Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

AbstractThe computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

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Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽

10.1007/s00453-020-00729-z ◽

2020 ◽

Vol 82 (11) ◽

pp. 3338-3389

Author(s):

Ankit Chauhan ◽

Tobias Friedrich ◽

Ralf Rothenberger

Keyword(s):

Random Graph ◽

Random Graphs ◽

Power Law ◽

Real World ◽

Degree Distribution ◽

High Probability ◽

Maximum Independent Set ◽

Graph Models ◽

Random Graph Models ◽

Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

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Relating Vertex and Global Graph Entropy in Randomly Generated Graphs

10.20944/preprints201805.0302.v1 ◽

2018 ◽

Author(s):

Philip Tee ◽

George Parisis ◽

Luc Berthouze ◽

Ian Wakeman

Keyword(s):

Computational Complexity ◽

Recent Work ◽

Random Graphs ◽

Real World ◽

Close Correlation ◽

Complete Problem ◽

Entropy Measures ◽

Vertex Set ◽

Np Complete ◽

The Greedy Algorithm

Combinatoric measures of entropy capture the complexity of a graph, but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of alternative vertex level measures of entropy that do not suffer from this pathological computational complexity. It can be demonstrated that they are still effective at quantifying graph complexity. It is intriguing to consider whether there is a fundamental link between local and global entropy measures. In this paper, we investigate the existence of correlation between vertex level and global measures of entropy, for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate close correlation for this subset of graphs and outline how this may arise theoretically.

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On paths, trails and closed trails in edge-colored graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Laurent Gourvès ◽

Adria Lyra ◽

Carlos A. Martinhon ◽

Jérôme Monnot

Keyword(s):

Polynomial Time ◽

Optimal Solution ◽

Colored Graph ◽

Edge Disjoint ◽

Colored Graphs ◽

International Audience ◽

Np Complete ◽

T Path ◽

Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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The Complexity of Checking Consistency of Pedigree Information and Related Problems

BRICS Report Series ◽

10.7146/brics.v10i17.21787 ◽

2003 ◽

Vol 10 (17) ◽

Cited By ~ 1

Author(s):

Luca Aceto ◽

Jens Alsted Hansen ◽

Anna Ingólfsdóttir ◽

Jacob Johnsen ◽

John Knudsen

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Input Data ◽

Single Gene ◽

Pedigree Information ◽

Consistency Checking ◽

Computational Problem ◽

Np Complete ◽

Mendelian Laws

Consistency checking is a fundamental computational problem in genetics. Given a pedigree and information on the genotypes (of some) of the individuals in it, the aim of consistency checking is to determine whether these data are consistent with the classic Mendelian laws of inheritance. This problem arose originally from the geneticists' need to filter their input data from erroneous information, and is well motivated from both a biological and a sociological viewpoint. This paper shows that consistency checking is NP-complete, even with focus on a single gene and in the presence of three alleles. Several other results on the computational complexity of problems from genetics that are related to consistency checking are also offered. In particular, it is shown that checking the consistency of pedigrees over two alleles, and of pedigrees without loops, can be done in polynomial time.

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Rigorous Estimation of Computational Complexity for OMV SAT Algorithm

Open Systems & Information Dynamics ◽

10.1142/s1230161208000158 ◽

2008 ◽

Vol 15 (02) ◽

pp. 173-187 ◽

Cited By ~ 8

Author(s):

Satoshi Iriyama ◽

Masanori Ohya

Keyword(s):

Computational Complexity ◽

Boolean Function ◽

Polynomial Time ◽

Quantum Algorithm ◽

Sat Problem ◽

Amplification Process ◽

Np Complete

For SAT problem, which is known to be NP-complete, Ohya and Masuda found a quantum algorithm calculating a given boolean function for all truth assignments. They showed that we can decide whether this boolean function is satisfiable or not in polynomial time if a certain superposed state can be detected physically, which turns out to be related to an amplification process. Then Ohya and Volovich realized this amplification by means of chaos dynamics [4, 5]. In this paper, we study the complexity of the SAT algorithm by rigorously counting the steps of OMV algorithm discussed previously in [1, 2, 4, 5].

