We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in ~ O(n 2+ ) time, where satisfies the equation !(1; ; 1) = 1 + 2 and !(1; ; 1) is the exponent of the multiplication of an n \Theta n matrix by an n \Theta n matrix. The currently best available bounds on !(1; ; 1), obtained by Coppersmith and Winograd, and by Huang and Pan, imply that ! 0:575. The running time of our algorithm is therefore O(n 2:575 ). Our algorithm improves on the ~ O(n (3+!)=2 ) time algorithm, where ! = !(1; 1; 1) ! 2:376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2:688 ). The second

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.

Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m non-zero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices.

We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. The new algorithms are instantiations of a generic algorithm that we present for finding a directed C k , i.e., a directed cycle of length k, in a digraph, for any fixed k 3. This algorithm partitions the prospective C k 's in the input digraph G = (V, E) into O(log V ) classes, according to the degrees of their vertices. For each cycle class we determine, in O(E log V ) time, whether G contains a C k from that class, where c k = c k (#) is a constant that depends only on #, the exponent of square matrix multiplication. The search for cycles from a given class is guided by the solution of a small dynamic programming problem. The total running time of the obtained deterministic algorithm is therefore O(E k+1 V ).

We provide simple, faster algorithms for the detection of cliques and dominating sets of fixed order. Our algorithms are based on reductions to rectangular matrix multiplication. We also describe an improved algorithm for diamonds detection.

Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bit-comvSW5[5Wv of the algorithm asymithmW5[5 n and thenorm of A. Existing approaches rely onnumBDWzv approximW[ comoximW[z5 on exactcomvB-[;5vSW or on both types of arithmvSW incom9zqvSWqD c 2003 Elsevier B.V. All rights reserved. Keywords: Determ9v9vm Bit-com9vmv-- Integer mteger Approxim59 comoxim599 Exact comv-5-5DvS Random-5D algorithm 1. I517251716 Com;vm9 the sign or the value of thedetermDvS; nm-Bqq A is a classicalproblem Numblem mmble are usually focused oncomB-W9v the sign via an accurateapproxim;B99 of the determvS;;5 Amer the applications areimvWD;qW problem ofcom9qWvS;;5[ geom9qW that can be reduced to the determ5vS; question; the readerma refer to [11,12,9,10,46,45] and to the bibliography therein. InsymDW;B comW;BvS5-5 theproblem ofcom-DzWv the exact value of the ThismisvqB; is based on work supported in part by the National Science Foundation under grants Nrs. DMS-9977392, CCR-9988177, and CCR-0113121 (Kaltofen) and by the Centre National de la Recherche Scienti#que, Actions Incitatives No. 5929 et STIC LINBOX 2001 (Villard).

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 H-SUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN H-SUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication. For vertex-weighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h) ) time algorithm for MIN H-SUBGRAPH 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 edge-weighted 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

This paper provides an adaptive version of the unblocked baby steps/giant steps algorithm [20, Section 2]. The result is most easily stated when b |#| where # is the determinant to be computed and # with 1 is not known. Note that by Hadamard's bound |#| # n(b + log 2 (n)/2), so # = 0 covers the worst case. We describe a Monte Carlo algorithm that produces # in (n bit operations, again with standard matrix arithmetic. The corresponding bit complexity of the early termination Gaussian elimination method is 4-# , which is always more, and that of the algorithm by [10] is (n 1+1/2