SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2089-2094, September 2024. Abstract. We revisit the Kahn–Kalai conjecture, recently proved in striking fashion by Park and Pham, and present a slightly reformulated simple proof which has a few advantages: (1) it works for nonuniform product measures, (2) it gives near-optimal bounds even for sampling probabilities close to 1, (3) it gives a clean bound of [math] for every [math]-bounded set system, [math].
{"title":"A Simple Proof of the Nonuniform Kahn–Kalai Conjecture","authors":"Bryan Park, Jan Vondrák","doi":"10.1137/23m1587075","DOIUrl":"https://doi.org/10.1137/23m1587075","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2089-2094, September 2024. <br/> Abstract. We revisit the Kahn–Kalai conjecture, recently proved in striking fashion by Park and Pham, and present a slightly reformulated simple proof which has a few advantages: (1) it works for nonuniform product measures, (2) it gives near-optimal bounds even for sampling probabilities close to 1, (3) it gives a clean bound of [math] for every [math]-bounded set system, [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2069-2088, September 2024. Abstract. We revisit the Stochastic Score Classification (SSC) problem introduced by Gkenosis et al. (ESA 2018): We are given [math] tests. Each test [math] can be conducted at cost [math], and it succeeds independently with probability [math]. Further, a partition of the (integer) interval [math] into [math] smaller intervals is known. The goal is to conduct tests so as to determine that interval from the partition in which the number of successful tests lies while minimizing the expected cost. Ghuge, Gupta, and Nagarajan (IPCO 2022) recently showed that a polynomial-time constant-factor approximation algorithm exists. We show that interweaving the two strategies that order tests increasingly by their [math] and [math] ratios, respectively—as already proposed by Gkensosis et al. for a special case—yields a small approximation ratio. We also show that the approximation ratio can be slightly decreased from 6 to [math] by adding in a third strategy that simply orders tests increasingly by their costs. The similar analyses for both algorithms are nontrivial but arguably clean. Finally, we complement the implied upper bound of [math] on the adaptivity gap with a lower bound of 3/2. Since the lower-bound instance is a so-called unit-cost [math]-of-[math] instance, we settle the adaptivity gap in this case.
{"title":"Simple Algorithms for Stochastic Score Classification with Small Approximation Ratios","authors":"Benedikt M. Plank, Kevin Schewior","doi":"10.1137/22m1523492","DOIUrl":"https://doi.org/10.1137/22m1523492","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2069-2088, September 2024. <br/> Abstract. We revisit the Stochastic Score Classification (SSC) problem introduced by Gkenosis et al. (ESA 2018): We are given [math] tests. Each test [math] can be conducted at cost [math], and it succeeds independently with probability [math]. Further, a partition of the (integer) interval [math] into [math] smaller intervals is known. The goal is to conduct tests so as to determine that interval from the partition in which the number of successful tests lies while minimizing the expected cost. Ghuge, Gupta, and Nagarajan (IPCO 2022) recently showed that a polynomial-time constant-factor approximation algorithm exists. We show that interweaving the two strategies that order tests increasingly by their [math] and [math] ratios, respectively—as already proposed by Gkensosis et al. for a special case—yields a small approximation ratio. We also show that the approximation ratio can be slightly decreased from 6 to [math] by adding in a third strategy that simply orders tests increasingly by their costs. The similar analyses for both algorithms are nontrivial but arguably clean. Finally, we complement the implied upper bound of [math] on the adaptivity gap with a lower bound of 3/2. Since the lower-bound instance is a so-called unit-cost [math]-of-[math] instance, we settle the adaptivity gap in this case.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2041-2068, September 2024. Abstract. This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors [math] and [math] between thin categories of relational structures are adjoint if for all structures [math] and [math], we have that [math] maps homomorphically to [math] if and only if [math] maps homomorphically to [math]. If this is the case, [math] is called the left adjoint to [math] and [math] the right adjoint to [math]. Foniok and Tardif [Discrete Math., 338 (2015), pp. 527–535] described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr [Reports of the Midwest Category Seminar IV, Lecture Notes in Math. 137, Springer, 1970, pp. 100–113]. We generalize results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction—it has been shown that such functors can be used as efficient reductions between these problems.
