Discrete Optimization最新文献

The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph $G$ and seeks a smallest vertex set $S$ that hits all cycles in $G$. This is one of Karp’s 21 $\mathrm{NP}$-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a $4^{k}k!n^{O\left(1\right)}$-time algorithm, where $k=\left|S\right|$.

Here we show fixed-parameter tractability of two generalizations of DFVS:

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  Find a smallest vertex set $S$ such that every strong component of $G-S$ has size at most $s$: we give an algorithm solving this problem in time $4^k(ks+k+s)!\cdot n^{O\left(1\right)}$. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.

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  Find a smallest vertex set $S$ such that every non-trivial strong component of $G-S$ is 1-out-regular: we give an algorithm solving this problem in time $2^{O\left(k^3\right)}\cdot n^{O\left(1\right)}$.

We consider the unconstrained traveling tournament problem, a sports timetabling problem that minimizes traveling of teams. Since its introduction about 20 years ago, most research was devoted to modeling and reformulation approaches. In this paper we carry out a polyhedral study for the cubic integer programming formulation by establishing the dimension of the integer hull as well as of faces induced by model inequalities. Moreover, we introduce a new class of inequalities and show that they are facet-defining. Finally, we evaluate the impact of these inequalities on the linear programming bounds.

In this paper we study the problem of partitioning a tree with $n$ weighted vertices into $p$ connected components. For each component, we measure its gap, that is, the difference between the maximum and the minimum weight of its vertices, with the aim of minimizing the sum of such differences. We present an $O\left(n^3{p}^2\right)$ time and $O\left(n^{3}p\right)$ space algorithm for this problem. Then, we generalize it, requiring a minimum of $ϵ\ge 1$ nodes in each connected component, and provide an $O\left(n^3{p}^2{ϵ}^2\right)$ time and $O\left(n^{3}pϵ\right)$ space algorithm to solve this new problem version. We provide a refinement of our analysis involving the topology of the tree and an improvement of the algorithms for the special case in which the weights of the vertices have a heap structure. All presented algorithms can be straightforwardly extended to other similar objective functions. Actually, for the problem of minimizing the maximum gap with a minimum number of nodes in each component, we propose an algorithm which is independent of $ϵ$ and requires $O(n^{2}logn{p}^2)$ time and $O\left(n^{2}p\right)$ space.

We provide a linear description of the unconstrained stable allocations problem by proving that the corresponding polytope is affinely congruent to the order polytope of a partially ordered set. The same holds for stable matchings hence simplifying the derivation of known polyhedral results. We also show that this congruence no longer holds for the constrained version of stable allocations. As side outcomes, we characterise the neighbouring vertices of the order polytope and the partially ordered set associated with stable allocations.

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties.

By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points of an orthogonal array (OA) defining integer linear description (ILD). Our results address the conjecture that if this polytope is non-empty, then it is full-dimensional within the affine space where all the feasible points of the ILD’s linear description (LD) relaxation lie, raised by Appa et al. (2006). In particular, our theoretical results provide a sufficient condition for this polytope to be full-dimensional within the LD relaxation affine space when it is non-empty. This sufficient condition implies all the known non-trivial values of the dimension of the $(k,s)$ assignment polytope. However, our results suggest that the conjecture mentioned above may not be true. More generally, we provide previously unknown restrictions on the feasible values of the dimension of the convex hull of all feasible points of our OA defining ILD. We also determine all possible corresponding sets of equality constraints up to equivalence that can potentially be implied by the integrality constraints of this ILD. Moreover, we find additional restrictions on the dimension of the convex hull of all feasible points, and larger sets of corresponding equality constraints for the $n=2$ and even $s$ cases. Each of these cases possesses symmetries that do not necessarily exist in the $3\le n$ or odd $s$ cases. Finally, we discuss how to decrease the number of possible values for the dimension of the convex hull of all feasible points of an arbitrary ILD as well as generate sets of corresponding equality constraints with the zero right hand side. These are the only sets of zero right hand side equality constraints up to equivalence that can potentially be implied by the integrality constraints of the ILD.

This paper is concerned with the design and analysis of algorithms for optimization problems in arc-dependent networks. A network is said to be arc-dependent if the cost of an arc $a$ depends upon the arc taken to enter $a$. These networks are fundamentally different from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. We first study the arc-dependent shortest path (ADSP) problem, which is also known as the suffix-1 path-dependent shortest path problem in the literature. This problem has a polynomial time solution if the shortest paths are not required to be simple. The ADSP problem finds applications in a number of domains, including highway engineering, turn penalties and prohibitions, and fare rebates. In this paper, we are interested in the ADSP problem when restricted to simple paths. We call this restricted version the simple arc-dependent shortest path (SADSP) problem. We show that the SADSP problem is NP-complete. We present inapproximability results and an exact exponential algorithm for this problem. We also extend our results for the longest path problem in arc-dependent networks. Additionally, we explore the problem of detecting negative cycles in arc-dependent networks and discuss its computational complexity. Our results include variants of the negative cycle detection problem such as longest, shortest, heaviest, and lightest negative simple cycles.²

We present a slight generalization of the result of Kamiyama and Kawase (2015) on packing time-respecting arborescences in acyclic pre-flow temporal networks. Our main contribution is to provide the first results on packing time-respecting arborescences in non-acyclic temporal networks. As negative results, we prove the NP-completeness of the decision problem of the existence of 2 arc-disjoint spanning time-respecting arborescences and of a related problem proposed in this paper.

This paper considers the bounded parallel-batching scheduling with two agents to minimize maximum cost of agent $A$ and makespan of agent $B$ simultaneously, in which all jobs of agent $A$ have equal processing time, the jobs from different agents can be processed in a common batch and the cost function of each agent is only determined by its own jobs. In the paper, we present a polynomial-time algorithm to generate all Pareto optimal points for the problem and determine a corresponding Pareto optimal schedule for each Pareto optimal point.