Journal of Combinatorial Optimization最新文献

{"title":"Computational complexity and algorithms for two scheduling problems under linear constraints","authors":"Kameng Nip, Peng Xie","doi":"10.1007/s10878-024-01122-0","DOIUrl":"https://doi.org/10.1007/s10878-024-01122-0","url":null,"abstract":"<p>This paper considers two different types of scheduling problems under linear constraints. The first is the single-machine scheduling problem with minimizing total completion time, while the second is the no-wait two-machine flow shop scheduling problem with minimizing makespan. For these two problems, a set of jobs is required to be scheduled to one or two machines. In contrast to the classic scheduling problems, the processing times of jobs are not fixed constants and are not predetermined. The decision-maker only knows that they should satisfy a system of given linear constraints. For both problems, the goal is to determine the processing time for each job and find the schedule that minimizes a particular criterion, namely, the total completion time or the makespan. First, we study the computational complexity and show that both the problems under linear constraints are NP-hard. These hardness results significantly differ from their traditional scheduling counterparts, as both of those are solvable in polynomial time. Then we propose polynomial time exact or approximation algorithms for various special cases. By utilizing the existing scheduling algorithms and the properties of linear programming, we demonstrate that both problems are polynomially solvable when the total number of linear constraints is a fixed constant.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140551886","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":"A tight max-flow min-cut duality theorem for nonlinear multicommodity flows","authors":"Matthew Broussard, Bala Krishnamoorthy","doi":"10.1007/s10878-024-01120-2","DOIUrl":"https://doi.org/10.1007/s10878-024-01120-2","url":null,"abstract":"<p>The Max-Flow Min-Cut theorem is the classical duality result for the <span>Max-Flow </span>problem, which considers flow of a single commodity. We study a multiple commodity generalization of <span>Max-Flow </span>in which flows are composed of real-valued <i>k</i>-vectors through networks with arc capacities formed by regions in <span>(mathbb {R}^k)</span>. Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the <i>mutual capacity</i> of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with <i>m</i> constraints in at most <span>(mathbb {R}^{(m-n+1)k})</span> space from one that requires <i>mn</i> constraints in <span>(mathbb {R}^{mk})</span> space for a network with <i>n</i> nodes and <i>m</i> edges. We present efficient algorithms that compute <span>(epsilon )</span>-approximations to both the ratio and the integer ratio maximum flow problems.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140544747","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":"Minimizing the expense transmission time from the source node to demand nodes","authors":"Mehdi Ghiyasvand, Iman Keshtkar","doi":"10.1007/s10878-024-01113-1","DOIUrl":"https://doi.org/10.1007/s10878-024-01113-1","url":null,"abstract":"<p>An undirected graph <span>(G=(V,A))</span> by a set <i>V</i> of <i>n</i> nodes, a set <i>A</i> of <i>m</i> edges, and two sets <span>(S, Dsubseteq V)</span> consists of source and demand nodes are given. This paper presents two new versions of location problems which are called the <span>(f(sigma ))</span>-location and <span>(g(sigma ))</span>-location problems. We define an <span>(f(sigma ))</span>-location of the network <i>N</i> as a node <span>(sin S)</span> with the property that the maximum expense transmission time from the node <i>s</i> to the destinations of <i>D</i> is as cheap as possible. The <span>(f(sigma ))</span>-location problem divides the range <span>((0,infty ))</span> into intervals <span>(displaystyle cup _{i}{(a_i,b_i)})</span> and finds a source <span>(s_iin S)</span>, for each interval <span>((a_i,b_i))</span>, such that <span>(s_i)</span> is a <span>(f(sigma ))</span>-location for each <span>(sigma in (a_i,b_i))</span>. Also, define a <span>(g(sigma ))</span>-location as a node <i>s</i> of <i>S</i> with the property that the sum of expense transmission times from the node <i>s</i> to all destinations of <i>D</i> is as cheap as possible. The <span>(g(sigma ))</span>-location problem divides the range <span>((0,infty ))</span> into intervals <span>(displaystyle cup _{i}{(a_i,b_i)})</span> and finds a source <span>(s_iin S)</span>, for each interval <span>((a_i,b_i))</span>, such that <span>(s_i)</span> is a <span>(g(sigma ))</span>-location for each <span>(sigma in (a_i,b_i))</span>. This paper presents two strongly polynomial time algorithms to solve <span>(f(sigma ))</span>-location and <span>(g(sigma ))</span>-location problems.