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Subcubic Equivalences Between Path, Matrix, and Triangle Problems
"... We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvab ..."
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We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The allpairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edgeweighted graph. • Finding a minimum weight cycle in a graph of nonnegative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also
Fast algorithms for (max,min)matrix multiplication and bottleneck shortest paths
 In Proc. 19th SODA
, 2009
"... Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realv ..."
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Cited by 14 (1 self)
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Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realvalued matrix. In this paper, we give a (max, min)matrix multiplication algorithm running in time O(n (3+ω)/2) ≤ O(n 2.688), where ω is the exponent of binary matrix multiplication. Our algorithm improves on a recent O(n 2+ω/3) ≤ O(n 2.792)time algorithm of Vassilevska, Williams, and Yuster. Although our algorithm is slower than the best APBP algorithm on vertex capacitated graphs, running in O(n 2.575) time, it is just as efficient as the best algorithm for computing the dominance product, a problem closely related to (max, min)matrix multiplication. Our techniques can be extended to give subcubic algorithms for related bottleneck problems. The allpairs bottleneck shortest paths problem (APBSP) asks for the maximum flow that can be routed along a shortest path. We give an APBSP algorithm for edgecapacitated graphs running in O(n (3+ω)/2) time and a slightly faster O(n 2.657)time algorithm for vertexcapactitated graphs. The second algorithm significantly improves on an O(n2.859)time APBSP algorithm of Shapira, Yuster, and Zwick. Our APBSP algorithms make use of new hybrid products we call the distancemaxmin product and dominancedistance product. 1
Nondecreasing paths in weighted graphs, or: how to optimally read a train schedule
 In Proc. SODA
, 2008
"... A travel booking office has timetables giving arrival and departure times for all scheduled trains, including their origins and destinations. A customer presents a starting city and demands a route with perhaps several train connections taking him to his destination as early as possible. The booking ..."
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Cited by 6 (2 self)
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A travel booking office has timetables giving arrival and departure times for all scheduled trains, including their origins and destinations. A customer presents a starting city and demands a route with perhaps several train connections taking him to his destination as early as possible. The booking office must find the best route for its customers. This problem was first considered in the theory of algorithms by George Minty [Min58], who reduced it to a problem on directed weighted graphs: find a path from a given source to a given target such that the consecutive weights on the path are nondecreasing and the last weight on the path is minimized. Minty gave the first algorithm for the single source version of the problem, in which one finds minimum last weight nondecreasing paths from the source to every other vertex. In this paper we give the first linear time algorithm for this problem. We also define an all pairs version for the problem and give a strongly polynomial truly subcubic algorithm for it. 1
Efficient algorithms on sets of permutations, dominance, and realweighted APSP
"... Sets of permutations play an important role in the design of some efficient algorithms. In this paper we design two algorithms that manipulate sets of permutations. Both algorithms, each solving a different problem, use fast matrix multiplication techniques to achieve a significant improvement in th ..."
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Sets of permutations play an important role in the design of some efficient algorithms. In this paper we design two algorithms that manipulate sets of permutations. Both algorithms, each solving a different problem, use fast matrix multiplication techniques to achieve a significant improvement in the running time over the naive solutions. For a set of permutations P ⊂ Sn we say that i kdominates j if the number of permutations π ∈ P for which π(i) < π(j) is k. The dominance matrix of P is the n × n matrix DP where DP (i, j) = k if and only if i kdominates j. We give an efficient algorithm for computing DP using fast rectangular matrix multiplication. In particular, when P  = n our algorithm runs in O(n2.684) time. Computing the dominance matrix of permutations is computationally equivalent to the dominance problem in computational geometry. Thus, our algorithm slightly improves upon a wellknown O(n2.688) time algorithm of Matousek for the dominance problem. Permutation dominance is used, together with several other ingredients, to obtain a truly subcubic algorithm for the All Pairs Shortest Paths (APSP) problem in realweighted directed graphs, where the number of distinct weights emanating from each vertex is O(n 0.338). A special case of this algorithm implies an O(n 2.842) time algorithm for real vertexweighted APSP, which slightly improves a recent result of Chan [STOC07]. A set of permutations P ⊂ Sn is fully expanding if the product of any two elements of P yields a distinct permutation. Stated otherwise, P 2  = P  2 where P 2 ⊂ Sn is the set of products of two elements of P. We present a randomized algorithm that computes P 2  and hence decides if P is fully expanding. The algorithm also produces a table that, for any σ1, σ2, σ3, σ4 ∈ P, answers the query σ1σ2 = σ3σ4 in Õ(1) time. The algorithm uses, among other ingredients, a combination of fast matrix multiplication and polynomial identity testing. In particular, for P  = n our algorithm runs in O(nω) time where ω < 2.376 is the matrix multiplication
Efficient Algorithms for Path Problems in Weighted Graphs
, 2008
"... Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shorte ..."
