Discrete Optimization最新文献

The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

The problem considered is the non-preemptive scheduling of independent jobs that consume a resource (which is non-renewable and replenished regularly) on parallel uniformly related machines. The input defines the speed of machines, size of jobs, the quantity of the resource required by the jobs, the replenished quantities, and replenishment dates of the resource. Every job can start processing only after the required quantity of the resource is allocated to the job. The objective function is a generalization of makespan minimization and minimization of the $l_p$-norm of the vector of loads of the machines. We present an EPTAS for this problem. Prior to our work only a PTAS was known in this non-renewable resource settings only for the special case of our problem of makespan minimization on identical machines.

The Grundy domination number, ${\gamma }_{\mathrm{gr}}\left(G\right)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\dots ,v_k)$ of vertices in $G$ such that for every $i\in \{2,\dots ,k\}$, the closed neighborhood $N\left[v_i\right]$ contains a vertex that does not belong to any closed neighborhood $N\left[v_j\right]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $\Gamma \left(G\right)$, which is in turn greater than or equal to the independence number $\alpha \left(G\right)$. In this paper, we initiate the study of the class of graphs $G$ with $\Gamma \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$ and its subclass consisting of graphs $G$ with $\alpha \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs $G$ with $\Gamma \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$ and present large families of such graphs.

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.

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.

In this paper, we investigate the problem of determining $s-t$ reachability in choice networks. In the traditional $s-t$ reachability problem, we are given a weighted network tuple $G=〈V,E,c,s,t〉$, with the goal of checking if there exists a path from $s$ to $t$ in $G$. In an optional choice network, we are given a choice set $S\subseteq E\times E$, in addition to the network tuple $G$. In the $s-t$ reachability problem in choice networks (OCR${}_D$), the goal is to find whether there exists a path from vertex $s$ to vertex $t$, with the caveat that at most one edge from each edge-pair $(x,y)\in S$ is used in the path. OCR${}_D$ finds applications in a number of domains, including routing in wireless networks and sensor placement. We analyze the computational complexities of the OCR${}_D$ problem and its variants from a number of algorithmic perspectives. We show that the problem is NP-complete in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is NPO PB-complete. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set $S$. In particular, we show that the problem can be solved in time $O^{\ast }(1.42^{\left|S\right|})$. We also consider weighted versions of the OCR${}_D$ problem and detail their computational complexities; in particular, the optimization version of the $WOC{R}_D$ problem is NPO-complete. While similar results have been obtained for related problems, our results improve