Consider n independent jobs and m identical machines in parallel. Job j has a processing time p(j) and a deadline (d) over bar (j). It must complete its processing before or at its deadline. All jobs are available for...
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Consider n independent jobs and m identical machines in parallel. Job j has a processing time p(j) and a deadline (d) over bar (j). It must complete its processing before or at its deadline. All jobs are available for processing at time t = 0 and preemptions are allowed. A set of jobs is said to be feasible if there exists a schedule that meets all the deadlines;such a schedule is called a feasible schedule. Given a feasible set of jobs, our goal is to find a schedule that minimizes the total completion time SigmaC(j). In the classical alpha|beta|gamma scheduling notation this problem is referred to as P|prmt, (d) over bar (j)\SigmaC(j). Lawler (Recent Results in the Theory of Machine Scheduling, in Mathematical Programming: The State of the Art, A. Bachem, M. Grotschel, and B. Korte, eds., Springer, Berlin, 1982, pp. 202 - 234) raised the question of whether or not the problem is NP-hard. In this paper we present a polynomial-time algorithm for every m greater than or equal to 2, and we show that the more general problem with m unrelated machines, i.e., R | prmt, (d) over bar (j)\SigmaC(j), is strongly NP-hard.
We call graphs of a fixed degree k sparse regular graphs and their complements dense regular graphs. Recently, it was conjectured that finding a maximum regular induced subgraph H in a 2P(3)-free graph can be solved i...
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We call graphs of a fixed degree k sparse regular graphs and their complements dense regular graphs. Recently, it was conjectured that finding a maximum regular induced subgraph H in a 2P(3)-free graph can be solved in polynomialtime if and only if H is sparse. In the present paper, we prove the "sparse" part of this conjecture, i.e., we show that for each fixed k, the problem of finding a maximum k-regular induced subgraph in a 2P(3)-free graph can be solved in polynomialtime. (C) 2013 Elsevier B.V. All rights reserved.
Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor-corrector algorithm for linear programming (LP...
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Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor-corrector algorithm for linear programming (LP) exhibits asymptotic quadratic convergence of the primal-dual gap to zero, without any assumptions concerning nondegeneracy, or the convergence of the iteration sequence. In this paper we prove a similar result for the monotone linear complementarity problem (LCP), assuming only that a strictly complementary solution exists. We also show by example that the existence of a strictly complementarity solution appears to be necessary to achieve superlinear convergence for the algorithm.
The present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functionals with finite support that are accessible only through some of t...
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The present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functionals with finite support that are accessible only through some of the values of their discrete Radon transform. It turns out that this task behaves quite differently from its well-studied companion problem involving 1-dimensional X-rays. Concentrating on the case of coordinate hyperplanes in R-d and on functionals psi : Z(d) --> D with D is an element of {{0, 1,...,r}, N-0} for some arbitrary but fixed r, we show in particular that the problem can be solved in polynomialtime if information is available for m such hyperplanes when m less than or equal to d - 1 but is NP-hard for m = d and D = {0, 1,...,r}. However, for D = N-0, a case that is relevant in the context of contingency tables, the problem is still in P. Similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem. (C) 2002 Elsevier Science B.V. All rights reserved.
Paths P-1, ..., P-k are mutually induced if any two distinct P-i and P-j have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The INDUCED DISJOINT PATHS problem is to decide if a gra...
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Paths P-1, ..., P-k are mutually induced if any two distinct P-i and P-j have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The INDUCED DISJOINT PATHS problem is to decide if a graph G with k pairs of specified vertices (s(i), t(i)) contains k mutually induced paths P-i such that each P-i connects s(i) and t(i). This problem is NP-complete even for k = 2. We prove that it can be solved in polynomialtime for AT-free graphs even when k is part of the input. Consequently, the problem of deciding if an AT-free graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. We show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter vertical bar V-H vertical bar, even for a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-IN-A-PATH and k-IN-A-TREE are polynomial-time solvable on AT-free graphs even if k is part of the input. (C) 2021 Elsevier Inc. All rights reserved.
We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama [Theoret. Comput. Sci., 809 (2...
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We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama [Theoret. Comput. Sci., 809 (2020), pp. 265--276] and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become coNP-hard even in the b-matching case.
An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a gra...
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An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solvable in polynomialtime for claw-free graphs. In the present article, we provide a polynomial-time algorithm to solve the dominating induced matching problem for graphs containing no long claw, that is, no induced subgraph obtained from the claw by subdividing each of its edges exactly once.
In this paper, we propose new algorithms for evacuation problems defined on dynamic flow networks. A dynamic flow network is a directed graph in which source nodes are given supplies and a single sink node is given a ...
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In this paper, we propose new algorithms for evacuation problems defined on dynamic flow networks. A dynamic flow network is a directed graph in which source nodes are given supplies and a single sink node is given a demand. The evacuation problem seeks a dynamic flow that sends all supplies from sources to the sink such that its demand is satisfied in the minimum feasible time horizon. For this problem, the current best algorithms are developed by Schl & ouml;ter (2018) and Kamiyama (2019), which run in strongly polynomialtime but with high-order polynomialtime complexity because they use submodular function minimization as a subroutine. In this paper, we propose new algorithms that do not explicitly execute submodular function minimization, and we prove that they are faster than the current best algorithms when an input network is restricted such that the sink has a small in-degree and every edge has the same capacity.
The generalized k-connectivity k(k) (G) of a graph G was introduced by Chartrand et al. in (Bull Bombay Math Colloq 2:1-6, 1984), which is a nice generalization of the classical connectivity. Recently, as a natural co...
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The generalized k-connectivity k(k) (G) of a graph G was introduced by Chartrand et al. in (Bull Bombay Math Colloq 2:1-6, 1984), which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we consider the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. We give a confirmative solution to a conjecture raised by S. Li in Ph.D. Thesis (2012). We also give a polynomial-time algorithm to a problem of generalized k-edge-connectivity.
Best match graphs (BMG) are a key intermediate in graph-based orthology detection and contain a large amount of information on the gene tree. We provide a near-cubic algorithm to determine whether a BMG is binary-expl...
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Best match graphs (BMG) are a key intermediate in graph-based orthology detection and contain a large amount of information on the gene tree. We provide a near-cubic algorithm to determine whether a BMG is binary-explainable, i.e., whether it can be explained by a fully resolved gene tree and, if so, to construct such a tree. Moreover, we show that all such binary trees are refinements of the unique binary-refinable tree (BRT), which in general is a substantial refinement of the also unique least resolved tree of a BMG. Finally, we show that the problem of editing an arbitrary vertex-colored graph to a binary-explainable BMG is NP-complete and provide an integer linear program formulation for this task.
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