{"title":"A polynomial-time algorithm for conformable coloring on regular bipartite and subcubic graphs","authors":"Luerbio Faria, Mauro Nigro, Diana Sasaki","doi":"10.1016/j.disopt.2024.100865","DOIUrl":"10.1016/j.disopt.2024.100865","url":null,"abstract":"<div><div>In 1988, Chetwynd and Hilton observed that a <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-total coloring induces a vertex coloring in the graph, they called it conformable. A <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-vertex coloring of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> is called <em>conformable</em> if the number of color classes of parity different from that of <span><math><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></math></span> is at most the deficiency <span><math><mrow><mo>def</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><mrow><mo>(</mo><mi>Δ</mi><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>, where <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> is the degree of a vertex <span><math><mi>v</mi></math></span> of <span><math><mi>V</mi></math></span>. In 1994, McDiarmid and Sánchez-Arroyo proved that deciding whether a graph <span><math><mi>G</mi></math></span> has <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-total coloring is NP-complete even when <span><math><mi>G</mi></math></span> is <span><math><mi>k</mi></math></span>-regular bipartite with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. However, the time-complexity of the problem of determining whether a graph admits a conformable coloring (<span>Conformability</span> problem) remains unknown. In this paper, we prove that <span>Conformability</span> problem is polynomial solvable for the class of <span><math><mi>k</mi></math></span>-regular bipartite and for the class of subcubic graphs.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100865"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized min-up/min-down polytopes","authors":"Cécile Rottner","doi":"10.1016/j.disopt.2024.100866","DOIUrl":"10.1016/j.disopt.2024.100866","url":null,"abstract":"<div><div>Consider a time horizon and a set of <span><math><mi>N</mi></math></span> possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an <span><math><mi>N</mi></math></span>-state system with <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. The minimum-time constraints enforce that if the system switches to state <span><math><mi>i</mi></math></span> at time <span><math><mi>t</mi></math></span>, then it must remain in state <span><math><mi>i</mi></math></span> for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state <span><math><mi>i</mi></math></span> is switched on at time <span><math><mi>t</mi></math></span>, the system must remain on states <span><math><mrow><mi>j</mi><mo>≥</mo><mi>i</mi></mrow></math></span> during a minimum time. Symmetrically, they also enforce that if state <span><math><mi>i</mi></math></span> is switched off at time <span><math><mi>t</mi></math></span>, the system must remain on states <span><math><mrow><mi>j</mi><mo><</mo><mi>i</mi></mrow></math></span> during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities are facet-defining and lead to a complete description of polynomial size for each polytope.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100866"},"PeriodicalIF":0.9,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Corrigendum to “Bilevel time minimizing transportation problem” [Discrete Optim.] 5 (4) (2008) 714–723","authors":"Sonia , Ankit Khandelwal","doi":"10.1016/j.disopt.2024.100863","DOIUrl":"10.1016/j.disopt.2024.100863","url":null,"abstract":"<div><div>This is a corrigendum to our research paper titled “Bilevel time minimizing transportation problem” published in 2008. We deeply regret a minor error in the formulation of an intermediate problem solved as part of the algorithm. The intermediate problem, <span><math><msubsup><mrow><mrow><mo>(</mo><mi>T</mi><mi>P</mi><mo>)</mo></mrow></mrow><mrow><mi>t</mi></mrow><mrow><mi>T</mi></mrow></msubsup></math></span>, used to iteratively generate the prospective solution pairs, was initially modeled as a linear programming problem. But the correct formulation of its objective function now involves a binary function, thus making it an NP-hard problem. The algorithm is no longer polynomially bound as it involves solving a finite number of mixed 0-1 programming problems. The manuscript’s original contribution stands correct and there is no change to the structure or the accuracy of the algorithm. The changes required to the original paper, due to this error, are presented in this corrigendum.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100863"},"PeriodicalIF":0.9,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pascale Bendotti , Luca Brunod Indrigo , Philippe Chrétienne , Bruno Escoffier
{"title":"Anchor-robust project scheduling with non-availability periods","authors":"Pascale Bendotti , Luca Brunod Indrigo , Philippe Chrétienne , Bruno Escoffier","doi":"10.1016/j.disopt.2024.100864","DOIUrl":"10.1016/j.disopt.2024.100864","url":null,"abstract":"<div><div>In large-scale scheduling applications, it is often decisive to find reliable schedules prior to the execution of the project. Most of the time however, data is affected by various sources of uncertainty. Robust optimization is used to overcome this imperfect knowledge. Anchor robustness, as introduced in the literature for processing time uncertainty, makes it possible to guarantee job starting times for a subset of jobs. In this paper, anchor robustness is extended to the case where uncertain non-availability periods must be taken into account. Three problems are considered in the case of budgeted uncertainty: checking that a given subset of jobs is anchored in a given schedule, finding a schedule of minimal makespan in which a given subset of jobs is anchored and finding an anchored subset of maximum weight in a given schedule. Polynomial time algorithms are proposed for the first two problems while an inapproximability result is given for the third one.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100864"},"PeriodicalIF":0.9,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Circuit and Graver walks and linear and integer programming","authors":"Shmuel Onn","doi":"10.1016/j.disopt.2024.100862","DOIUrl":"10.1016/j.disopt.2024.100862","url":null,"abstract":"<div><div>We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100862"},"PeriodicalIF":0.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour
{"title":"Approximation schemes for Min-Sum k-Clustering","authors":"Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour","doi":"10.1016/j.disopt.2024.100860","DOIUrl":"10.1016/j.disopt.2024.100860","url":null,"abstract":"<div><p>We consider the Min-Sum <span><math><mi>k</mi></math></span>-Clustering (<span><math><mi>k</mi></math></span>-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> and a parameter <span><math><mi>k</mi></math></span>, the goal is to partition the points <span><math><mi>V</mi></math></span> into <span><math><mi>k</mi></math></span> clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.</p><p>The <span><math><mi>k</mi></math></span>-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span> in polynomial time for general metrics and an approximation ratio <span><math><mrow><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span> in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for <span><math><mi>k</mi></math></span>-MSC (when <span><math><mi>k</mi></math></span> is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from <span><math><mi>k</mi></math></span>-MSC to the balanced <span><math><mi>k</mi></math></span>-median problem and design approximation algorithms for the latter to obtain an approximation for <span><math><mi>k</mi></math></span>-MSC.</p><p>In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for <span><math><mi>k</mi></math></span>-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced <span><math><mi>k</mi></math></span>-median and is a trade off between center-based clustering problems (such as balanced <span><math><mi>k</mi></math></span>-median) and pair-wise clustering (such as Min-Sum <span><math><mi>k</mi></math></span>-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100860"},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000392/pdfft?md5=0298fb9c3c75e407870e412a1aae1a26&pid=1-s2.0-S1572528624000392-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner
{"title":"Mostar index and bounded maximum degree","authors":"Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner","doi":"10.1016/j.disopt.2024.100861","DOIUrl":"10.1016/j.disopt.2024.100861","url":null,"abstract":"<div><p>Došlić et al. defined the Mostar index of a graph <span><math><mi>G</mi></math></span> as <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where, for an edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, the term <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of vertices of <span><math><mi>G</mi></math></span> that have a smaller distance in <span><math><mi>G</mi></math></span> to <span><math><mi>u</mi></math></span> than to <span><math><mi>v</mi></math></span>. For a graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> and maximum degree at most <span><math><mi>Δ</mi></math></span>, we show <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> only depends on <span><math><mi>Δ</mi></math></span> and the <span><math><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> term only depends on <span><math><mi>n</mi></math></span>. Furthermore, for integers <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><mi>Δ</mi></math></span> at least 3, we show the existence of a <span><math><mi>Δ</mi></math></span>-regular graph of order <span><math><mi>n</mi></math></span> at least <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> with <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><mat","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100861"},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000409/pdfft?md5=e34071dea61722ee4baab21c7039f3bf&pid=1-s2.0-S1572528624000409-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Easy and hard separation of sparse and dense odd-set constraints in matching","authors":"Brady Hunsaker, Craig Tovey","doi":"10.1016/j.disopt.2024.100849","DOIUrl":"10.1016/j.disopt.2024.100849","url":null,"abstract":"<div><p>We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. <span><math><mrow><mo>≤</mo><mn>9</mn><mo>+</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>/</mo><mn>6</mn></mrow></math></span>) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, it is NP-complete to separate over the class of constraints on odd sets of size <span><math><mrow><mn>2</mn><mrow><mo>⌊</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> or <span><math><mrow><mo>≥</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100849"},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-set inequalities for the binary knapsack polyhedra","authors":"Todd Easton , Jennifer Tryon , Fabio Vitor","doi":"10.1016/j.disopt.2024.100859","DOIUrl":"10.1016/j.disopt.2024.100859","url":null,"abstract":"<div><p>This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100859"},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Revisiting some classical linearizations of the quadratic binary optimization problem and linkages with constraint aggregations","authors":"Abraham P. Punnen, Navpreet Kaur Dhanda","doi":"10.1016/j.disopt.2024.100858","DOIUrl":"10.1016/j.disopt.2024.100858","url":null,"abstract":"<div><p>In this paper, we examine explicit linearization models associated with the quadratic unconstrained binary optimization problem (QUBO). After reviewing some well-known basic linearization techniques, we present a new explicit linearization technique (PK) for QUBO with interesting properties. In particular, PK yields the same LP relaxation value as that of the standard linearization of QUBO while employing less number of constraints. We then show that using weighted aggregations of selected constraints of the basic linearization models, several new and effective linearization models of QUBO can be developed. Although aggregation of constraints has been studied in the past for solving systems of Diophantine equations in non-negative variables, none of them resulted in practical models, particularly due to the large size of the associated multipliers. For our aggregation based models, the multipliers can be of any positive real number. Moreover, we show that by choosing the multipliers appropriately, the LP relaxations of the resulting linearizations have optimal objective function values identical to that of the corresponding non-aggregated models. Theoretical and experimental comparisons of the new and existing explicit linearization models are also provided. Although our discussions were focused primarily on QUBO, the models and results we obtained extend naturally to the quadratic binary optimization problem with linear constraints.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100858"},"PeriodicalIF":0.9,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000379/pdfft?md5=88f5ed7ffa74e8ecf9bb69fd52845013&pid=1-s2.0-S1572528624000379-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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