{"title":"Functors on Relational Structures Which Admit Both Left and Right Adjoints","authors":"Víctor Dalmau, Andrei Krokhin, Jakub Opršal","doi":"10.1137/23m1555223","DOIUrl":"https://doi.org/10.1137/23m1555223","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2041-2068, September 2024. <br/> Abstract. This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors [math] and [math] between thin categories of relational structures are adjoint if for all structures [math] and [math], we have that [math] maps homomorphically to [math] if and only if [math] maps homomorphically to [math]. If this is the case, [math] is called the left adjoint to [math] and [math] the right adjoint to [math]. Foniok and Tardif [Discrete Math., 338 (2015), pp. 527–535] described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr [Reports of the Midwest Category Seminar IV, Lecture Notes in Math. 137, Springer, 1970, pp. 100–113]. We generalize results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction—it has been shown that such functors can be used as efficient reductions between these problems.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2005-2040, September 2024. Abstract. A [math]-dimensional framework is a pair [math], where [math] is a graph and [math] maps the vertices of [math] to points in [math]. The edges of [math] are mapped to the corresponding line segments. A graph [math] is said to be globally rigid in [math] if every generic [math]-dimensional framework [math] is determined, up to congruence, by its edge lengths. A finer property is global linkedness: we say that a vertex pair [math] of [math] is globally linked in [math] in [math] if in every generic [math]-dimensional framework [math] the distance between [math] and [math] is uniquely determined by the edge lengths. In this paper we investigate globally linked pairs in graphs in [math]. We give several characterizations of those rigid graphs [math] in which a pair [math] is globally linked if and only if there exist [math] internally disjoint paths from [math] to [math] in [math]. We call these graphs [math]-joined. Among others, we show that [math] is [math]-joined if and only if for each pair of generic frameworks of [math] with the same edge lengths, one can be obtained from the other by a sequence of partial reflections along hyperplanes determined by [math]-separators of [math]. We also show that the family of [math]-joined graphs is closed under edge addition, as well as under gluing along [math] or more vertices. As a key ingredient to our main results, we prove that rigid graphs in [math] contain no crossing [math]-separators. Our results give rise to new families of graphs for which global linkedness (and global rigidity) in [math] can be tested in polynomial time.
{"title":"Partial Reflections and Globally Linked Pairs in Rigid Graphs","authors":"Dániel Garamvölgyi, Tibor Jordán","doi":"10.1137/23m157065x","DOIUrl":"https://doi.org/10.1137/23m157065x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2005-2040, September 2024. <br/> Abstract. A [math]-dimensional framework is a pair [math], where [math] is a graph and [math] maps the vertices of [math] to points in [math]. The edges of [math] are mapped to the corresponding line segments. A graph [math] is said to be globally rigid in [math] if every generic [math]-dimensional framework [math] is determined, up to congruence, by its edge lengths. A finer property is global linkedness: we say that a vertex pair [math] of [math] is globally linked in [math] in [math] if in every generic [math]-dimensional framework [math] the distance between [math] and [math] is uniquely determined by the edge lengths. In this paper we investigate globally linked pairs in graphs in [math]. We give several characterizations of those rigid graphs [math] in which a pair [math] is globally linked if and only if there exist [math] internally disjoint paths from [math] to [math] in [math]. We call these graphs [math]-joined. Among others, we show that [math] is [math]-joined if and only if for each pair of generic frameworks of [math] with the same edge lengths, one can be obtained from the other by a sequence of partial reflections along hyperplanes determined by [math]-separators of [math]. We also show that the family of [math]-joined graphs is closed under edge addition, as well as under gluing along [math] or more vertices. As a key ingredient to our main results, we prove that rigid graphs in [math] contain no crossing [math]-separators. Our results give rise to new families of graphs for which global linkedness (and global rigidity) in [math] can be tested in polynomial time.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Noga Alon, Jonathan D. Cohen, Thomas L. Griffiths, Pasin Manurangsi, Daniel Reichman, Igor Shinkar, Tal Wagner
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 2001-2003, June 2024. Abstract. We correct an error in the appendix of [N. Alon et al., SIAM J. Discrete Math., 34 (2020), pp. 885–903] and prove that it is NP-hard to approximate the size of a maximum induced matching of a bipartite graph within any constant factor.
SIAM 离散数学杂志》，第 38 卷第 2 期，第 2001-2003 页，2024 年 6 月。 摘要。我们纠正了 [N. Alon et al., SIAM J. Discrete Math., 34 (2020), pp.