\u0000</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140352210","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":"n-fold L(2, 1)-labelings of Cartesian product of paths and cycles","authors":"Fei-Huang Chang, Ma-Lian Chia, Shih-Ang Jiang, David Kuo, Jing-Ho Yan","doi":"10.1007/s10878-024-01119-9","DOIUrl":"https://doi.org/10.1007/s10878-024-01119-9","url":null,"abstract":"<p>For two sets of nonnegative integers <i>A</i> and <i>B</i>, the distance between these two sets, denoted by <i>d</i>(<i>A</i>, <i>B</i>), is defined by <span>(d(A,B)=min {|a-b|:ain A,bin B})</span>. For a positive integer <i>n</i>, let <span>(S_{n})</span> denote the family <span>( {X:Xsubseteq {mathbb {N}} cup {0},|X|=n})</span>. Given a graph <i>G</i> and positive integers <i>n</i>, <i>p</i> and <i>q</i>, an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> is a function <span>(f:V(G)rightarrow S_{n} )</span> satisfies <span>(d(f(u),f(v))ge p)</span> if <span>(d_{G}(u,v)=1)</span>, and <span>( d(f(u),f(v))ge q)</span> if <span>(d_{G}(u,v)=2)</span>. An <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling <i>f</i> of <i>G</i> is an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> with the property that <span>(max {a:ain bigcup _{uin V(G)}f(u)}le k)</span>. The smallest number <i>k</i> to guarantee that <i>G</i> has an <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling is called the <i>n</i> -fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling number of <i>G</i> and is denoted by <span>(lambda _{p,q}^{n}(G))</span>. When <span>(p=2, )</span> <span>(q=1,)</span> we use <span>(lambda ^{n}(G))</span> to replace <span>( lambda _{2,1}^{n}(G))</span> for simplicity. We study the <i>n</i>-fold <i>L</i>(2, 1) -labeling numbers of Cartesian product of paths and cycles in this paper. We give a necessary and sufficient condition for <span>(lambda ^{n}(C_{m}square P_{2}))</span> equals <span>(4n+1.)</span> Based on this, we determine the exact value of <span>( lambda ^{2}(C_{m}square P_{2}))</span> (except for <span>(m=5,6)</span> and 9) and <span>(lambda ^{3}(C_{m}square P_{2}))</span> (except for <span>(m=5,6,9,10,13)</span> and 17). We also give bounds for <span>(lambda ^{n}(C_{m}square P_{k}))</span> when <i>n</i>, <i>m</i> satisfy certain conditions, and from this, we obtain the exact value of <span>(lambda ^{2}(P_{m}square P_{k}))</span> (except for the case <span>(P_{4}square P_{3})</span>).</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140352162","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":"The average size of maximal matchings in graphs","authors":"Alain Hertz, Sébastien Bonte, Gauvain Devillez, Hadrien Mélot","doi":"10.1007/s10878-024-01144-8","DOIUrl":"https://doi.org/10.1007/s10878-024-01144-8","url":null,"abstract":"<p>We investigate the ratio <span>(mathcal {I}(G))</span> of the average size of a maximal matching to the size of a maximum matching in a graph <i>G</i>. If many maximal matchings have a size close to <span>(nu (G))</span>, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, <span>(mathcal {I}(G))</span> approaches <span>(frac{1}{2})</span>. We propose a general technique to determine the asymptotic behavior of <span>(mathcal {I}(G))</span> for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of <span>(mathcal {I}(G))</span> which were typically obtained using generating functions, and we then determine the asymptotic value of <span>(mathcal {I}(G))</span> for other families of graphs, highlighting the spectrum of possible values of this graph invariant between <span>(frac{1}{2})</span> and 1.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349442","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":"An exact borderline between the NP-hard and polynomial-time solvable cases of flow shop scheduling with job-dependent storage requirements","authors":"Alexander Kononov, Marina Pakulich","doi":"10.1007/s10878-024-01121-1","DOIUrl":"https://doi.org/10.1007/s10878-024-01121-1","url":null,"abstract":"<p>We consider two versions of two-machine flow shop scheduling problems, where each job requires an additional resource from the start of its first operation till the end of its second operation. We refer to this resource as storage space. The storage requirement of each job is determined by the processing time of its first operation. The two problems differ from each other in the way resources are allocated for each job. In the first case, the job captures all the necessary units of storage space at the beginning of processing its first operation. In the second case, the job takes up storage space gradually as its first operation is performed. In both problems, the goal is to minimize the makespan. In our paper, we establish the exact borderline between the NP-hard and polynomial-time solvable instances of the problems with respect to the ratio between the storage size and the maximum operation length.\u0000</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349406","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":"Randomized approximation schemes for minimizing the weighted makespan on identical parallel machines","authors":"Ruiqing Sun","doi":"10.1007/s10878-024-01118-w","DOIUrl":"https://doi.org/10.1007/s10878-024-01118-w","url":null,"abstract":"<p>In this paper, we discuss scheduling problems with <i>m</i> identical machines and <i>n</i> jobs where each job has to be assigned to some machine. The objective is to minimize the weighted makespan of jobs, i.e., the maximum weighted completion time of jobs. This scheduling problem is a generalization of minimizing the makespan on parallel machine scheduling problem. We present a (<span>(2-frac{1}{m})</span>)-approximation algorithm and a randomized efficient polynomial time approximation scheme (EPTAS) for the problem. We also design a randomized fully polynomial time approximation scheme (FPTAS) for the special case when the number of machines is fixed.