Improved time bounds for the maximum flow problem

Also issued as: Working paper (Sloan School of Management) ; WP no. 1966-87Includes bibliographical references (p. 18-19).Research supported by the National Science Foundation. DCR-8605962 Research supported by the Office of Naval Research. NOOO14-87-K-0467by Ravindra K. Ahuja, James B. Orlin and Robert E. Tarjan

Max s,ts,t-Flow Oracles and Negative Cycle Detection in Planar Digraphs

We study the maximum $s,t$-flow oracle problem on planar directed graphs where the goal is to design a data structure answering max $s,t$-flow value (or equivalently, min $s,t$-cut value) queries for arbitrary source-target pairs $(s,t)$. For the case of polynomially bounded integer edge capacities, we describe an exact max $s,t$-flow oracle with truly subquadratic space and preprocessing, and sublinear query time. Moreover, if $(1-\epsilon)$-approximate answers are acceptable, we obtain a static oracle with near-linear preprocessing and $\tilde{O}(n^{3/4})$ query time and a dynamic oracle supporting edge capacity updates and queries in $\tilde{O}(n^{6/7})$ worst-case time. To the best of our knowledge, for directed planar graphs, no (approximate) max $s,t$-flow oracles have been described even in the unweighted case, and only trivial tradeoffs involving either no preprocessing or precomputing all the $n^2$ possible answers have been known. One key technical tool we develop on the way is a sublinear (in the number of edges) algorithm for finding a negative cycle in so-called dense distance graphs. By plugging it in earlier frameworks, we obtain improved bounds for other fundamental problems on planar digraphs. In particular, we show: (1) a deterministic $O(n\log(nC))$ time algorithm for negatively-weighted SSSP in planar digraphs with integer edge weights at least $-C$. This improves upon the previously known bounds in the important case of weights polynomial in $n$, and (2) an improved $O(n\log{n})$ bound on finding a perfect matching in a bipartite planar graph.Comment: Extended abstract to appear in SODA 202

Improved Bounds for Shortest Paths in Dense Distance Graphs

We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques. Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra\u27s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds. At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck

Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs

In the edge(vertex)-disjoint path problem we are given a graph $G$ and a set ${\cal T}$ of connection requests. Every connection request in ${\cal T}$ is a vertex pair $(s_i,t_i),$ $1 \leq i \leq K.$ The objective is to connect a maximum number of the pairs via edge(vertex)-disjoint paths. The edge-disjoint path problem can be generalized to the multiple-source unsplittable flow problem where connection request $i$ has a demand $\rho_i$ and every edge $e$ a capacity $u_e.$ All these problems are NP-hard and have a multitude of applications in areas such as routing, scheduling and bin packing. Given the hardness of the problem, we study polynomial-time approximation algorithms. In this context, a $\rho$-approximation algorithm is able to route at least a $1/\rho$ fraction of the connection requests. Although the edge- and vertex-disjoint path problems, and more recently the unsplittable flow generalization, have been extensively studied, they remain notoriously hard to approximate with a bounded performance guarantee. For example, even for the simple edge-disjoint path problem, no $o(\sqrt{|E|})$-approximation algorithm is known. Moreover some of the best existing approximation ratios are obtained through sophisticated and non-standard randomized rounding schemes. In this paper we introduce techniques which yield algorithms for a wide range of disjoint-path and unsplittable flow problems. For the general unsplittable flow problem, even with weights on the commodities, our techniques lead to the first approximation algorithm and obtain an approximation ratio that matches, to within logarithmic factors, the $O(\sqrt{|E|})$ approximation ratio for the simple edge-disjoint path problem. In addition to this result and to improved bounds for several disjoint-path problems, our techniques simplify and unify the derivation of many existing approximation results. We use two basic techniques. First, we propose simple greedy algorithms for edge- and vertex-disjoint paths and second, we propose the use of a framework based on packing integer programs for more general problems such as unsplittable flow. A packing integer program is of the form maximize $c^{T}\cdot x,$ subject to $Ax \leq b,$ $A,b,c \geq 0.$ As part of our tools we develop improved approximation algorithms for a class of packing integer programs, a result that we believe is of independent interest