A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$$ As a corollary, we conclude that the transitive closure of a directed graph $G$ can be computed by a combinatorial randomized algorithm in the aforementioned time. $\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$ We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time $\widetilde{O}(n^{2.75})$

Studies in Efficient Discrete Algorithms

This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity

A QPTAS for the Base of the Number of Triangulations of a Planar Point Set

The number of triangulations of a planar n point set is known to be $c^n$, where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm for counting triangulations of a planar n point set runs in $O^*(2^n)$ time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in $n.$ We present the first quasi-polynomial approximation scheme for the base of the number of triangulations of a planar point set

An Output-Sensitive Algorithm for All-Pairs Shortest Paths in Directed Acyclic Graphs

First, we present a new algorithm for the single-source shortest paths problem (SSSP) in edge-weighted directed graphs, with n vertices, m edges, and both positive and negative real edge weights. Given a positive integer parameter t, in O(tm) time the algorithm finds for each vertex v a path distance from the source to v not exceeding that yielded by the shortest path from the source to v among the so called t+ light paths. A directed path between two vertices is t+ light if it contains at most t more edges than the minimum edge-cardinality directed path between these vertices. For t= O(n), our algorithm yields an O(nm)-time solution to SSSP in directed graphs with real edge weights matching that of Bellman and Ford. Our main contribution is a new, output-sensitive algorithm for the all-pairs shortest paths problem (APSP) in directed acyclic graphs (DAGs) with positive and negative real edge weights. The running time of the algorithm depends on such parameters as the number of leaves in (lexicographically first) shortest-paths trees, and the in-degrees in the input graph. If the trees are sufficiently thin on the average, the algorithm is substantially faster than the best known algorithm. Finally, we discuss an extension of hypothetical improved upper time-bounds for APSP in non-negatively edge-weighted DAGs to include directed graphs with a polynomial number of large directed cycles

A QPTAS for the Base of the Number of Crossing-Free Structures on a Planar Point Set

The number of triangulations of a planar n point set S is known to be c^n, where the base c lies between 2.43 and 30. Similarly, the number of spanning trees on S is known to be d^n, where the base d lies between 6.75 and 141.07. The fastest known algorithm for counting triangulations of S runs in O^∗(2^n) time while that for counting spanning trees runs in O^∗(7.125^n) time. The fastest known arbitrarily close approximation algorithms for the base of the number of triangulations of S and the base of the number of spanning trees of S, respectively, run in time subexponential in n. We present the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of spanning trees on S, respectively

A QPTAS for the base of the number of crossing-free structures on a planar point set

The number of triangulations of a planar n point set S is known to be cn, where the base c lies between 2.43 and 30. Similarly, the number of crossing-free spanning trees on S is known to be dn, where the base d lies between 6.75 and 141.07. The fastest known algorithm for counting triangulations of S runs in 2(1+o(1))nlog n time while that for counting crossing-free spanning trees runs in O (7.125n) time. The fastest known, non-trivial approximation algorithms for the number of triangulations of S and the number of crossing-free spanning trees of S, respectively, run in time subexponential in n. We present the first non-trivial approximation algorithms for these numbers running in quasi-polynomial time. They yield the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of crossing-free spanning trees on S, respectively

Optimal Cuts and Partitions in Tree Metrics in Polynomial Time

We present a polynomial time dynamic programming algorithm for optimal partitions in the shortest path metric induced by a tree. This resolves, among other things, the exact complexity status of the optimal partition problems in one dimensional geometric metric settings. Our method of solution could be also of independent interest in other applications. We discuss also an extension of our method to the class of metrics induced by the bounded treewidth graphs

Iterative merging heuristics for correlation clustering

A straightforward natural iterative heuristic for correlation clustering in the general setting is to start from singleton clusters and whenever merging two clusters improves the current quality score merge them into a single cluster. We analyse the approximation and complexity aspects of this heuristic and its three simple deterministic or random refinements

Bounds for semi-disjoint bilinear forms in a unit-cost computational model

We study the complexity of the so called semi-disjoint bilin-ear forms over different semi-rings, in particular the n-dimensional vector convolution and n × n matrix product. We consider a powerful unit-cost computational model over the ring of integers allowing for several addi-tional operations and generation of large integers. We show the following dichotomy for such a powerful model: while almost all arithmetic semi-disjoint bilinear forms have the same asymptotic time complexity as that yielded by naive algorithms, matrix multiplication, the so called distance matrix product, and vector convolution can be solved in a linear number of steps. It follows in particular that in order to obtain a non-trivial lower bounds for these three basic problems one has to assume restrictions on the set of allowed operations and/or the size of used integers

L'algorithme de coût min-max de Lawler : une variante avec incertitude

National audience  Nous considérons l'ordonnancement de n tâches sur une machine. La tâche j, j = 1, . . . , n, est caractérisée par le temps d'exécution pj ≥ 0 et par une fonction croissante Φj(t) qui indique le coût à payer si la tâche j termine à l'instant t. Des contraintes de précédence sont décrites par un graphe orienté acyclique G = (V, E ). L'objectif est de trouver un ordonnancement admissible, c'est-à-dire une permutation π = (π(1),..., π(n)), qui minimise le coût maximum Φmax. L'algorithme de Lawler résout le problème correspondant 1|prec|Φmax en temps O(n2). Cet algorithme peut se résumer comme suit : La permutation est construite de droite à gauche. A chaque étape on place une tâche disponible (c'est-à-dire une tâche terminale dans le graphe restant de précédence) qui possède le coût minimum. Nous spécialisons la fonction coût. Le coût d'une tâche j dépend d'une fonction croissante φ(t), commune pour toutes les tâches, et d'un paramètre λj. La valeur de λj n'est pas connue en avance et peut prendre n'importe quelle valeur dans un intervalle [λ−j , λ+j ]. Posons Φj (t) = φ(t) + λj où λj ∈ [λ−j , λ+j ] et nous obtenons le problème avec incertitude 1|prec ; Φj(t)=φ(t)+λj, λj ∈[λ−j ,λ+j ]|Φmax. Ce problème est inspiré par une application pour laquelle λj donne le temps de transport d'une pièce j du site de production jusqu'au client. Le temps de trajet est incertain à cause du trafic. Le problème est résolu par une permutation qui vérifie le critère du regret min-max. Il est montré que cette solution s'obtient en O(n2) par l'algorithme de Lawler et une fonction coût particulière. Cette approche permet aussi de décrire les instances pour lesquelles une permutation optimale pour tous les scénarios possibles existe. (voir : https://cahiersleibniz.g-scop.grenoble-inp.fr ; N. 209)  </p