arXiv subjects

Karl Bringmann

Publications and source records attributed to Karl Bringmann.

At least 91 records · Page 5Linked to original sources

arXiv2017

Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs

We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with $ n $ nodes and $ m $ edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time $ \tilde O (m^{3/4} n^{3/2}) $, which gives the first improvement over Megiddo's $ \tilde O (n^3) $ algorithm [JACM'83] for sparse graphs. We further demonstrate how to obtain both an algorithm with running time $ n^3 / 2^{Ω{(\sqrt{\log n})}} $ on general graphs and an algorithm with running time $ \tilde O (n) $ on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.

cs.DS↗

arXiv2017

Tree Edit Distance Cannot be Computed in Strongly Subcubic Time (unless APSP can)

The edit distance between two rooted ordered trees with $n$ nodes labeled from an alphabet~$Σ$ is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. Tree edit distance is a well known generalization of string edit distance. The fastest known algorithm for tree edit distance runs in cubic $O(n^3)$ time and is based on a similar dynamic programming solution as string edit distance. In this paper we show that a truly subcubic $O(n^{3-\varepsilon})$ time algorithm for tree edit distance is unlikely: For $|Σ| = Ω(n)$, a truly subcubic algorithm for tree edit distance implies a truly subcubic algorithm for the all pairs shortest paths problem. For $|Σ| = O(1)$, a truly subcubic algorithm for tree edit distance implies an $O(n^{k-\varepsilon})$ algorithm for finding a maximum weight $k$-clique. Thus, while in terms of upper bounds string edit distance and tree edit distance are highly related, in terms of lower bounds string edit distance exhibits the hardness of the strong exponential time hypothesis [Backurs, Indyk STOC'15] whereas tree edit distance exhibits the hardness of all pairs shortest paths. Our result provides a matching conditional lower bound for one of the last remaining classic dynamic programming problems.

cs.DS↗

arXiv2017

Greedy Routing and the Algorithmic Small-World Phenomenom

The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s, was theoretically explained by Kleinberg in 2000. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks. In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length θ(loglog n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps. These results also address the question of Krioukov et al. whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer.

cs.SI↗

arXiv2017

A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum

Given a set $Z$ of $n$ positive integers and a target value $t$, the Subset Sum problem asks whether any subset of $Z$ sums to $t$. A textbook pseudopolynomial time algorithm by Bellman from 1957 solves Subset Sum in time $O(nt)$. This has been improved to $O(n \max Z)$ by Pisinger [J. Algorithms'99] and recently to $\tilde O(\sqrt{n} t)$ by Koiliaris and Xu [SODA'17]. Here we present a simple randomized algorithm running in time $\tilde O(n+t)$. This improves upon a classic algorithm and is likely to be near-optimal, since it matches conditional lower bounds from Set Cover and k-Clique. We then use our new algorithm and additional tricks to improve the best known polynomial space solution from time $\tilde O(n^3 t)$ and space $\tilde O(n^2)$ to time $\tilde O(nt)$ and space $\tilde O(n \log t)$, assuming the Extended Riemann Hypothesis. Unconditionally, we obtain time $\tilde O(n t^{1+\varepsilon})$ and space $\tilde O(n t^\varepsilon)$ for any constant $\varepsilon > 0$.

cs.DS↗

arXiv2016

A Dichotomy for Regular Expression Membership Testing

We study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today. Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem. In this paper we complete their work as follows: 1) We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. 2) We classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth. 3) For the Word Break problem we give an improved $\tilde{O}(n m^{1/3} + m)$ algorithm. Surprisingly, we also prove a matching conditional lower bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem. In total, we prove matching upper and lower bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression membership testing.

cs.DS↗

arXiv2016

Hitting Set for hypergraphs of low VC-dimension

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014). In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time. To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.

cs.DS↗

arXiv2015

Parameterized Complexity Dichotomy for Steiner Multicut

The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). We present two proofs: one using the randomized contractions technique of Chitnis et al, and one relying on new structural lemmas that decompose the Steiner cut into important separators and minimal s-t cuts. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).

