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تسجيل مستخدم جديدNew Unconditional Hardness Results for Dynamic and Online Problems
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نشر من قبل
Kasper Green Larsen
تاريخ النشر
2015
مجال البحث
الهندسة المعلوماتية
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English
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There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjointness known as Patrascus Multiphase Problem. We give improved unconditional lower bounds for these problems as well as introducing new proof techniques of independent interest. These include a technique capable of proving strong threshold lower bounds of the following form: If we insist on having a very fast query time, then the update time has to be slow enough to compute a lookup table with the answer to every possible query. This is the first time a lower bound of this type has been proven.
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y $sigma$ to indices at most $i$. This problem has been studied in the literature for various special cases of the cost function $f$, and in a general setting for a submodular or supermodular cost $f$ [ITT2012]. Though MLOP was known to be NP-hard for general submodular functions, it was unknown whether the special case of graphic matroid MLOP (with $f$ being the matroid rank function of a graph) was polynomial-time solvable. Following this motivation, we explore related classes of linear ordering problems, including symmetric submodular MLOP, minimum latency vertex cover, and minimum sum vertex cover. We show that the most special cases of our problem, graphic matroid MLOP and minimum latency vertex cover, are both NP-hard. We further expand the toolkit for approximating MLOP variants: using the theory of principal partitions, we show a $2-frac{1+ell_{f}}{1+|E|}$ approximation to monotone submodular MLOP, where $ell_{f}=frac{f(E)}{max_{xin E}f({x})}$ satisfies $1 leq ell_f leq |E|$. Thus our result improves upon the best known bound of $2-frac{2}{1+|E|}$ by Iwata, Tetali, and Tripathi [ITT2012]. This leads to a $2-frac{1+r(E)}{1+|E|}$ approximation for the matroid MLOP, corresponding to the case when $r$ is the rank function of a given matroid. Finally, we show that MLVC can be $4/3$ approximated, matching the integrality gap of its vanilla LP relaxation.
In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $mathcal{M}={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is
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