scholarly journals Finding hidden cliques in linear time

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Uriel Feige ◽  
Dorit Ron
Keyword(s):  
High Probability ◽  
Linear Time ◽  
Maximum Clique ◽  
Spectral Approach ◽  
Input Graph ◽  
Random Choice ◽  
Spectral Techniques ◽  
Vertex Graph ◽  
International Audience ◽  
Linear Time Algorithms

International audience In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.

scholarly journals Finding Hidden Cliques in Linear Time with High Probability

2013 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
YAEL DEKEL ◽  
ORI GUREL-GUREVICH ◽  
YUVAL PERES
Keyword(s):  
Random Graph ◽  
Failure Probability ◽  
High Probability ◽  
Linear Time ◽  
Success Probability ◽  
Graph Model ◽  
General Setting ◽  
Random Graph Model ◽  
Spectral Techniques ◽  
Random Subset

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

scholarly journals Spanning connectedness and Hamiltonian thickness of graphs and interval graphs

2015 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Interval Graphs ◽  
Spanning Subgraph ◽  
Linear Forest ◽  
Connectedness Property ◽  
Vertex Ordering ◽  
International Audience ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

International audience A spanning connectedness property is one which involves the robust existence of a spanning subgraph which is of some special form, say a Hamiltonian cycle in which a sequence of vertices appear in an arbitrarily given ordering, or a Hamiltonian path in the subgraph obtained by deleting any three vertices, or three internally-vertex-disjoint paths with any given endpoints such that the three paths meet every vertex of the graph and cover the edges of an almost arbitrarily given linear forest of a certain fixed size. Let π = π1 · · · πn be an ordering of the vertices of an n-vertex graph G. For any positive integer k ≤ n − 1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i ∈ {1,. .. , n − 1} that πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} is at least min{n − i, k}; For any nonnegative integer k, we say that π is a −k-thick Hamiltonian vertex ordering of G provided |{i : πiπi+1 / ∈ E(G), 1 ≤ i ≤ n − 1}| ≤ k + 1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.

scholarly journals A four-sweep LBFS recognition algorithm for interval graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Structure Theory ◽  
Time Interval ◽  
Breadth First Search ◽  
Input Graph ◽  
Graph Recognition ◽  
Unit Interval Graph ◽  
International Audience

Graph Theory International audience In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

scholarly journals Almost Linear Time Algorithms for Minsum k-Sink Problems on Dynamic Flow Path Networks

2021 ◽  
Author(s):  
Yuya Higashikawa ◽  
Naoki Katoh ◽  
Junichi Teruyama ◽  
Koji Watase
Keyword(s):  
Linear Time ◽  
Flow Path ◽  
Dynamic Flow ◽  
Linear Time Algorithms

AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

2000 ◽  
Vol 11 (03) ◽  
pp. 365-371 ◽  
Author(s):  
LJUBOMIR PERKOVIĆ ◽  
BRUCE REED
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Tree Decomposition ◽  
Linear Time Algorithm ◽  
Input Graph ◽  
Primary Motivation ◽  
Small Width ◽  
Tree Decompositions ◽  
Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

scholarly journals Linear-Time Algorithms for Tree Root Problems

Algorithmica ◽  
2013 ◽  
Vol 71 (2) ◽  
pp. 471-495 ◽  
Author(s):  
Maw-Shang Chang ◽  
Ming-Tat Ko ◽  
Hsueh-I Lu
Keyword(s):  
Linear Time ◽  
Linear Time Algorithms

FAULT TOLERANT ROUTING IN HYPERCUBES AND STAR GRAPHS

1996 ◽  
Vol 06 (01) ◽  
pp. 127-136 ◽  
Author(s):  
QIAN-PING GU ◽  
SHIETUNG PENG
Keyword(s):  
Free Path ◽  
Fault Tolerant ◽  
Linear Time ◽  
Time Efficiency ◽  
Routing Problem ◽  
Star Graphs ◽  
Linear Time Algorithms ◽  
Better Than

In this paper, we give two linear time algorithms for node-to-node fault tolerant routing problem in n-dimensional hypercubes Hn and star graphs Gn. The first algorithm, given at most n−1 arbitrary fault nodes and two non-fault nodes s and t in Hn, finds a fault-free path s→t of length at most [Formula: see text] in O(n) time, where d(s, t) is the distance between s and t. Our second algorithm, given at most n−2 fault nodes and two non-fault nodes s and t in Gn, finds a fault-free path s→t of length at most d(Gn)+3 in O(n) time, where [Formula: see text] is the diameter of Gn. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

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