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140333488","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":"Selecting intervals to optimize the design of observational studies subject to fine balance constraints","authors":"","doi":"10.1007/s10878-024-01116-y","DOIUrl":"https://doi.org/10.1007/s10878-024-01116-y","url":null,"abstract":"<h3>Abstract</h3> <p>Motivated by designing observational studies using matching methods subject to fine balance constraints, we introduce a new optimization problem. This problem consists of two phases. In the first phase, the goal is to cluster the values of a continuous covariate into a limited number of intervals. In the second phase, we find the optimal matching subject to fine balance constraints with respect to the new covariate we obtained in the first phase. We show that the resulting optimization problem is NP-hard. However, it admits an FPT algorithm with respect to a natural parameter. This FPT algorithm also translates into a polynomial time algorithm for the most natural special cases of the problem.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140333484","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":"Online car-sharing problem with variable booking times","authors":"Haodong Liu, Kelin Luo, Yinfeng Xu, Huili Zhang","doi":"10.1007/s10878-024-01114-0","DOIUrl":"https://doi.org/10.1007/s10878-024-01114-0","url":null,"abstract":"<p>In this paper, we address the problem of online car-sharing with variable booking times (CSV for short). In this scenario, customers submit ride requests, each specifying two important time parameters: the booking time and the pick-up time (start time), as well as two location parameters—the pick-up location and the drop-off location within a graph. For each request, it’s important to note that it must be booked before its scheduled start time. The booking time can fall within a specific interval prior to the request’s starting time. Additionally, each car is capable of serving only one request at any given time. The primary objective of the scheduler is to optimize the utilization of <i>k</i> cars to serve as many requests as possible. As requests arrive at their booking times, the scheduler faces an immediate decision: whether to accept or decline the request. This decision must be made promptly upon request submission, precisely at the booking time. We prove that no deterministic online algorithm can achieve a competitive ratio smaller than <span>(L+1)</span> even on a special case of a path (where <i>L</i> denotes the ratio between the largest and the smallest request travel time). For general graphs, we give a Greedy Algorithm that achieves <span>((3L+1))</span>-competitive ratio for CSV. We also give a Parted Greedy Algorithm with competitive ratio <span>((frac{5}{2}L+10))</span> when the number of cars <i>k</i> is no less than <span>(frac{5}{4}L+20)</span>; for CSV on a special case of a path, the competitive ratio of Parted Greedy Algorithm is <span>((2L+10))</span> when <span>(kge L+20)</span>.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140331200","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":"Star covers and star partitions of double-split graphs","authors":"Joyashree Mondal, S. Vijayakumar","doi":"10.1007/s10878-024-01112-2","DOIUrl":"https://doi.org/10.1007/s10878-024-01112-2","url":null,"abstract":"<p>A graph that is isomorphic to the complete bipartite graph <span>(K_{1,r})</span> for some <span>(rge 0)</span> is called a <i>star</i>. A collection <span>(mathcal {C} = {V_1, ldots , V_k})</span> of subsets of the vertex set of a graph <span>(G = (V, E))</span> is called a <i>star cover</i> of <i>G</i> if each set in the collection induces a star and has <span>(V_1cup ldots cup V_k = V)</span>. A star cover <span>(mathcal {C})</span> of a graph <span>(G = (V, E))</span> is called a <i>star partition</i> of <i>G</i> if <span>(mathcal {C})</span> is also a partition of <i>V</i>. The problem <span>Star Cover</span> takes a graph <i>G</i> as input and asks for a star cover of <i>G</i> of minimum size. The problem <span>Star Partition</span> takes a graph <i>G</i> as input and asks for a star partition of <i>G</i> of minimum size. From Shalu et al. (Discrete Appl Math 319:81–91, 2022), it follows that both these problems are NP-hard even for bipartite graphs. In this paper, we show that both <span>Star Cover</span> and <span>Star Partition</span> have <span>(O(n^7))</span> time exact algorithms for double-split graphs. Proving that our algorithms indeed have running time <span>(varOmega (n^7))</span> necessitates the construction of an intricate infinite family of double-split graphs meeting several requirements. Other contributions of the paper are a simple linear time recognition algorithm for double-split graphs and a useful succinct matrix representation for double-split graphs.\u0000</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196083","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}