TY - CONF AB - Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with π‘œ(π‘šπ‘›) total update time, where π‘š is the number of edges and 𝑛 is the number of nodes in the graph [Henzinger et al. STOC 2014]. The algorithm is a combination of several different algorithms, each for a different π‘š vs. 𝑛 trade-off. For the case of π‘š=Θ(𝑛1.5) the running time is 𝑂(𝑛2.47), just barely below π‘šπ‘›=Θ(𝑛2.5). In this paper we simplify the previous algorithm using new algorithmic ideas and achieve an improved running time of 𝑂̃ (min(π‘š7/6𝑛2/3,π‘š3/4𝑛5/4+π‘œ(1),π‘š2/3𝑛4/3+π‘œ(1)+π‘š3/7𝑛12/7+π‘œ(1))). This gives, e.g., 𝑂(𝑛2.36) for the notorious case π‘š=Θ(𝑛1.5). We obtain the same upper bounds for the problem of maintaining the strongly connected components of a directed graph undergoing edge deletions. Our algorithms are correct with high probabililty against an oblivious adversary. AU - Henzinger, Monika H AU - Krinninger, Sebastian AU - Nanongkai, Danupon ID - 11785 SN - 0302-9743 T2 - 42nd International Colloquium on Automata, Languages and Programming TI - Improved algorithms for decremental single-source reachability on directed graphs VL - 9134 ER - TY - CONF AB - We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs with m edges and n vertices only rather simple O(m n)-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time 𝑂(𝑛2). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an 𝑂(π‘š2/log𝑛)-time algorithm for 2-edge strongly connected components, and thus improve over the O(m n) running time also when π‘š=𝑂(𝑛). Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of 𝑂(𝑛2log𝑛) for k-edge-connectivity and 𝑂(𝑛3) for k-vertex-connectivity. AU - Henzinger, Monika H AU - Krinninger, Sebastian AU - Loitzenbauer, Veronika ID - 11787 SN - 0302-9743 T2 - 2nd International Colloquium on Automata, Languages and Programming TI - Finding 2-edge and 2-vertex strongly connected components in quadratic time VL - 9134 ER - TY - CONF AB - Ad exchanges are becoming an increasingly popular way to sell advertisement slots on the internet. An ad exchange is basically a spot market for ad impressions. A publisher who has already signed contracts reserving advertisement impressions on his pages can choose between assigning a new ad impression for a new page view to a contracted advertiser or to sell it at an ad exchange. This leads to an online revenue maximization problem for the publisher. Given a new impression to sell decide whether (a) to assign it to a contracted advertiser and if so to which one or (b) to sell it at the ad exchange and if so at which reserve price. We make no assumptions about the distribution of the advertiser valuations that participate in the ad exchange and show that there exists a simple primal-dual based online algorithm, whose lower bound for the revenue converges to 𝑅𝐴𝐷𝑋+𝑅𝐴(1βˆ’1/𝑒), where 𝑅𝐴𝐷𝑋 is the revenue that the optimum algorithm achieves from the ad exchange and 𝑅𝐴 is the revenue that the optimum algorithm achieves from the contracted advertisers. AU - DvoΕ™Γ‘k, Wolfgang AU - Henzinger, Monika H ID - 11788 SN - 0302-9743 T2 - 12th International Workshop of Approximation and Online Algorithms TI - Online ad assignment with an ad exchange VL - 8952 ER - TY - CONF AB - In this paper, we develop a dynamic version of the primal-dual method for optimization problems, and apply it to obtain the following results. (1) For the dynamic set-cover problem, we maintain an 𝑂(𝑓2)-approximately optimal solution in 𝑂(𝑓⋅log(π‘š+𝑛)) amortized update time, where 𝑓 is the maximum β€œfrequency” of an element, 𝑛 is the number of sets, and π‘š is the maximum number of elements in the universe at any point in time. (2) For the dynamic 𝑏-matching problem, we maintain an 𝑂(1)-approximately optimal solution in 𝑂(log3𝑛) amortized update time, where 𝑛 is the number of nodes in the graph. AU - Bhattacharya, Sayan AU - Henzinger, Monika H AU - Italiano, Giuseppe F. ID - 11786 SN - 0302-9743 T2 - 42nd International Colloquium on Automata, Languages and Programming TI - Design of dynamic algorithms via primal-dual method VL - 9134 ER - TY - JOUR AB - Phylogenetic diversity (PD) is a measure of biodiversity based on the evolutionary history of species. Here, we discuss several optimization problems related to the use of PD, and the more general measure split diversity (SD), in conservation prioritization. Depending on the conservation goal and the information available about species, one can construct optimization routines that incorporate various conservation constraints. We demonstrate how this information can be used to select sets of species for conservation action. Specifically, we discuss the use of species' geographic distributions, the choice of candidates under economic pressure, and the use of predator–prey interactions between the species in a community to define viability constraints. Despite such optimization problems falling into the area of NP hard problems, it is possible to solve them in a reasonable amount of time using integer programming. We apply integer linear programming to a variety of models for conservation prioritization that incorporate the SD measure. We exemplarily show the results for two data sets: the Cape region of South Africa and a Caribbean coral reef community. Finally, we provide user-friendly software at http://www.cibiv.at/software/pda. AU - Chernomor, Olga AU - Minh, Bui Quang AU - Forest, FΓ©lix AU - Klaere, Steffen AU - Ingram, Travis AU - Henzinger, Monika H AU - von Haeseler, Arndt ID - 11845 IS - 1 JF - Methods in Ecology and Evolution TI - Split diversity in constrained conservation prioritization using integer linear programming VL - 6 ER -