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COMPUTATIONAL COMPLEXITY OF THE PERFECT MATCHING PROBLEM IN HYPERGRAPHS WITH SUBCRITICAL DENSITY

International Journal of Foundations of Computer Science ◽

10.1142/s0129054110007635 ◽

2010 ◽

Vol 21 (06) ◽

pp. 905-924 ◽

Cited By ~ 11

Author(s):

MAREK KARPIŃSKI ◽

ANDRZEJ RUCIŃSKI ◽

EDYTA SZYMAŃSKA

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Perfect Matching ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Existence Problem ◽

Matching Problem ◽

Uniform Hypergraphs ◽

The Status ◽

Np Complete

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for [Formula: see text] and trivial for c ≥ ½, leaving the status of the problem with c in the interval [Formula: see text] widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

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The Computational Complexity of Some Problems of Linear Algebra

BRICS Report Series ◽

10.7146/brics.v3i33.20013 ◽

1996 ◽

Vol 3 (33) ◽

Cited By ~ 1

Author(s):

Jonathan F. Buss ◽

Gudmund Skovbjerg Frandsen ◽

Jeffery O. Shallit

Keyword(s):

Computational Complexity ◽

Commutative Ring ◽

Linear Algebra ◽

Polynomial Time ◽

Matrix Rank ◽

Maximum Rank ◽

The Matrix ◽

Np Complete

We consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt) with entries chosen from E union {x1, x2, ..., xt}, we want to determine maxrankS(M) = max rank M(a1, a2, ... , at) and minrankS(M) = min rank M(a1, a2, ..., at). There are also variants of these problems that specify more about the structure of M, or instead of asking for the minimum or maximum rank, ask if there is some substitution of the variables that makes the matrix invertible or noninvertible. Depending on E, S, and on which variant is studied, the complexity of these problems can range from polynomial-time solvable to random polynomial-time solvable to NP-complete to PSPACE-solvable to unsolvable.

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Analysis of E-Commerce Product Graphs

10.36227/techrxiv.12814244.v1 ◽

2020 ◽

Author(s):

Shalin Shah

Keyword(s):

Random Graph ◽

Random Graphs ◽

Power Law ◽

Real World ◽

Degree Distribution ◽

Graph Model ◽

Graph Analysis ◽

Random Graph Model ◽

Product Graphs ◽

Clustering Coefficients

Consumer behavior in retail stores gives rise to product graphs based on copurchasingor co-viewing behavior. These product graphs can be analyzed usingthe known methods of graph analysis. In this paper, we analyze the product graphat Target Corporation based on the Erd˝os-Renyi random graph model. In particular,we compute clustering coefficients of actual and random graphs, and we find thatthe clustering coefficients of actual graphs are much higher than random graphs.We conduct the analysis on the entire set of products and also on a per categorybasis and find interesting results. We also compute the degree distribution andwe find that the degree distribution is a power law as expected from real worldnetworks, contrasting with the ER random graph.

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Checking Consistency of Pedigree Information is NP-complete (Preliminary Report)

BRICS Report Series ◽

10.7146/brics.v9i42.21757 ◽

2002 ◽

Vol 9 (42) ◽

Cited By ~ 1

Author(s):

Luca Aceto ◽

Jens Alsted Hansen ◽

Anna Ingólfsdóttir ◽

Jacob Johnsen ◽

John Knudsen

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Preliminary Report ◽

Input Data ◽

Pedigree Information ◽

Consistency Checking ◽

Computational Problem ◽

Np Complete ◽

Mendelian Laws

Consistency checking is a fundamental computational problem in genetics. Given a pedigree and information on the genotypes of some of the individuals in it, the aim of consistency checking is to determine whether these data are consistent with the classic Mendelian laws of inheritance. This problem arose originally from the geneticists' need to filter their input data from erroneous information, and is well motivated from both a biological and a sociological viewpoint. This paper shows that consistency checking is NP-complete, even in the presence of three alleles. Several other results on the computational complexity of problems from genetics that are related to consistency checking are also offered. In particular, it is shown that checking the consistency of pedigrees over two alleles can be done in polynomial time.

Download Full-text