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Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Abstract

Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n×n matrix M and will receive n column-vectors of size n, denoted by v_1, ..., v_n, one by one. After seeing each vector v_i, we have to output the product Mv_i before we can see the next vector. A naive algorithm can solve this problem using O(n^3) time in total, and its running time can be slightly improved to O(n^3/\log^2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n^{3-ɛ})) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, d-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term ``combinatorial algorithms'' is interpreted as ``Strassen-like algorithms'' [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures - such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase - thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fully-dynamic densest subgraph and diameter problems.

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Citation
Category
Paper in Conference Proceedings or in Workshop Proceedings (Full Paper in Proceedings)
Event Title
47th ACM Symposium on Theory of Computing (STOC 2015)
Divisions
Theory and Applications of Algorithms
Subjects
Datenstrukturen
Theoretische Informatik
Event Location
Portland, OR, USA
Event Type
Conference
Event Dates
June 14 - June 17
Date
June 2015
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