Haim Kaplan

Reachability and Distance Queries via 2-Hop Labels

Reachability and distance queries in graphs are fundamental to numerous applications, ranging from geographic navigation systems to Internet routing. Some of these applications involve huge graphs and yet require fast query answering. We propose a new data structure for representing all distances in a graph. The data structure is distributed in the sense that it may be viewed as assigning labels to the vertices, such that a query involving vertices u and v may be answered using only the labels of u and v.Our labels are based on 2-hop covers of the shortest paths, or of all paths, in a graph. For shortest paths, such a cover is a collection S of shortest paths such that for every two vertices u and v, there is a shortest path from u to v that is a concatenation of two paths from S. We describe an efficient algorithm for finding an almost optimal 2-hop cover of a given collection of paths. Our approach is general and can be applied to directed or undirected graphs, exact or approximate shortest paths, or to reachability queries.We study the proposed data structure using a combination of theoretical and experimental means. We implemented our algorithm and checked the size of the resulting data structure on several real-life networks from different application areas. Our experiments show that the total size of the labels is typically not much larger than the network itself, and is usually considerably smaller than an explicit representation of the transitive closure of the network.

A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals

We give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation. Our algorithm is faster than the previous algorithm of Hannenhalli and Pevzner and its faster implementation by Berman and Hannenhalli. The algorithm is conceptually simple and does not require special data structures. Our study also considerably simplifies the combinatorial structures used by the analysis.

Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

A directed multigraph is said to be d-regular if the indegree and outdegree of every vertex is exactly d. By Halls theorem, one can represent such a multigraph as a combination of at most n2 cycle covers, each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than ⌊d/2⌋ copies of any 2-cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2-cycle occurs in at most one component of each pair. Our proof is constructive and gives a polynomial algorithm to find such a decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give an alternative, simpler algorithm to extract a single such pair.This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum Traveling Salesman Problem (TSP) problem. The first stage of the rounding procedure obtains two cycle covers that do not share a 2-cycle with weight at least twice the weight of the optimal solution. Then we show how to extract a tour from the 2 cycle covers, whose weight is at least 2/3 of the weight of the longest tour. This improves upon the previous 5/8 approximation with a simpler algorithm. Utilizing a reduction from maximum TSP to the shortest superstring problem, we obtain a 2.5-approximation algorithm for the latter problem, which is again much simpler than the previous one.For minimum asymmetric TSP, the same technique gives two cycle covers, not sharing a 2-cycle, with weight at most twice the weight of the optimum. Assuming triangle inequality, we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0.842 log2 n larger than optimal. This improves upon a previous approximation algorithm with approximation guarantee of 0.999 log2 n. Other applications of the rounding procedure are approximation algorithms for maximum 3-cycle cover (factor 2/3, previously 3/5) and maximum asymmetric TSP with triangle inequality (factor 10/13, previously 3/4).

Graph sandwich problems

Abstract The graph sandwich problem for property Π is defined as follows: Given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) such that E 1 ⊆ E 2 , is there a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 which satisfies property Π? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NP-completeness of others. We describe polynomial time algorithms for threshold graphs, split graphs, and cographs. For the sandwich problem for threshold graphs, the only case in which a previous algorithm existed, we obtain a faster algorithm. NP-completeness proofs are given for comparability graphs, permutation graphs, and several other families. For Eulerian graphs; one Version of the problem is polynomial and another is NP-complete.

Four Strikes Against Physical Mapping of DNA

Physical mapping is a central problem in molecular biology and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NP-complete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the k-consecutive ones problem for k > or = 2. These models have been chosen to reflect various features typical in biological data, including false-negative and positive errors, small width of the map, and chimericism.

Optimal oblivious routing in polynomial time

A recent seminal result of Racke is that for any undirected network there is an oblivious routing algorithm with a polylogarithmic competitive ratio with respect to congestion. Unfortunately, Rackes construction is not polynomial time. We give a polynomial time construction that guarantees Rackes bounds, and more generally gives the true optimal ratio for any (undirected or directed) network.

Compact Labeling Scheme for Ancestor Queries

We consider the following problem. Given a rooted tree

Prefetching the means for document transfer: a new approach for reducing Web latency

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Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs

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Nearest common ancestors: a survey and a new distributed algorithm

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