approximate nearest neighbor under $\ell_\infty$ with polynomial preproc…
Improved Approximation Ratios for the Shortest Common Superstring Problem with Reverse Complements
Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学研究科计算机科学系）；Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, The University of Tokyo（东京大学医学科学研究所人类基因组中心医疗数据信息学分部）
We present the first $m\,\text{polylog}(n)$ work, $\text{polylog}(n)$ time algorithm in the PRAM model that computes $(1+ε)$-approximate single-source shortest paths on weighted, undirected graphs. This improves upon the breakthrough result of Cohen~[JACM'00] that achieves $O(m^{1+ε_0})$ work and $\text{polylog}(n)$ time. While most previous approaches, including Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent developments in \emph{continuous optimization}, studying the shortest path problem from the lens of the closely-related \emph{minimum transshipment} problem…
