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### Enumeration of Enumeration Algorithms [article]

Kunihiro Wasa
2016 arXiv   pre-print
In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.  ...  Reference  Enumeration of all maximal independent sets in a hypergraph of bounded intersections Input A hypergraph H ∈ A(k, r), where k + r ≤ constant. Output All maximal independent sets in H.  ...  Comment This algorithm uses the enumeration algorithm for maximal independent sets. Reference  1.6.6 Enumeration of all triangulation paths of a point set Input A point set P .  ...

### Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs [chapter]

Yoshio Okamoto, Takeaki Uno, Ryuhei Uehara
2005 Lecture Notes in Computer Science
With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant  ...  We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of  ...  Shankar Ram for pointing out a paper  . References  ...

### Parallel Batch-Dynamic k-Clique Counting [article]

Laxman Dhulipala, Quanquan C. Liu, Julian Shun, Shangdi Yu
2020 arXiv   pre-print
Using a recently developed parallel k-clique counting algorithm, we also obtain a simple batch-dynamic algorithm for k-clique counting on graphs with arboricity α running in O(Δ(m+Δ)α^k-4) expected work  ...  Assuming that a parallel fast matrix multiplication algorithm exists with parallel matrix multiplication constant ω_p, the same algorithm solves dynamic k-clique counting with O(min(Δ m^(2k - 1)ω_p/3(ω_p  ...  We thank Josh Alman, Nicole Wein, and Virginia Vassilevska Williams for helpful discussions on various aspects of our paper. We also thank anonymous reviewers for their helpful suggestions.  ...

### Towards optimal lower bounds for clique and chromatic number

Lars Engebretsen, Jonas Holmerin
2003 Theoretical Computer Science
It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n 1− , for any constant ¿ 0, unless NP = ZPP.  ...  log n) 3=2 ) ), neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n 1− (n) where ∈ O((log log n) −1=2 ).  ...  Acknowledgements We would like to thank Alex Samorodnitsky and Luca Trevisan for providing us with preliminary versions of their paper  .  ...

### Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximativ (Dagstuhl Seminar 18421)

Henning Fernau, Petr. A. Golovach, Marie-France Sagot, Michael Wagner
2019 Dagstuhl Reports
Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph or particular satisfying assignments of logical expressions.  ...  Enumeration problems arise in a natural way in various fields of Computer Science, as, e.g., Artificial Intelligence and Data Mining, in Natural Sciences Engineering, Social Sciences, and Biology.  ...  Starting with a paper by Couturier Heggernes, Kratsch, van't Hoff (2013) matching upper and lower bounds for the number of minimal dominating sets in n-vertex graphs of a certain graph class have been  ...

### An Experimental Study of Polylogarithmic, Fully Dynamic, Connectivity Algorithms

Raj Iyer, David Karger, Hariharan Rahul, Mikkel Thorup
2001 ACM Journal of Experimental Algorithmics
The lower bound holds even for the restricted case where the graph is a set of disjoint paths, and it allows for both randomization Dynamic Connectivity Algorithms 3 and amortization.  ...  Thus, our algorithms with heuristics still obey the polylogarithmic time bounds proven for the original algorithms, while being faster in many cases.  ...

### On Finding and Enumerating Maximal and Maximum k-Partite Cliques in k-Partite Graphs

Charles Phillips, Kai Wang, Erich Baker, Jason Bubier, Elissa Chesler, Michael Langston
2019 Algorithms
Let k denote an integer greater than 2, let G denote a k-partite graph, and let S denote the set of all maximal k-partite cliques in G.  ...  The task of identifying a vertex-maximum element of S is also considered and, in contrast with the k = 2 case, shown to be NP-hard for every k ≥ 3.  ...  Appendix C We provide a counterexample to the claim in  that some fact in  guarantees that at least half the clauses in any one-in-three SAT instance can be simultaneously one-in-three satisfied  ...

### Scalable algorithms for association mining

M.J. Zaki
2000 IEEE Transactions on Knowledge and Data Engineering
The items are organized into a subset lattice search space, which is decomposed into small independent chunks or sublattices, which can be solved in memory.  ...  In this paper, we present efficient algorithms for the discovery of frequent itemsets which forms the compute intensive phase of the task.  ...  For sparse graphs, of bounded arboricity i, all maximal bipartite cliques can be enumerated in time yi Q Á P Pi Á .  ...

