Results 1  10
of
16
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
Abstract

Cited by 54 (3 self)
 Add to MetaCart
In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
Finding, minimizing, and counting weighted subgraphs
 In Proceedings of the FourtyFirst Annual ACM Symposium on the Theory of Computing
, 2009
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of cop ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) nodeweight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edgeweighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edgeweighted problem is much harder than its nodeweighted version. 1
Finding the smallest Hsubgraph in real weighted graphs and related problems
 In Proc. of ICALP, SpringerVerlag LNCS 4051:262–273
, 2006
"... Abstract. Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN HSUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
Abstract. Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN HSUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN HSUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication. For vertexweighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h) ) time algorithm for MIN HSUBGRAPH in case H is a fixed graph with h vertices and ω < 2.376 is the exponent of matrix multiplication. The value of t(ω, h) is determined by solving a small integer program. In particular, the smallest triangle can be found in O(n 2+1/(4−ω) ) ≤ o(n 2.616) time, the smallest K4 in O(n ω+1) time, the smallest K7 in O(n 4+3/(4−ω) ) time. As h grows, t(ω, h) converges to 3h/(6 − ω) < 0.828h. Interestingly, only for h = 4, 5, 8 the running time of our algorithm essentially matches that of the (unweighted) Hsubgraph detection problem. Already for triangles, our results improve upon the main result of [VW06]. Using rectangular matrix multiplication, the value of t(ω, h) can be improved; for example, the runtime for triangles becomes O(n 2.575). We also present an algorithm whose running time is a function of m, the number of edges. In particular, the smallest triangle can be found in O(m (18−4ω)/(13−3ω) ) ≤ o(m 1.45) time. For edgeweighted graphs we present an O(m 2−1/k log n) time algorithm that finds the smallest cycle of length 2k or 2k − 1. This running time is identical, up to a logarithmic factor, to the running time of the algorithm of Alon et al. for the unweighted case. Using the color coding method and a recent algorithm of Chan for distance products, we obtain an O(n 3 / log n) time randomized algorithm for finding the smallest cycle of any fixed length. 1
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 solvable ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
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
Algorithms and Resource Requirements for Fundamental Problems
, 2007
"... no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.
AllPairs Bottleneck Paths For General Graphs in Truly SubCubic Time
 STOC'07
, 2007
"... In the allpairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real nonnegative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can b ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
In the allpairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real nonnegative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and allpairs 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. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.
Dynamic programming and fast matrix multiplication
 of LNCS
, 2006
"... Abstract. We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Abstract. We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Dominating Set and Longest Path. Our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the currently fastest algorithms for Planar Vertex Cover of runtime O(2 2.52 √ n), for Planar Dominating Set of runtime exact O(2 3.99 √ n) and parameterized O(2 11.98 √ k) · n O(1) , and for Planar Longest Path of runtime O(2 5.58 √ n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376). 1
AllPairs Bottleneck Paths in Vertex Weighted Graphs
 In Proc. of SODA, 978–985
, 2007
"... Let G = (V, E, w) be a directed graph, where w: V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Let G = (V, E, w) be a directed graph, where w: V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the AllPairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first subcubic algorithm for this problem. Unlike the subcubic algorithm for the allpairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights. The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on
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 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
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