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Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture
University of Vienna.
University of Vienna.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-4468-2675
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-3694-740X
2015 (English)In: STOC '15 Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, ACM Press, 2015, p. 21-30Conference paper, Published paper (Refereed)
Abstract [en]

Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n x n matrix M and will receive n column-vectors of size n, denoted by v1, ..., vn, one by one. After seeing each vector vi, we have to output the product Mvi before we can see the next vector. A naive algorithm can solve this problem using O(n3) time in total, and its running time can be slightly improved to O(n3/log2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n3-ε)) 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.

Place, publisher, year, edition, pages
ACM Press, 2015. p. 21-30
National Category
Computer Sciences
Identifiers
Scopus ID: 2-s2.0-84958762655OAI: oai:DiVA.org:kth-165846DiVA, id: diva2:808913
Conference
STOC 2015: 47th Annual Symposium on the Theory of Computing,Portland, OR, June 15 - June 17 2015
Note

QC 20150811

Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2018-07-24Bibliographically approved
In thesis
1. Dynamic algorithms: new worst-case and instance-optimal bounds via new connections
Open this publication in new window or tab >>Dynamic algorithms: new worst-case and instance-optimal bounds via new connections
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis studies a series of questions about dynamic algorithms which are algorithms for quickly maintaining some information of an input data undergoing a sequence of updates. The first question asks \emph{how small the update time for handling each update can be} for each dynamic problem. To obtain fast algorithms, several relaxations are often used including allowing amortized update time, allowing randomization, or even assuming an oblivious adversary. Hence, the second question asks \emph{whether these relaxations and assumptions can be removed} without sacrificing the speed. Some dynamic problems are successfully solved by fast dynamic algorithms without any relaxation. The guarantee of such algorithms, however, is for a worst-case scenario. This leads to the last question which asks for \emph{an algorithm whose cost is nearly optimal for every scenario}, namely an instance-optimal algorithm. This thesis shows new progress on all three questions.

For the first question, we give two frameworks for showing the inherent limitations of fast dynamic algorithms. First, we propose a conjecture called the Online Boolean Matrix-vector Multiplication Conjecture (OMv). Assuming this conjecture, we obtain new \emph{tight} conditional lower bounds of update time for more than ten dynamic problems even when algorithms are allowed to have large polynomial preprocessing time. Second, we establish the first analogue of NP-completeness'' for dynamic problems, and show that many natural problems are NP-hard'' in the dynamic setting. This hardness result is based on the hardness of all problems in a huge class that includes a number of natural and hard dynamic problems. All previous conditional lower bounds for dynamic problems are based on hardness of specific problems/conjectures.

For the second question, we give an algorithm for maintaining a minimum spanning forest in an $n$-node graph undergoing edge insertions and deletions using $n^{o(1)}$ worst-case update time with high probability. This significantly improves the long-standing $O(\sqrt{n})$ bound by {[}Frederickson STOC'83, Eppstein, Galil, Italiano and Nissenzweig FOCS'92{]}. Previously, a spanning forest (possibly not minimum) can be maintained in polylogarithmic update time if either amortized update is allowed or an oblivious adversary is assumed. Therefore, our work shows how to eliminate these relaxations without slowing down updates too much.

For the last question, we show two main contributions on the theory of instance-optimal dynamic algorithms. First, we use the forbidden submatrix theory from combinatorics to show that a binary search tree (BST) algorithm called \emph{Greedy} has almost optimal cost when its input \emph{avoids a pattern}. This is a significant progress towards the Traversal Conjecture {[}Sleator and Tarjan JACM'85{]} and its generalization. Second, we initialize the theory of instance optimality of heaps by showing a general transformation between BSTs and heaps and then transferring the rich analogous theory of BSTs to heaps. Via the connection, we discover a new heap, called the \emph{smooth heap}, which is very simple to implement, yet inherits most guarantees from BST literature on being instance-optimal on various kinds of inputs. The common approach behind all our results is about making new connections between dynamic algorithms and other fields including fine-grained and classical complexity theory, approximation algorithms for graph partitioning, local clustering algorithms, and forbidden submatrix theory.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 51
Series
TRITA-EECS-AVL ; 2018:51
National Category
Computer Sciences
Computer Science
Identifiers
urn:nbn:se:kth:diva-232471 (URN)978-91-7729-865-6 (ISBN)
Public defence
2018-08-27, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Note

QC 20180725

Available from: 2018-07-25 Created: 2018-07-24 Last updated: 2018-07-25Bibliographically approved

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Cite
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