Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
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{"title"=>"Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm", "type"=>"journal", "authors"=>[{"first_name"=>"José Antonio", "last_name"=>"Martín H.", "scopus_author_id"=>"55665862600"}], "year"=>2013, "source"=>"PLoS ONE", "identifiers"=>{"scopus"=>"2-s2.0-84872341102", "pmid"=>"23349711", "issn"=>"19326203", "sgr"=>"84872341102", "doi"=>"10.1371/journal.pone.0053437", "pui"=>"368116093"}, "id"=>"5590b107-e5c6-3f86-a778-a6b3fdbdb2c1", "abstract"=>"Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways. In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present if and only if it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to \"efficiently\" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter α∈N. Nevertheless, here it is proved that the probability of requiring a value of α>k to obtain a solution for a random graph decreases exponentially: P(α>k)≤2(-(k+1)), making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.", "link"=>"http://www.mendeley.com/research/solving-hard-computational-problems-efficiently-asymptotic-parametric-complexity-3coloring-algorithm", "reader_count"=>6, "reader_count_by_academic_status"=>{"Professor > Associate Professor"=>1, "Researcher"=>2, "Student > Ph. D. Student"=>1, "Student > Master"=>1, "Student > Bachelor"=>1}, "reader_count_by_user_role"=>{"Professor > Associate Professor"=>1, "Researcher"=>2, "Student > Ph. D. Student"=>1, "Student > Master"=>1, "Student > Bachelor"=>1}, "reader_count_by_subject_area"=>{"Agricultural and Biological Sciences"=>2, "Business, Management and Accounting"=>1, "Physics and Astronomy"=>1, "Computer Science"=>2}, "reader_count_by_subdiscipline"=>{"Physics and Astronomy"=>{"Physics and Astronomy"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>2}, "Computer Science"=>{"Computer Science"=>2}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}}, "reader_count_by_country"=>{"Germany"=>3}, "group_count"=>1}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/508502"], "description"=>"<p>Runtime analysis of the backtracking (brute force plus heuristic) () vs. the proposed parametric algorithm () over random planar graphs between 10 and 100 vertices. Plot (a) shows the running times as a function of the number of vertices for both kinds of instance types and for both algorithms. Plot (b) shows the proportion of 3-colorable and non-3-colorable graphs over the total number of graphs per number of vertices. Plots (c) and (d) also show the running times as a function of the number of vertices discriminated by the instance types (yes-or-no) so that subtle differences can be observed. Plots (e) and (f) also show the running times but as a function of the average degree and instance type.</p>", "links"=>[], "tags"=>["biotechnology", "Computational biology", "computer science", "mathematics"], "article_id"=>178996, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g002", "stats"=>{"downloads"=>0, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Backtracking_vs_proposed_algorithm_/178996", "title"=>"Backtracking vs. proposed algorithm.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:09:27"}
  • {"files"=>["https://ndownloader.figshare.com/files/509018"], "description"=>"<p>Graphs are sampled from the well-known Erdős-Rënyi random graphs distribution. The sample consist of graph instances for 100 vertices, generating in total 10000 graphs. For each number of vertices, the average degree is varied from 3 to 6.</p>", "links"=>[], "tags"=>["graphs"], "article_id"=>179526, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.t005", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_sample_used_in_the_random_graphs_test_/179526", "title"=>"Parameters of the sample used in the random graphs test.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-19 17:12:23"}
  • {"files"=>["https://ndownloader.figshare.com/files/509056"], "description"=>"<p>Random planar graph instances from 10 to 100 vertices incremented by 1 and generating 100 graphs for each number of vertices, i.e., 9000 graphs in total.</p>", "links"=>[], "tags"=>["scalability", "backtracking"], "article_id"=>179555, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.t002", "stats"=>{"downloads"=>5, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_scalability_test_between_backtracking_and_the_proposed_algorithm_/179555", "title"=>"Parameters of the scalability test between backtracking and the proposed algorithm.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-19 17:12:30"}
  • {"files"=>["https://ndownloader.figshare.com/files/508429"], "description"=>"<p>(A) Complete graph . (B) Complete 3-partite graph . (C) Tadpole graph , which imposes a binary constraint on 3-coloring: either or . (D) Path of length two . (E) graph that imposes a binary constraint on 3-coloring: either or .</p>", "links"=>[], "tags"=>["named", "graphs"], "article_id"=>178932, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g001", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Specially_named_graphs_used_in_the_article_/178932", "title"=>"Specially named graphs used in the article.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:09:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/508612"], "description"=>"<p>Runtime analysis over random planar graphs between 100 and 1000 vertices (incremented by 100 and generating 1000 graphs for each number). Plot (a) shows the running times as a function of the number of vertices for both kinds of instance types. Plot (b) shows the proportion of 3-colorable and non-3-colorable graphs over the total number of graphs per number of vertices.</p>", "links"=>[], "tags"=>["planar"], "article_id"=>179118, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g003", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_for_planar_graphs_/179118", "title"=>"Results for planar graphs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:10:05"}