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Cited by 5 (0 self)
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Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the allpairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions
All Pairs Bottleneck Paths and MaxMin Matrix Products in Truly Subcubic Time
, 2009
"... In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP pro ..."
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Cited by 5 (0 self)
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In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all pairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max,min)product of two arbitrary matrices over R ∪ {∞,−∞} in O(n 2+ω/3) ≤ O(n 2.792) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Maxmin products can be used to compute the maximum bottleneck values for all pairs of vertices together with a “successor matrix” from which one can extract an explicit maximum bottleneck path for any pair of vertices in time linear in the length of the path.
BoundedLeg Distance and Reachability Oracles
"... In a weighted, directed graph an Lbounded leg path is one whose constituent edges have length at most L. For any fixed L, computing Lbounded leg shortest paths is just as easy as the standard shortest path algorithm. In this paper we study approximate distance oracles (and reachability oracles) fo ..."
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Cited by 2 (2 self)
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In a weighted, directed graph an Lbounded leg path is one whose constituent edges have length at most L. For any fixed L, computing Lbounded leg shortest paths is just as easy as the standard shortest path algorithm. In this paper we study approximate distance oracles (and reachability oracles) for bounded leg path problems, where the leg bound L is not known in advance, but forms part of the query. Boundedleg path problems are more complicated than standard shortest path problems because the number of distinct shortest paths between two vertices (over all leg bounds) could be as large as the number of edges in the graph. The bounded leg constraint models situations where there is some limited resource that must be spent when traversing an edge. For example, the size of a fuel tank or the life of a battery places a hard limit on how far a vehicle can travel in one leg before refueling or recharging. Someone making a long road trip may place a hard limit on how many hours they are willing to drive in any one day. Our main result is a nearly optimal algorithm for preprocessing a directed graph in order to answer approximate bounded leg distance and bounded leg shortest path queries. In particular, we can preprocess any graph in Õ(n3) time, producing a data structure with size Õ(n2) that answers (1 + ɛ)approximate bounded leg distance queries in O(log log n) time. If the corresponding (1 + ɛ)approximate shortest path has l edges it can be returned in O(l log log n) time. These bounds are all within polylog(n) factors of the best standard allpairs shortest path algorithm and improve substantially the previous best bounded leg shortest path algorithm, whose preprocessing time and space are O(n 4) and Õ(n 2.5). We also consider bounded leg oracles in other situations. In the context of planar directed graphs we give a timespace tradeoff for answering bounded leg reachability queries. For any k ≥ 2 we can build a data structure with size O(kn 1+1/k) that answers reachability queries in time
Triangle Detection Versus Matrix Multiplication: A Study of Truly Subcubic Reducibility
"... It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that s ..."
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It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that such a reduction is impossible: for example, triangle detection returns one bit, while a BMM algorithm returns n 2 bits. Similar reasoning goes for other matrix products and their corresponding triangle problems. We show this intuition is false, and present a new generic strategy for efficiently computing matrix products over algebraic structures used in optimization. We say an algorithm on n × n matrices (or nnode graphs) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0, where M is the absolute value of the largest entry (or the largest edge weight). We prove an equivalence between the existence of truly subcubic algorithms for any (min, ⊙) matrix product, the corresponding matrix product verification problem, and a corresponding triangle detection problem. Our work simplifies and unifies prior work, and has some new consequences: • The following problems either all have truly subcubic algorithms, or none of them do: