16,295 research outputs found
CLIPPER+: A Fast Maximal Clique Algorithm for Robust Global Registration
We present CLIPPER+, an algorithm for finding maximal cliques in unweighted
graphs for outlier-robust global registration. The registration problem can be
formulated as a graph and solved by finding its maximum clique. This
formulation leads to extreme robustness to outliers; however, finding the
maximum clique is an NP-hard problem, and therefore approximation is required
in practice for large-size problems. The performance of an approximation
algorithm is evaluated by its computational complexity (the lower the runtime,
the better) and solution accuracy (how close the solution is to the maximum
clique). Accordingly, the main contribution of CLIPPER+ is outperforming the
state-of-the-art in accuracy while maintaining a relatively low runtime.
CLIPPER+ builds on prior work (CLIPPER [1] and PMC [2]) and prunes the graph by
removing vertices that have a small core number and cannot be a part of the
maximum clique. This will result in a smaller graph, on which the maximum
clique can be estimated considerably faster. We evaluate the performance of
CLIPPER+ on standard graph benchmarks, as well as synthetic and real-world
point cloud registration problems. These evaluations demonstrate that CLIPPER+
has the highest accuracy and can register point clouds in scenarios where over
of associations are outliers. Our code and evaluation benchmarks are
released at https://github.com/ariarobotics/clipperp
On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs
Clique-Helly and hereditary clique-Helly graphs are polynomial-time recognizable. Recently, we presented a proof that the clique graph recognition problem is NP-complete [L. Alcón, L. Faria, C.M.H. de Figueiredo, M. Gutierrez, Clique graph recognition is NP-complete, in: Proc. WG 2006, in: Lecture Notes in Comput. Sci., vol. 4271, Springer, 2006, pp. 269-277]. In this work, we consider the decision problems: given a graph G = (V, E) and an integer k ≥ 0, we ask whether there exists a subset V ′ ⊆ V with | V ′ | ≥ k such that the induced subgraph G [V ′ ] of G is, variously, a clique, clique-Helly or hereditary clique-Helly graph. The first problem is clearly NP-complete, from the above reference; we prove that the other two decision problems mentioned are NP-complete, even for maximum degree 6 planar graphs. We consider the corresponding maximization problems of finding a maximum induced subgraph that is, respectively, clique, clique-Helly or hereditary clique-Helly. We show that these problems are Max SNP-hard, even for maximum degree 6 graphs. We show a general polynomial-time frac(1, Δ + 1)-approximation algorithm for these problems when restricted to graphs with fixed maximum degree Δ. We generalize these results to other graph classes. We exhibit a polynomial 6-approximation algorithm to minimize the number of vertices to be removed in order to obtain a hereditary clique-Helly subgraph.Facultad de Ciencias Exacta
On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs
Clique-Helly and hereditary clique-Helly graphs are polynomial-time recognizable. Recently, we presented a proof that the clique graph recognition problem is NP-complete [L. Alcón, L. Faria, C.M.H. de Figueiredo, M. Gutierrez, Clique graph recognition is NP-complete, in: Proc. WG 2006, in: Lecture Notes in Comput. Sci., vol. 4271, Springer, 2006, pp. 269-277]. In this work, we consider the decision problems: given a graph G = (V, E) and an integer k ≥ 0, we ask whether there exists a subset V ′ ⊆ V with | V ′ | ≥ k such that the induced subgraph G [V ′ ] of G is, variously, a clique, clique-Helly or hereditary clique-Helly graph. The first problem is clearly NP-complete, from the above reference; we prove that the other two decision problems mentioned are NP-complete, even for maximum degree 6 planar graphs. We consider the corresponding maximization problems of finding a maximum induced subgraph that is, respectively, clique, clique-Helly or hereditary clique-Helly. We show that these problems are Max SNP-hard, even for maximum degree 6 graphs. We show a general polynomial-time frac(1, Δ + 1)-approximation algorithm for these problems when restricted to graphs with fixed maximum degree Δ. We generalize these results to other graph classes. We exhibit a polynomial 6-approximation algorithm to minimize the number of vertices to be removed in order to obtain a hereditary clique-Helly subgraph.Facultad de Ciencias Exacta
Solving Maximum Clique Problem for Protein Structure Similarity
A basic assumption of molecular biology is that proteins sharing close
three-dimensional (3D) structures are likely to share a common function and in
most cases derive from a same ancestor. Computing the similarity between two
protein structures is therefore a crucial task and has been extensively
investigated. Evaluating the similarity of two proteins can be done by finding
an optimal one-to-one matching between their components, which is equivalent to
identifying a maximum weighted clique in a specific "alignment graph". In this
paper we present a new integer programming formulation for solving such clique
problems. The model has been implemented using the ILOG CPLEX Callable Library.
In addition, we designed a dedicated branch and bound algorithm for solving the
maximum cardinality clique problem. Both approaches have been integrated in
VAST (Vector Alignment Search Tool) - a software for aligning protein 3D
structures largely used in NCBI (National Center for Biotechnology
Information). The original VAST clique solver uses the well known Bron and
Kerbosh algorithm (BK). Our computational results on real life protein
alignment instances show that our branch and bound algorithm is up to 116 times
faster than BK for the largest proteins
A structure theorem for graphs with no cycle with a unique chord and its consequences
We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.Cycle with a unique chord, decomposition, structure, detection, recognition, Heawood graph, Petersen graph, coloring.
Maximum Cliques in Graphs with Small Intersection Number and Random Intersection Graphs
In this paper, we relate the problem of finding a maximum clique to the
intersection number of the input graph (i.e. the minimum number of cliques
needed to edge cover the graph). In particular, we consider the maximum clique
problem for graphs with small intersection number and random intersection
graphs (a model in which each one of labels is chosen independently with
probability by each one of vertices, and there are edges between any
vertices with overlaps in the labels chosen).
We first present a simple algorithm which, on input finds a maximum
clique in time steps, where is an
upper bound on the intersection number and is the number of vertices.
Consequently, when the running time of this algorithm is
polynomial.
We then consider random instances of the random intersection graphs model as
input graphs. As our main contribution, we prove that, when the number of
labels is not too large (), we can use the label
choices of the vertices to find a maximum clique in polynomial time whp. The
proof of correctness for this algorithm relies on our Single Label Clique
Theorem, which roughly states that whp a "large enough" clique cannot be formed
by more than one label. This theorem generalizes and strengthens other related
results in the state of the art, but also broadens the range of values
considered.
As an important consequence of our Single Label Clique Theorem, we prove that
the problem of inferring the complete information of label choices for each
vertex from the resulting random intersection graph (i.e. the \emph{label
representation of the graph}) is \emph{solvable} whp. Finding efficient
algorithms for constructing such a label representation is left as an
interesting open problem for future research
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