cs.DS↗

arXiv2015

Quadratic Conditional Lower Bounds for String Problems and Dynamic Time Warping

Classic similarity measures of strings are longest common subsequence and Levenshtein distance (i.e., the classic edit distance). A classic similarity measure of curves is dynamic time warping. These measures can be computed by simple $O(n^2)$ dynamic programming algorithms, and despite much effort no algorithms with significantly better running time are known. We prove that, even restricted to binary strings or one-dimensional curves, respectively, these measures do not have strongly subquadratic time algorithms, i.e., no algorithms with running time $O(n^{2-\varepsilon})$ for any $\varepsilon > 0$, unless the Strong Exponential Time Hypothesis fails. We generalize the result to edit distance for arbitrary fixed costs of the four operations (deletion in one of the two strings, matching, substitution), by identifying trivial cases that can be solved in constant time, and proving quadratic-time hardness on binary strings for all other cost choices. This improves and generalizes the known hardness result for Levenshtein distance [Backurs, Indyk STOC'15] by the restriction to binary strings and the generalization to arbitrary costs, and adds important problems to a recent line of research showing conditional lower bounds for a growing number of quadratic time problems. As our main technical contribution, we introduce a framework for proving quadratic-time hardness of similarity measures. To apply the framework it suffices to construct a single gadget, which encapsulates all the expressive power necessary to emulate a reduction from satisfiability. Finally, we prove quadratic-time hardness for longest palindromic subsequence and longest tandem subsequence via reductions from longest common subsequence, showing that conditional lower bounds based on the Strong Exponential Time Hypothesis also apply to string problems that are not necessarily similarity measures.

cs.CC↗

arXiv2014

Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds

The Fréchet distance is a well-studied and very popular measure of similarity of two curves. The best known algorithms have quadratic time complexity, which has recently been shown to be optimal assuming the Strong Exponential Time Hypothesis (SETH) [Bringmann FOCS'14]. To overcome the worst-case quadratic time barrier, restricted classes of curves have been studied that attempt to capture realistic input curves. The most popular such class are c-packed curves, for which the Fréchet distance has a $(1+ε)$-approximation in time $\tilde{O}(c n /ε)$ [Driemel et al. DCG'12]. In dimension $d \ge 5$ this cannot be improved to $O((cn/\sqrtε)^{1-δ})$ for any $δ> 0$ unless SETH fails [Bringmann FOCS'14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with runtime $\tilde{O}(cn/\sqrtε)$. This is optimal in high dimensions apart from lower order factors unless SETH fails. Our main new ingredients are as follows: For filling the classical free-space diagram we project short subcurves onto a line, which yields one-dimensional separated curves with roughly the same pairwise distances between vertices. Then we tackle this special case in near-linear time by carefully extending a greedy algorithm for the Fréchet distance of one-dimensional separated curves.

cs.CG↗

arXiv2014

Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails

The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n^2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).

cs.CG↗

arXiv2014

Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The distance between two nodes is then the length of a shortest path (with respect to the weights drawn) that connects these nodes. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the k-median problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.

cs.DS↗

arXiv2014

Counting Triangulations and other Crossing-Free Structures Approximately

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of $P$ can be computed in $O^{*}(2^{n})$ time, which is less than the lower bound of $Ω(2.43^{n})$ on the number of triangulations of any point set. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by $Λ$ the output of our algorithm, and by $c^{n}$ the exact number of triangulations of $P$, for some positive constant $c$, we prove that $c^{n}\leqΛ\leq c^{n}\cdot 2^{o(n)}$. This is the first algorithm that in sub-exponential time computes a $(1+o(1))$-approximation of the base of the number of triangulations, more precisely, $c\leqΛ^{\frac{1}{n}}\leq(1 + o(1))c$. Our algorithm can be adapted to approximately count other crossing-free structures on $P$, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.

cs.CG↗

arXiv2014

Balls into bins via local search: cover time and maximum load

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m = n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m > n.