{"title":"Erratum: Multitasking Capacity: Hardness Results and Improved Constructions","authors":"Noga Alon, Jonathan D. Cohen, Thomas L. Griffiths, Pasin Manurangsi, Daniel Reichman, Igor Shinkar, Tal Wagner","doi":"10.1137/23m1603856","DOIUrl":"https://doi.org/10.1137/23m1603856","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 2001-2003, June 2024. <br/> Abstract. We correct an error in the appendix of [N. Alon et al., SIAM J. Discrete Math., 34 (2020), pp. 885–903] and prove that it is NP-hard to approximate the size of a maximum induced matching of a bipartite graph within any constant factor.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"73 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, Leon Schiller
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1943-2000, June 2024. Abstract. A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach nongeometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.
{"title":"Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs","authors":"Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, Leon Schiller","doi":"10.1137/23m157394x","DOIUrl":"https://doi.org/10.1137/23m157394x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1943-2000, June 2024. <br/> Abstract. A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach nongeometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1930-1942, June 2024. Abstract. We prove that the number of tropical critical points of an affine matroid [math] is equal to the beta invariant of [math]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [math] and the inverted Bergman fan of [math], where [math] is an element of [math] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [math] on [math], this is the number of ways to find weights [math] on [math] and [math] on [math] with [math] such that, on each circuit of [math] (resp., [math]), the minimum [math]-weight (resp., [math]-weight) occurs at least twice. This answers a question of Sturmfels.
{"title":"The Tropical Critical Points of an Affine Matroid","authors":"Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao","doi":"10.1137/23m1556174","DOIUrl":"https://doi.org/10.1137/23m1556174","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1930-1942, June 2024. <br/> Abstract. We prove that the number of tropical critical points of an affine matroid [math] is equal to the beta invariant of [math]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [math] and the inverted Bergman fan of [math], where [math] is an element of [math] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [math] on [math], this is the number of ways to find weights [math] on [math] and [math] on [math] with [math] such that, on each circuit of [math] (resp., [math]), the minimum [math]-weight (resp., [math]-weight) occurs at least twice. This answers a question of Sturmfels.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"108 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1915-1929, June 2024. Abstract. It is proved that the Weisfeiler–Leman dimension of the class of permutation graphs is at most 18. Previously, it was only known that this dimension is finite (B. Grußien, Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017, pp. 1–12).
SIAM 离散数学杂志》，第 38 卷，第 2 期，第 1915-1929 页，2024 年 6 月。 摘要。证明了置换图类的 Weisfeiler-Leman 维度最多为 18。在此之前，人们只知道这个维度是有限的（B. Grußien, Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017, pp.）
{"title":"On the Weisfeiler–Leman Dimension of Permutation Graphs","authors":"Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko","doi":"10.1137/23m1575019","DOIUrl":"https://doi.org/10.1137/23m1575019","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1915-1929, June 2024. <br/> Abstract. It is proved that the Weisfeiler–Leman dimension of the class of permutation graphs is at most 18. Previously, it was only known that this dimension is finite (B. Grußien, Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017, pp. 1–12).","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1902-1914, June 2024. Abstract. We consider the symmetric difference of two graphs on the same vertex set [math], which is the graph on [math] whose edge set consists of all edges that belong to exactly one of the two graphs. Let [math] be a class of graphs, and let [math] denote the maximum possible cardinality of a family [math] of graphs on [math] such that the symmetric difference of any two members in [math] belongs to [math]. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379–403] with the aim of providing a new graphic approach to coding theory. In particular, [math] denotes the maximum possible size of this code. Existing results show that as the graph class [math] changes, [math] can vary from [math] to [math]. We study several phase transition problems related to [math] in general settings and present a partial solution to a recent problem posed by Alon et al.
{"title":"Phase Transitions of Structured Codes of Graphs","authors":"Bo Bai, Yu Gao, Jie Ma, Yuze Wu","doi":"10.1137/23m1614572","DOIUrl":"https://doi.org/10.1137/23m1614572","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1902-1914, June 2024. <br/> Abstract. We consider the symmetric difference of two graphs on the same vertex set [math], which is the graph on [math] whose edge set consists of all edges that belong to exactly one of the two graphs. Let [math] be a class of graphs, and let [math] denote the maximum possible cardinality of a family [math] of graphs on [math] such that the symmetric difference of any two members in [math] belongs to [math]. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379–403] with the aim of providing a new graphic approach to coding theory. In particular, [math] denotes the maximum possible size of this code. Existing results show that as the graph class [math] changes, [math] can vary from [math] to [math]. We study several phase transition problems related to [math] in general settings and present a partial solution to a recent problem posed by Alon et al.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].
{"title":"On Graphs Coverable by [math] Shortest Paths","authors":"Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca","doi":"10.1137/23m1564511","DOIUrl":"https://doi.org/10.1137/23m1564511","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. <br/> Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号