### Parallel Batch-Dynamic Algorithms for k-Core Decomposition and Related Graph Problems [article]

Quanquan C. Liu, Jessica Shi, Shangdi Yu, Laxman Dhulipala, Julian Shun
2022 arXiv   pre-print
Given a graph with n vertices and a batch of updates ℬ, our algorithm provably maintains a (2 + ε)-approximation of the coreness values of all vertices (for any constant ε > 0) in O(|ℬ|log^2 n) amortized  ...  We demonstrate the usefulness of our low out-degree orientation algorithm by presenting a new framework to formally study batch-dynamic algorithms in bounded-arboricity graphs.  ...  #CCF-1845763, Google Faculty Research Award, Google Research Scholar Award, FinTech@CSAIL Initiative, DARPA SDH Award #HR0011-18-3-0007, and Applications Driving Architectures (ADA) Research Center, a  ...

### New Algorithms for Enumerating All Maximal Cliques [chapter]

Kazuhisa Makino, Takeaki Uno
2004 Lecture Notes in Computer Science
For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques.  ...  We propose two algorithms for enumerating all maximal cliques.  ...  Then all maximal cliques of G can be enumerated with amortized O((∆ * ) 3 (∆ * + θ) + θ 3 ) time delay and in O((n + N * )θ + m) space, where N * denotes the number of all maximal cliques in G[V * ], and  ...

### Theoretically and Practically Efficient Parallel Nucleus Decomposition [article]

Jessica Shi, Laxman Dhulipala, Julian Shun
2022 arXiv   pre-print
The key to the theoretical efficiency of our algorithm is the use of a theoretically-efficient parallel algorithms for clique listing and bucketing.  ...  On a 30-core machine with two-way hyper-threading on real-world graphs, we achieve up to a 55x speedup over the state-of-the-art parallel nucleus decomposition algorithm by Sariyuce et al., and up to a  ...  We make use of the fact that the total number of 𝑐-cliques in 𝐺 is bounded by 𝑂 (𝑚𝛼 𝑐−2 ), which follows from the 𝑐-clique enumeration algorithm  .  ...

### Recent Advances in Fully Dynamic Graph Algorithms – A Quick Reference Guide

Kathrin Hanauer, Monika Henzinger, Christian Schulz
2022 ACM Journal of Experimental Algorithmics
Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms.  ...  In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms.  ...  The author presented a space-eicient algorithm with linear space complexity. Lastly, Chehreghani et al.  ...

### Higher Lower Bounds from the 3SUM Conjecture [article]

Tsvi Kopelowitz, Seth Pettie, Ely Porat
2019 arXiv   pre-print
matrix multiplication exponent is ω=2). - Any static data structure for SetDisjointness that answers queries in constant time must spend Ω(N^2-o(1)) time in preprocessing, where N is the size of the set  ...  in a graph with arboricity/degeneracy α is essentially optimal, for any α. - Bjørklund, Pagh, Williams, and Zwick's algorithm (ICALP 2014) for listing t triangles is essentially optimal (assuming the  ...  The maximum number of triangles in a graph is O(m 3/2 ). (For example, consider a clique.)  ...

### Customizable Contraction Hierarchies [article]

Julian Dibbelt, Ben Strasser, Dorothea Wagner
2015 arXiv   pre-print
We consider the problem of quickly computing shortest paths in weighted graphs given auxiliary data derived in an expensive preprocessing phase.  ...  We provide an in-depth experimental analysis on large road and game maps that clearly shows that Customizable Contraction Hierarchies are a very practicable solution in scenarios where edge weights often  ...  Acknowledgment: We would like to thank Ignaz Rutter and Tim Zeitz for very inspiring conversations.  ...

### Parallel Batch-Dynamic Algorithms for k-Core Decomposition and Related Graph Problems

Quanquan C. Liu, Jessica Shi, Shangdi Yu, Laxman Dhulipala, Julian Shun
2022 Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures
Given a graph with 𝑛 vertices and a batch of updates B, our algorithm provably maintains a (2 + 𝜀)-approximation of the coreness values of all vertices (for any constant 𝜀 > 0) in 𝑂 (|B| log 2 𝑛)  ...  amortized work and 𝑂 (log 2 𝑛 log log 𝑛) depth (parallel time) with high probability.  ...  #CCF-1845763, Google Faculty Research Award, Google Research Scholar Award, FinTech@CSAIL Initiative, DARPA SDH Award #HR0011-18-3-0007, and Applications Driving Architectures (ADA) Research Center, a  ...
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