Research Projects, Ryan Williams

PROJECTS AND THEOREMS

In reverse chronological order.
Please observe the copyrights of these papers, whenever it is necessary to observe them.

2011

S. Buss and R. Williams. Limits on Alternation-Trading Proofs for Time-Space Lower Bounds. Submitted.
Preprint available on ECCC.
R. Williams. Non-Uniform ACC Circuit Lower Bounds.
In 26th IEEE Conference on Computational Complexity (CCC 2011).
[Full version] [Conference version] [Slides from the conference talk, in pptx]
Best Paper Award
B. Kimelfeld, J. Vondrak, and R. Williams. Maximizing conjunctive views in deletion propagation.
In ACM Symposium on Principles of Database Systems (PODS 2011), 187--198, 2011.

2010

V. Vassilevska Williams and R. Williams. Subcubic Equivalences Between Path, Matrix, and Triangle Problems.
In 51st IEEE Symposium on Foundations of Computer Science (FOCS 2010).
A preliminary version: [pdf]
Summary: We show that many long-studied problems with natural cubic time algorithms are all equivalent under a new notion of "subcubic reducibility", meaning that either all of them require about cubic time or none of them do. There are several surprises: for example, finding a negative edge-weighted triangle in truly subcubic time implies that all-pairs shortest paths is in truly subcubic time(!). Compare with our STOC'06 paper, which gives a truly subcubic algorithm for the triangle problem in the node-weighted case.
J. Blocki and R. Williams. Resolving the complexity of some data privacy problems.
In 37th International Colloquium on Automata, Languages, and Programming (ICALP 2010).
Full version: [arXiv]
R. Impagliazzo and R. Williams. Communication complexity with synchronized clocks.
In 25th IEEE Conference on Computational Complexity (CCC 2010).
Conference version: [pdf]
R. Williams. Improving exhaustive search implies superpolynomial lower bounds.
In 42nd ACM Symposium on Theory of Computing (STOC 2010).
Invited to the STOC'10 special issue of SIAM Journal on Computing.
Full version (draft): [pdf]
R. Williams. Alternation-Trading Proofs, Linear Programming, and Lower Bounds.
In 27th Symposium on Theoretical Aspects of Computer Science (STACS 2010).
Full version (draft): [pdf]
An older version that is more focused on computer searches (entitled "Automated Proofs of Time Lower Bounds") can be found here: [pdf]
Some Maple code for proof searches can be found HERE (as a text file) and HERE (as a Maple worksheet).
M. Pătraşcu and R. Williams. On the Possibility of Faster SAT Algorithms.
In 20th ACM-SIAM Symposium on Discrete Algorithms (SODA 2010).
Preliminary full version: [pdf]

2009

N. Bansal and R. Williams. Regularity Lemmas and Combinatorial Algorithms.
In 50th IEEE Symposium on Foundations of Computer Science (FOCS 2009).
Conference version: [pdf]
Full version (draft): [pdf]
L. Fortnow, R. Santhanam, and R. Williams. Fixed Polynomial Size Circuit Lower Bounds.
In 24th IEEE Conference on Computational Complexity (CCC 2009). [pdf]
S. Diehl, D. van Melkebeek, and R. Williams. An Improved Time-Space Lower Bound for Tautologies.
In International Computing and Combinatorics Conference (COCOON 2009).
Invited to the special issue of Journal of Combinatorial Optimization for COCOON'09.
Draft of journal version: [pdf]
I. Koutis and R. Williams. Limits and applications of group algebras for parameterized problems.
In 36th International Colloquium on Automata, Languages, and Programming (ICALP 2009).
Conference version: [pdf]
Full version coming soon.
Erratum: there's a bug in the algorithm for k-leaf (Thm 2.3). We are currently working to fix it.
V. Vassilevska and R. Williams. Finding, Minimizing, and Counting Weighted Subgraphs.
In 41st ACM Symposium on Theory of Computing (STOC 2009).
Full version: [pdf]
R. Williams. Finding Paths of Length k in O*(2^k) Time. IPL 109(6):315--318, 2009. [arXiv]

2008

R. Williams. Applying Practice to Theory. SIGACT News December 2008. [arXiv]
R. Williams. Maximum 2-satisfiability. Encyclopedia of Algorithms, 2008. [pdf]
R. Williams. Non-Linear Time Lower Bound for (Succinct) Quantified Boolean Formulas. ECCC [pdf]
G. Blelloch, V. Vassilevska, and R. Williams. A New Combinatorial Approach to Sparse Graph Problems. [pdf]
In 35th International Colloquium on Automata, Languages, and Programming (ICALP 2008).

2007

R. Williams. Algorithms and Resource Requirements for Fundamental Problems. Ph.D Thesis. [pdf]
NOTE: My thesis improves upon several of the below papers, namely ICALP'04, CCC'05, and CCC'07. It gives an overview of time-space tradeoff lower bounds, introduces an automated theorem-proving strategy for discovering new lower bounds, generalizes my exact algorithm for 2-CSP in several ways, and shows how algorithmic progress on some problems in parameterized complexity would yield new algorithms for the satisfiability problem (which eventually turned into the SODA'10 paper with Patrascu).
V. Vassilevska, R. Williams, and R. Yuster. All-Pairs Bottleneck Paths For General Graphs in Truly Sub-Cubic Time.
In 39th ACM Symposium on Theory of Computing (STOC 2007). [pdf]
R. Williams. Time-Space Tradeoffs for Counting NP Solutions Modulo Integers.
In 22th IEEE Conference on Computational Complexity (CCC 2007).
Journal of Computational Complexity 17(2), 2008. [pdf] Slides: [pdf]
Special issue for CCC'07, received Ron Book Prize for Best Student Paper.
R. Williams. Matrix-Vector Multiplication in Sub-Quadratic Time (Some Preprocessing Required).
In 17th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007). Full version: [pdf] Slides: [pdf]

2006

V. Vassilevska, R. Williams, and R. Yuster. Finding the smallest H-subgraph in real weighted graphs and related problems.
In 33rd International Colloquium on Automata, Languages, and Programming (ICALP 2006). [pdf]
Journal version to apper in ACM Transactions on Algorithms [arXiv].
V. Vassilevska and R. Williams. Finding a Maximum Weight Triangle in O(n^(3-delta)) Time, With Applications.
In 38th ACM Symposium on Theory of Computing (STOC 2006). [pdf]
V. Vassilevska, R. Williams, and S. L. M. Woo. Confronting Hardness Using a Hybrid Approach.
In 16th ACM-SIAM Symposium on Discrete Algorithms (SODA 2006). [pdf]

2005

V. Shkapenyuk, R. Williams, S. Harizopoulos, and A. Ailamaki. Deadlock Resolution in Pipelined Query Graphs. [pdf]
CMU Technical Report CMU-CS-05-122, March 2005.
R. Williams. Parallelizing Time With Polynomial Circuits.
In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2005).
Invited to the SPAA'05 special issue of Theory of Computing Systems.
Full version in Theory of Computing Systems, 2009. [pdf] Slides: [pdf]

R. Williams. Better Time-Space Lower Bounds for SAT and Related Problems.[pdf] Slides: [pdf]
In 20th IEEE Conference on Computational Complexity (CCC 2005).
Full version in Journal of Computational Complexity 15(4), 2006.
Special issue for CCC'05, received Ron Book Prize for Best Student Paper.
NOTE: This work is basically obsolete. Essentially all results here have been improved in my Ph.D Thesis and the paper "Alternation-Trading Proofs, Linear Programming, and Lower Bounds" in STACS 2010.
Carla Gomes and Ryan Williams. Approximation Algorithms.
Book chapter in: Introduction to Optimization, Decision Support and Search Methodologies, Burke and Kendall (eds.), Springer.

2004

R. Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. [pdf]
Theoretical Computer Science 348(2-3): 357--365, 2005.
Preliminary version in International Colloquium on Automata, Languages, and Programming (ICALP 2004).
Special issue for ICALP'04, received Best Student ICALP Paper Award.
NOTE: This work is basically obsolete. Essentially all results in this paper have been improved in Chapter 6 of my Ph.D. thesis.

A. Meyerson and R. Williams. On The Complexity of Optimal K-Anonymity. [pdf]
In ACM Symposium on Principles of Database Systems (PODS 2004).

2003

R. Williams. On Computing k-CNF Formula Properties. [ps] [pdf]
Older version in International Conference on Theory and Applications of Satisfiability Testing (SAT 2003).
R. Williams, C. Gomes, and B. Selman. Backdoors To Typical Case Complexity. [ps][pdf]
In International Joint Conference on Artificial Intelligence (IJCAI 2003).
Slides available here.
C. Gomes, and B. Selman, and R. Williams. On the connections between backdoors and heavy-tails on combinatorial search. [ps]
In International Conference on Theory and Applications of Satisfiability Testing (SAT 2003).

2002

R. Williams. Algorithms for Quantified Boolean Formulas. [pdf]
In 13th ACM-SIAM Symposium on Discrete Algorithms (SODA 2002). The above version is slightly revised from the original, correcting some typos.

The Junk Drawer: Old Manuscripts, Tech Reports, etc.

R. Williams. Defying Hardness With a Hybrid Approach. [pdf] CMU Technical Report CMU-CS-04-159, August 2004.
A. Meyerson and R. Williams. General k-Anonymization is Hard. [ps] [pdf] CMU Technical Report CMU-CS-03-113.
R. Williams. Abstract Complexity in Reflexive Type Theory. [ps]
Accepted in Implicit Computational Complexity (ICC 2002). Unfortunately, I could not attend.
Access Complexity. [pdf] The title stems from the emphasis on the accessibility of mass storage in characterizing complexity classes. Senior thesis in mathematics. Advisors: Juris Hartmanis and Anil Nerode. (Note: If you actually read this, please beware of bugs, typos, etc.)
Brute-Force Search and Oracle-Based Computations. [ps] Spin-off of the thesis which discusses a deterministic "brute-force" search model that captures exactly P^{NP}, with implications. Unpublished manuscript.
Space Efficient Reversible Simulations. [pdf]
Work done while at a DIMACS REU and the PCMI Summer School at Institute for Advanced Study, Summer 2000. Please note that the above paper is more recent than the paper on my (very old) DIMACS page. Buhrman, Tromp and Vitanyi independently found similar results, though they are a bit less general. For this reason, I'll only post the work here. Below is their ICALP paper, which cites the above.
H. Buhrman, J. Tromp, P. Vitanyi. Time and Space Bounds for Reversible Simulation. ICALP 2001.