SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024. Abstract. We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let [math] be the largest eigenvalue of the adjacency matrix of a graph [math] and [math] be the complement of [math]. A nice conjecture states that the graph on [math] vertices maximizing [math] is the join of a clique and an independent set with [math] and [math] (also [math] and [math] if [math]) vertices, respectively. We resolve this conjecture for sufficiently large [math] using analytic methods. Our second result concerns the [math]-spread of a graph [math], which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of [math]. It was conjectured by Cvetković, Rowlinson, and Simić [Publ. Inst. Math., 81 (2007), pp. 11–27] that the unique [math]-vertex connected graph of maximum [math]-spread is the graph formed by adding a pendant edge to [math]. We confirm this conjecture for sufficiently large [math].
{"title":"Graph Limits and Spectral Extremal Problems for Graphs","authors":"Lele Liu","doi":"10.1137/22m1508807","DOIUrl":"https://doi.org/10.1137/22m1508807","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024. <br/> Abstract. We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let [math] be the largest eigenvalue of the adjacency matrix of a graph [math] and [math] be the complement of [math]. A nice conjecture states that the graph on [math] vertices maximizing [math] is the join of a clique and an independent set with [math] and [math] (also [math] and [math] if [math]) vertices, respectively. We resolve this conjecture for sufficiently large [math] using analytic methods. Our second result concerns the [math]-spread of a graph [math], which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of [math]. It was conjectured by Cvetković, Rowlinson, and Simić [Publ. Inst. Math., 81 (2007), pp. 11–27] that the unique [math]-vertex connected graph of maximum [math]-spread is the graph formed by adding a pendant edge to [math]. We confirm this conjecture for sufficiently large [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657700","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 1, Page 529-565, March 2024. Abstract. We introduce a new class of balanced allocation processes which are primarily characterized by “filling” underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, and if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is [math] w.h.p. for any number of balls [math]. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of [math] on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar, and Shah [43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 2002, pp. 799–808].
{"title":"The Power of Filling in Balanced Allocations","authors":"Dimitrios Los, Thomas Sauerwald, John Sylvester","doi":"10.1137/23m1552231","DOIUrl":"https://doi.org/10.1137/23m1552231","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 529-565, March 2024. <br/> Abstract. We introduce a new class of balanced allocation processes which are primarily characterized by “filling” underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, and if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is [math] w.h.p. for any number of balls [math]. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of [math] on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar, and Shah [43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 2002, pp. 799–808].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584054","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}
Michel Habib, Lalla Mouatadid, Éric Sopena, Mengchuan Zou
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 566-589, March 2024. Abstract. Modular decomposition focuses on repeatedly identifying a module [math] (a collection of vertices that shares exactly the same neighborhood outside of [math]) and collapsing it into a single vertex. This notion of exactitude of neighborhood is very strict, especially when dealing with real-world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an [math]-module, a relaxation that maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal [math]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [math]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [math], but unfortunately we give evidence that computing such a decomposition tree can be difficult.
{"title":"[math]-Modules in Graphs","authors":"Michel Habib, Lalla Mouatadid, Éric Sopena, Mengchuan Zou","doi":"10.1137/21m1443534","DOIUrl":"https://doi.org/10.1137/21m1443534","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 566-589, March 2024. <br/> Abstract. Modular decomposition focuses on repeatedly identifying a module [math] (a collection of vertices that shares exactly the same neighborhood outside of [math]) and collapsing it into a single vertex. This notion of exactitude of neighborhood is very strict, especially when dealing with real-world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an [math]-module, a relaxation that maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal [math]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [math]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [math], but unfortunately we give evidence that computing such a decomposition tree can be difficult.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584516","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}
Nicolas Bousquet, Quentin Deschamps, Lucas De Meyer, Théo Pierron
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 504-528, March 2024. Abstract. The discharging method is a powerful proof technique, especially for graph coloring problems. Its major downside is that it often requires lengthy case analyses, which are sometimes given to a computer for verification. However, it is much less common to use a computer to actively look for a discharging proof. In this paper, we use a linear programming approach to automatically look for a discharging proof. While our system is not entirely autonomous, we manage to make some progress toward Wegner’s conjecture for distance-2 coloring of planar graphs by showing that 12 colors are sufficient to color at distance 2 every planar graph with maximum degree 4.
{"title":"Square Coloring Planar Graphs with Automatic Discharging","authors":"Nicolas Bousquet, Quentin Deschamps, Lucas De Meyer, Théo Pierron","doi":"10.1137/22m1492623","DOIUrl":"https://doi.org/10.1137/22m1492623","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 504-528, March 2024. <br/> Abstract. The discharging method is a powerful proof technique, especially for graph coloring problems. Its major downside is that it often requires lengthy case analyses, which are sometimes given to a computer for verification. However, it is much less common to use a computer to actively look for a discharging proof. In this paper, we use a linear programming approach to automatically look for a discharging proof. While our system is not entirely autonomous, we manage to make some progress toward Wegner’s conjecture for distance-2 coloring of planar graphs by showing that 12 colors are sufficient to color at distance 2 every planar graph with maximum degree 4.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"147 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500164","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 1, Page 485-503, March 2024. Abstract. The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids. Oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally unimodular matrices based on Seymour’s famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra as well as the construction of edge walks that may take a “detour" to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.