  • {"files"=>["https://ndownloader.figshare.com/files/509075"], "description"=>"<p>Uniformly random planar graph instances from 100 to 1000 vertices incremented by 100 and generating 1000 graphs for each number of vertices, i.e., 10000 graphs in total.</p>", "links"=>[], "tags"=>["planar", "graphs"], "article_id"=>179581, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.t003", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_sample_used_in_the_random_planar_graphs_test_/179581", "title"=>"Parameters of the sample used in the random planar graphs test.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-19 17:12:38"}
  • {"files"=>["https://ndownloader.figshare.com/files/508711"], "description"=>"<p>Runtime analysis over random planar graphs considering instance type and average degree. Plots (a) and (b) also show the running times as a function of the number of vertices but discriminated by the average degree.</p>", "links"=>[], "tags"=>["planar"], "article_id"=>179214, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g004", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_for_planar_graphs_/179214", "title"=>"Results for planar graphs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:10:35"}
  • {"files"=>["https://ndownloader.figshare.com/files/508999"], "description"=>"<p>Graphs are sampled from an ad-hoc distribution over the 4-regular planar graphs. The sample consist of graph instances from 100 to 1000 vertices incremented by 100 and generating 1000 graphs for each number of vertices, i.e., 10000 graphs in total. For the exact meaning of each graph transformation operation, i.e., see Ref. <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0053437#pone.0053437-Broersma1\" target=\"_blank\">[45]</a>.</p>", "links"=>[], "tags"=>["planar", "4-regular", "graphs"], "article_id"=>179499, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.t004", "stats"=>{"downloads"=>4, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Parameters_of_the_sample_used_in_the_random_planar_4_regular_graphs_test_/179499", "title"=>"Parameters of the sample used in the random planar 4-regular graphs test.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-19 17:12:14"}
  • {"files"=>["https://ndownloader.figshare.com/files/508879"], "description"=>"<p>The behavior of the proposed algorithm over the well-known Erdős-Rënyi random graphs distribution. Plot (a) shows the quantity of 3-colorable and non-3-colorable graphs as a function of the average degree, i.e., a phase transition plot, in this case, occurring at around . Plot (b) shows the quantity of graphs corresponding to each value. As predicted by the theory, the proportion of graphs decreases exponentially as a function of below the line of . Plots (c) and (d) show the running times as a function of the average degree.</p>", "links"=>[], "tags"=>["algorithm"], "article_id"=>179385, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g006", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Runtime_analysis_of_the_algorithm_for_random_graphs_/179385", "title"=>"Runtime analysis of the algorithm for random graphs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:11:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/508795"], "description"=>"<p>Runtime analysis over random 4-regular planar graphs between 100 and 1000 vertices. Plot (a) shows the running times as a function of the number of vertices for both kinds of instance types. Plot (b) shows the proportion of 3-colorable and non-3-colorable graphs over the total number of graphs per number of vertices.</p>", "links"=>[], "tags"=>["4-regular", "planar"], "article_id"=>179301, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.g005", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Results_for_4_regular_planar_graphs_/179301", "title"=>"Results for 4-regular planar graphs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-19 17:11:03"}
  • {"files"=>["https://ndownloader.figshare.com/files/509106"], "description"=>"<p>Average case (expected) performance of the algorithm with respect to the density of the graph, i.e., above/below the phase transition threshold () and the type of instance (Yes/No).</p>", "links"=>[], "tags"=>["priori", "algorithm", "3-colorability", "graph"], "article_id"=>179606, "categories"=>["Biotechnology", "Mathematics", "Biological Sciences", "Information And Computing Sciences"], "users"=>["José Antonio Martín H."], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0053437.t001", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_A_priori_expected_performance_of_the_algorithm_with_respect_to_3_colorability_and_graph_density_parameters_/179606", "title"=>"A priori expected performance of the algorithm with respect to 3-colorability and graph density parameters.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2013-02-19 17:12:46"}

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Relative Metric

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