math.PR↗

arXiv2013

Counting Triangulations and other Crossing-free Structures via Onion Layers

Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number of onion layers of $P$. In particular, for $k = O(1)$ our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than $O^{*}(3.1414^{n})$, even when $k = Θ(n)$. Given that there are several well-studied configurations of points with at least $Ω(3.464^{n})$ triangulations, and some even with $Ω(8^{n})$ triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances. In fact, it is widely believed that any set of $n$ points must have at least $Ω(3.464^{n})$ triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the "Restricted-Triangulation-Counting-Problem", which we prove to be $W[2]$-hard in the parameter $k$. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

cs.CG↗

arXiv2013

A Simple Sweep Line Algorithm for Counting Triangulations and Pseudo-triangulations

Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms run in time $O^{*}(t(P))$ and $O^{*}(pt(P))$ respectively, where $t(P)$ and $pt(P)$ are the largest number of triangulation paths (T-paths) and pseudo-triangulations paths (PT-paths), respectively, that the algorithms encounter during their execution. Moreover, we show that $t(P) = O^{*}(9^{n})$, which is the first non-trivial bound on $t(P)$ to be known. While there already are algorithms that count triangulations in $O^{*}\left(2^n\right)$, and $O^{*}\left(3.1414^{n}\right)$, there are sets of points where the number of T-paths is $O(2^{n})$. In such cases the algorithm herein presented could potentially be faster. Furthermore, it is not clear whether the already-known algorithms can be modified to count pseudo-triangulations so that their running times remain $O^{*}(c^n)$, for some small constant $c\in\mathbb{R}$. Therefore, for counting pseudo-triangulations (and possibly other similar structures) our approach seems better.

cs.CG↗

arXiv2013

Bringing Order to Special Cases of Klee's Measure Problem

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].

cs.CG↗

arXiv2013

Online Checkpointing with Improved Worst-Case Guarantees

In the online checkpointing problem, the task is to continuously maintain a set of k checkpoints that allow to rewind an ongoing computation faster than by a full restart. The only operation allowed is to replace an old checkpoint by the current state. Our aim are checkpoint placement strategies that minimize rewinding cost, i.e., such that at all times T when requested to rewind to some time t <= T the number of computation steps that need to be redone to get to t from a checkpoint before t is as small as possible. In particular, we want that the closest checkpoint earlier than t is not further away from t than q_k times the ideal distance T / (k+1), where q_k is a small constant. Improving over earlier work showing 1 + 1/k <= q_k <= 2, we show that q_k can be chosen asymptotically less than 2. We present algorithms with asymptotic discrepancy q_k <= 1.59 + o(1) valid for all k and q_k <= ln(4) + o(1) <= 1.39 + o(1) valid for k being a power of two. Experiments indicate the uniform bound p_k <= 1.7 for all k. For small k, we show how to use a linear programming approach to compute good checkpointing algorithms. This gives discrepancies of less than 1.55 for all k < 60. We prove the first lower bound that is asymptotically more than one, namely q_k >= 1.30 - o(1). We also show that optimal algorithms (yielding the infimum discrepancy) exist for all k.

cs.DS↗

arXiv2012

Remarks on Category-Based Routing in Social Networks

It is well known that individuals can route messages on short paths through social networks, given only simple information about the target and using only local knowledge about the topology. Sociologists conjecture that people find routes greedily by passing the message to an acquaintance that has more in common with the target than themselves, e.g. if a dentist in Saarbrücken wants to send a message to a specific lawyer in Munich, he may forward it to someone who is a lawyer and/or lives in Munich. Modelling this setting, Eppstein et al. introduced the notion of category-based routing. The goal is to assign a set of categories to each node of a graph such that greedy routing is possible. By proving bounds on the number of categories a node has to be in we can argue about the plausibility of the underlying sociological model. In this paper we substantially improve the upper bounds introduced by Eppstein et al. and prove new lower bounds.

cs.SI↗