{"title":"On the Combinatorial Diameters of Parallel and Series Connections","authors":"Steffen Borgwardt, Weston Grewe, Jon Lee","doi":"10.1137/22m1490508","DOIUrl":"https://doi.org/10.1137/22m1490508","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 485-503, March 2024. <br/> Abstract. The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids. Oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally unimodular matrices based on Seymour’s famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra as well as the construction of edge walks that may take a “detour\" to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"206 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555088","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 1, Page 453-484, March 2024. Abstract. We present an isomorphism test for graphs of Euler genus [math] running in time [math]. Our algorithm provides the first explicit upper bound on the dependence on [math] for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time [math] for some function [math] (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude [math] as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, our algorithm relies on the notion of [math]-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler–Leman algorithm. This concept may be of independent interest.
{"title":"Isomorphism Testing Parameterized by Genus and Beyond","authors":"Daniel Neuen","doi":"10.1137/22m1514076","DOIUrl":"https://doi.org/10.1137/22m1514076","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 453-484, March 2024. <br/> Abstract. We present an isomorphism test for graphs of Euler genus [math] running in time [math]. Our algorithm provides the first explicit upper bound on the dependence on [math] for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time [math] for some function [math] (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude [math] as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, our algorithm relies on the notion of [math]-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler–Leman algorithm. This concept may be of independent interest.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"255 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555109","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 1, Page 380-411, March 2024. Abstract. Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species (“taxa”), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34–46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity [math], defined as the number of convex characters on [math] in which each state appears on at least 2 taxa. We use this to give an algorithm with running time [math], where [math] is the golden ratio and [math] is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by [math] we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to [math] and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees.
{"title":"Convex Characters, Algorithms, and Matchings","authors":"Steven Kelk, Ruben Meuwese, Stephan Wagner","doi":"10.1137/21m1463999","DOIUrl":"https://doi.org/10.1137/21m1463999","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 380-411, March 2024. <br/> Abstract. Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species (“taxa”), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34–46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity [math], defined as the number of convex characters on [math] in which each state appears on at least 2 taxa. We use this to give an algorithm with running time [math], where [math] is the golden ratio and [math] is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by [math] we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to [math] and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"212 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500083","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 1, Page 412-452, March 2024. Abstract. In the distributional Twenty Questions game, Bob chooses a number [math] from 1 to [math] according to a distribution [math], and Alice (who knows [math]) attempts to identify [math] using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for [math], yet this strategy could potentially uses all [math] possible questions. Dagan et al. constructed a set of [math] questions which suffice to construct an optimal strategy for all [math], and showed that this number is optimal (up to subexponential factors) for infinitely many [math]. We determine the optimal size of such a set of questions for all [math] (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the [math]-ary setting, obtaining similar results with 1.25 replaced by [math].
{"title":"Optimal Sets of Questions for Twenty Questions","authors":"Yuval Filmus, Idan Mehalel","doi":"10.1137/21m1424494","DOIUrl":"https://doi.org/10.1137/21m1424494","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 412-452, March 2024. <br/> Abstract. In the distributional Twenty Questions game, Bob chooses a number [math] from 1 to [math] according to a distribution [math], and Alice (who knows [math]) attempts to identify [math] using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for [math], yet this strategy could potentially uses all [math] possible questions. Dagan et al. constructed a set of [math] questions which suffice to construct an optimal strategy for all [math], and showed that this number is optimal (up to subexponential factors) for infinitely many [math]. We determine the optimal size of such a set of questions for all [math] (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the [math]-ary setting, obtaining similar results with 1.25 replaced by [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500167","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 1, Page 348-379, March 2024. Abstract. We consider classes of objective functions of cardinality-constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of [math]-[math]-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, [math]-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient—as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of [math] on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for [math]-augmentable functions. In particular, as a by-product, we close a gap in [A. Bernstein et al., Math. Program., 191 (2022), pp. 953–979] by obtaining a tight lower bound for [math]-augmentable functions for all [math]. For weighted rank functions of independence systems, our tight bound becomes [math], which recovers the known bound of [math] for independence systems of rank quotient at least [math].
{"title":"Unified Greedy Approximability beyond Submodular Maximization","authors":"Yann Disser, David Weckbecker","doi":"10.1137/22m1526952","DOIUrl":"https://doi.org/10.1137/22m1526952","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 348-379, March 2024. <br/> Abstract. We consider classes of objective functions of cardinality-constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of [math]-[math]-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, [math]-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient—as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of [math] on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for [math]-augmentable functions. In particular, as a by-product, we close a gap in [A. Bernstein et al., Math. Program., 191 (2022), pp. 953–979] by obtaining a tight lower bound for [math]-augmentable functions for all [math]. For weighted rank functions of independence systems, our tight bound becomes [math], which recovers the known bound of [math] for independence systems of rank quotient at least [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500095","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, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].
{"title":"Invertibility of Digraphs and Tournaments","authors":"Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer","doi":"10.1137/23m1547135","DOIUrl":"https://doi.org/10.1137/23m1547135","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. <br/> Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500323","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}
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