Discrete Optimization最新文献

{"title":"Reachability in choice networks","authors":"Piotr Wojciechowski , K. Subramani , Alvaro Velasquez","doi":"10.1016/j.disopt.2023.100761","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100761","url":null,"abstract":"<div><p>In this paper, we investigate the problem of determining <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability in <strong>choice networks</strong>. In the traditional <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability problem, we are given a weighted network tuple <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>〈</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>〉</mo></mrow></mrow></math></span>, with the goal of checking if there exists a path from <span><math><mi>s</mi></math></span> to <span><math><mi>t</mi></math></span> in <span><math><mi>G</mi></math></span>. In an optional choice network, we are given a choice set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>×</mo><mi>E</mi></mrow></math></span>, in addition to the network tuple <span><math><mi>G</mi></math></span>. In the <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability problem in choice networks (OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span>), the goal is to find whether there exists a path from vertex <span><math><mi>s</mi></math></span> to vertex <span><math><mi>t</mi></math></span>, with the caveat that at most one edge from each edge-pair <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><mi>S</mi></mrow></math></span> is used in the path. OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> finds applications in a number of domains, including <strong>routing in wireless networks</strong> and <strong>sensor placement</strong>. We analyze the computational complexities of the OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> problem and its variants from a number of algorithmic perspectives. We show that the problem is <strong>NP-complete</strong> in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is <strong>NPO PB-complete</strong>. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set <span><math><mi>S</mi></math></span>. In particular, we show that the problem can be solved in time <span><math><mrow><msup><mrow><mi>O</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>.</mo><mn>4</mn><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></msup><mo>)</mo></mrow></mrow></math></span>. We also consider weighted versions of the OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> problem and detail their computational complexities; in particular, the optimization version of the <span><math><mrow><mi>W</mi><mi>O</mi><mi>C</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></math></span> problem is <strong>NPO-complete</strong>. While similar results have been obtained for related problems, our results improve ","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100761"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809017","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 complexity of 2-vertex-connected orientation in mixed graphs","authors":"Florian Hörsch ,&nbsp;Zoltán Szigeti","doi":"10.1016/j.disopt.2023.100774","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100774","url":null,"abstract":"<div><p>We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph <span><math><mrow><mi>D</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>\u0000<span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected for some <span><math><mrow><mi>T</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> if <span><math><mi>D</mi></math></span> is 2-arc-connected and <span><math><mrow><mi>D</mi><mo>−</mo><mi>v</mi></mrow></math></span> is strongly connected for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. We deduce a characterization of the graphs admitting a <span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected orientation from the theorem of Thomassen.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100774"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809020","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 polytope of binary sequences with bounded variation","authors":"Christoph Buchheim,&nbsp;Maja Hügging","doi":"10.1016/j.disopt.2023.100776","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100776","url":null,"abstract":"<div><p><span>We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length </span><span><math><mi>n</mi></math></span><span> with bounded variation<span>, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary optimal control problems<span> subject to a bounded total variation. We study two variants of the problem. In the first one, the variation of the binary vector is penalized in the objective function, while in the second one, it is bounded by a hard constraint. We show that the first variant is easy to deal with while the second variant turns out to be more complex, but still tractable. For the latter case, we present a complete polyhedral description of the convex hull of feasible solutions by facet-inducing inequalities and devise an exact linear-time separation algorithm. The proof of completeness also yields a new exact primal algorithm with a running time of </span></span></span><span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.</p><p>A preliminary version of this article has been published in the Proceedings of the 7th International Symposium on Combinatorial Optimization (ISCO 2022) (Buchheim and Hügging, 2022).</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100776"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716352","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":"Minimum gradation in greyscales of graphs","authors":"Natalia de Castro ,&nbsp;María A. Garrido-Vizuete ,&nbsp;Rafael Robles ,&nbsp;María Trinidad Villar-Liñán","doi":"10.1016/j.disopt.2023.100773","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100773","url":null,"abstract":"<div><p>In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100773"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809019","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":"Approximating single- and multi-objective nonlinear sum and product knapsack problems","authors":"Jan Boeckmann ,&nbsp;Clemens Thielen ,&nbsp;Ulrich Pferschy","doi":"10.1016/j.disopt.2023.100771","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100771","url":null,"abstract":"<div><p>We present a fully polynomial-time approximation scheme (FPTAS) for a very general version of the well-known knapsack problem. This generalization covers, with few exceptions, all versions of knapsack problems that have been studied in the literature so far and allows for an objective function consisting of sums or products of possibly nonlinear, separable item profits, while the knapsack constraint states an upper bound on the sum of possibly nonlinear, separable item weights. Moreover, we extend our FPTAS to a multi-objective fully polynomial-time approximation scheme (MFPTAS) for the multi-objective version of the problem.