Pub Date : 2022-12-07DOI: 10.1007/s00224-022-10103-x
Ararat Harutyunyan, Michael Lampis, Nikolaos Melissinos
<p>In <i>k</i>-<span>Digraph Coloring</span> we are given a digraph and are asked to partition its vertices into at most <i>k</i> sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) <i>k</i>-<span>Coloring</span>, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is “almost” acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense. In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set. It is already known that, for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is NP-hard on digraphs of directed feedback vertex set of size at most <i>k</i> + 4. We strengthen this result to show that, for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is already NP-hard for directed feedback vertex set of size exactly <i>k</i>. This immediately provides a dichotomy, as <i>k</i>-<span>Digraph Coloring</span> is trivial if directed feedback vertex set has size at most <i>k</i> − 1. Refining our reduction we obtain three further consequences: (i) 2-<span>Digraph Coloring</span> is NP-hard for oriented graphs of directed feedback vertex set at most 3; (ii) for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is NP-hard for graphs of feedback <i>arc</i> set of size at most <i>k</i><sup>2</sup>; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by <i>k</i> if feedback arc set has size at most <i>k</i><sup>2</sup> − 1; (iii) <i>k</i>-<span>Digraph Coloring</span> is NP-hard for graphs of directed feedback vertex <i>k</i>, even if the maximum degree <i>Δ</i> is at most 4<i>k</i> − 1; we show that this is also <i>almost</i> tight, as the problem becomes FPT for digraphs of directed feedback vertex set of size <i>k</i> and <i>Δ</i> ≤ 4<i>k</i> − 3. Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that <i>k</i>-<span>Digraph Coloring</span> admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)<i>k</i><sup>tw</sup>. Since this is considerably worse than the <i>k</i><sup>tw</sup> dependence of (undirected) <i>k</i>-<span>Coloring</span>, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for <i>k</i> = 2. Specifically, we show that an FPT algorithm solving 2-<span>Digraph Coloring</span> with dependence td<sup><i>o</i>(td)</sup> would contradict the ETH. Then, we consider the class of tournaments. It is known that
{"title":"Digraph Coloring and Distance to Acyclicity","authors":"Ararat Harutyunyan, Michael Lampis, Nikolaos Melissinos","doi":"10.1007/s00224-022-10103-x","DOIUrl":"https://doi.org/10.1007/s00224-022-10103-x","url":null,"abstract":"<p>In <i>k</i>-<span>Digraph Coloring</span> we are given a digraph and are asked to partition its vertices into at most <i>k</i> sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) <i>k</i>-<span>Coloring</span>, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is “almost” acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense. In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set. It is already known that, for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is NP-hard on digraphs of directed feedback vertex set of size at most <i>k</i> + 4. We strengthen this result to show that, for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is already NP-hard for directed feedback vertex set of size exactly <i>k</i>. This immediately provides a dichotomy, as <i>k</i>-<span>Digraph Coloring</span> is trivial if directed feedback vertex set has size at most <i>k</i> − 1. Refining our reduction we obtain three further consequences: (i) 2-<span>Digraph Coloring</span> is NP-hard for oriented graphs of directed feedback vertex set at most 3; (ii) for all <i>k</i> ≥ 2, <i>k</i>-<span>Digraph Coloring</span> is NP-hard for graphs of feedback <i>arc</i> set of size at most <i>k</i><sup>2</sup>; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by <i>k</i> if feedback arc set has size at most <i>k</i><sup>2</sup> − 1; (iii) <i>k</i>-<span>Digraph Coloring</span> is NP-hard for graphs of directed feedback vertex <i>k</i>, even if the maximum degree <i>Δ</i> is at most 4<i>k</i> − 1; we show that this is also <i>almost</i> tight, as the problem becomes FPT for digraphs of directed feedback vertex set of size <i>k</i> and <i>Δ</i> ≤ 4<i>k</i> − 3. Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that <i>k</i>-<span>Digraph Coloring</span> admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)<i>k</i><sup>tw</sup>. Since this is considerably worse than the <i>k</i><sup>tw</sup> dependence of (undirected) <i>k</i>-<span>Coloring</span>, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for <i>k</i> = 2. Specifically, we show that an FPT algorithm solving 2-<span>Digraph Coloring</span> with dependence td<sup><i>o</i>(td)</sup> would contradict the ETH. Then, we consider the class of tournaments. It is known that","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"47 18","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.1007/s00224-022-10108-6
I. Olkhovsky, A. Okhotin
{"title":"On the Transformation of LL(k)-linear to LL(1)-linear Grammars","authors":"I. Olkhovsky, A. Okhotin","doi":"10.1007/s00224-022-10108-6","DOIUrl":"https://doi.org/10.1007/s00224-022-10108-6","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"234-262"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43374939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-29DOI: 10.1007/s00224-022-10097-6
Yefim Dinitz, S. Dolev, Manish Kumar
{"title":"Local Deal-Agreement Algorithms for Load Balancing in Dynamic General Graphs","authors":"Yefim Dinitz, S. Dolev, Manish Kumar","doi":"10.1007/s00224-022-10097-6","DOIUrl":"https://doi.org/10.1007/s00224-022-10097-6","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"348-382"},"PeriodicalIF":0.5,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41694025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-28DOI: 10.1007/s00224-022-10099-4
Yaokun Wu, Yinfeng Zhu
{"title":"Correction to: Submodular Functions and Rooted Trees","authors":"Yaokun Wu, Yinfeng Zhu","doi":"10.1007/s00224-022-10099-4","DOIUrl":"https://doi.org/10.1007/s00224-022-10099-4","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"290"},"PeriodicalIF":0.5,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46494445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-06DOI: 10.1007/s00224-022-10102-y
Yusong Du, Xiao Ma
{"title":"On the Rejection Rate of Exact Sampling Algorithm for Discrete Gaussian Distributions over the Integers","authors":"Yusong Du, Xiao Ma","doi":"10.1007/s00224-022-10102-y","DOIUrl":"https://doi.org/10.1007/s00224-022-10102-y","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"66 1","pages":"1099 - 1122"},"PeriodicalIF":0.5,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41469554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-22DOI: 10.1007/s00224-022-10101-z
Nils Morawietz, Niels Grüttemeier, Christian Komusiewicz, Frank Sommer
{"title":"Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs","authors":"Nils Morawietz, Niels Grüttemeier, Christian Komusiewicz, Frank Sommer","doi":"10.1007/s00224-022-10101-z","DOIUrl":"https://doi.org/10.1007/s00224-022-10101-z","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"66 1","pages":"1019 - 1045"},"PeriodicalIF":0.5,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45610239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-12DOI: 10.1007/s00224-022-10095-8
Chryssis Georgiou, M. Mavronicolas, B. Monien
{"title":"(In)Existence of Equilibria for 2-Player, 2-Value Games with Semistrictly Quasiconcave Cost Functions","authors":"Chryssis Georgiou, M. Mavronicolas, B. Monien","doi":"10.1007/s00224-022-10095-8","DOIUrl":"https://doi.org/10.1007/s00224-022-10095-8","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"66 1","pages":"957 - 995"},"PeriodicalIF":0.5,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47124002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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