</p><p>As applications of our general algorithms, we obtain the first FPTAS for the recently-introduced 0–1 time-bomb knapsack problem as well as FPTASs for a variety of robust knapsack problems. Moreover, we extend our FPTAS to the minimization version of our general problem, which, in particular, allows us to explicitly state an FPTAS for the classical minimization knapsack problem, which has been missing in the literature so far.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100771"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809024","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 improved greedy algorithm for stochastic online scheduling on unrelated machines","authors":"Sven Jäger","doi":"10.1016/j.disopt.2022.100753","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100753","url":null,"abstract":"<div><p>Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>-point scheduling on each machine. In this way we obtain a <span><math><mrow><mrow><mo>(</mo><mn>3</mn><mo>+</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mi>Δ</mi><mo>)</mo></mrow></mrow></math></span>-competitive deterministic and an <span><math><mrow><mo>(</mo><mn>8</mn><mo>+</mo><mn>4</mn><mi>Δ</mi><mo>)</mo></mrow></math></span>-competitive randomized stochastic online scheduling policy, where <span><math><mi>Δ</mi></math></span> is an upper bound on the squared coefficients of variation of the processing times. We also give constant performance guarantees for these policies within the class of all fixed-assignment policies. The <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>-point scheduling on a single machine can be enhanced when the upper bound <span><math><mi>Δ</mi></math></span> is known a priori or the processing times are known to be <span><math><mi>δ</mi></math></span>-NBUE for some <span><math><mrow><mi>δ</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span>. This implies improved competitive ratios for unrelated machines but may also be of independent interest.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100753"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731728","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":"CHAMP: A multipass algorithm for Max Sat based on saver variables","authors":"Daniel Berend ,&nbsp;Shahar Golan ,&nbsp;Yochai Twitto","doi":"10.1016/j.disopt.2023.100760","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100760","url":null,"abstract":"<div><p>In this paper, we introduce the concept of saver variables in Max Sat and demonstrate their contribution to the performance of solvers for this problem. We present two types of saver variables: high-rank savers and consensual savers. We show how to incorporate them in various ways into an iterated algorithm, CHAMP, for Max Sat. We conduct an extensive empirical evaluation on two collections of instances — instances from a past Max Sat competition and random instances. It turns out that, by using savers, the number of unsatisfied clauses may be reduced by more than 70% in some families. Moreover, a refined version CHAMP＋ of CHAMP improves the results even further. We show that by combining CHAMP＋ with CCLS, a state-of-the-art solver, we obtain better solutions for many Max Sat instances.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100760"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49715046","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 polyhedral study of lifted multicuts","authors":"Bjoern Andres ,&nbsp;Silvia Di Gregorio ,&nbsp;Jannik Irmai ,&nbsp;Jan-Hendrik Lange","doi":"10.1016/j.disopt.2022.100757","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100757","url":null,"abstract":"<div><p>Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from 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> to an augmented graph <span><math><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>∪</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs <span><math><mrow><mi>F</mi><mo>⊆</mo><mfenced><mfrac><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced><mo>∖</mo><mi>E</mi></mrow></math></span> of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>E</mi><mo>∪</mo><mi>F</mi></mrow></msup></math></span> whose vertices are precisely the characteristic vectors of multicuts of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span> lifted from <span><math><mi>G</mi></math></span>, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100757"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731914","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":"Constant factor approximation for tracking paths and fault tolerant feedback vertex set","authors":"Václav Blažej,&nbsp;Pratibha Choudhary,&nbsp;Dušan Knop,&nbsp;Jan Matyáš Křišťan,&nbsp;Ondřej Suchý,&nbsp;Tomáš Valla","doi":"10.1016/j.disopt.2022.100756","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100756","url":null,"abstract":"<div><p>Consider a vertex-weighted graph <span><math><mi>G</mi></math></span> with a source <span><math><mi>s</mi></math></span> and a target <span><math><mi>t</mi></math></span>. <span>Tracking Paths</span> requires finding a minimum weight set of vertices (<em>trackers</em>) such that the sequence of trackers in each path from <span><math><mi>s</mi></math></span> to <span><math><mi>t</mi></math></span> is unique. In this work, we derive a factor 6-approximation algorithm for <span>Tracking Paths</span> in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related <em>r</em>-<span>Fault Tolerant Feedback Vertex Set</span> problem. There, for a fixed integer <span><math><mi>r</mi></math></span> and a given vertex-weighted graph <span><math><mi>G</mi></math></span>, the task is to find a minimum weight set of vertices intersecting every cycle of <span><math><mi>G</mi></math></span> in at least <span><math><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. We give a factor <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> approximation algorithm for <em>r</em>-<span>Fault Tolerant Feedback Vertex Set</span> if <span><math><mi>r</mi></math></span> is a constant.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100756"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731732","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":"Fractional Decomposition Tree Algorithm: A tool for studying the integrality gap of Integer Programs","authors":"Robert Carr ,&nbsp;Arash Haddadan ,&nbsp;Cynthia A. Phillips","doi":"10.1016/j.disopt.2022.100746","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100746","url":null,"abstract":"<div><p><span><span>We present a new algorithm, Fractional Decomposition Tree (FDT), for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap of an instance’s </span>polyhedron, independent of objective function, is bounded. The algorithm gives a construction for Carr and Vempala’s theorem that any feasible solution to the IP’s linear-programming relaxation, when scaled by the instance integrality gap, dominates a </span>convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the Best-of-Many Christofides Algorithm (BOMC).</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100746"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49714804","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}