Discrete Optimization最新文献

We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set $\{1,\dots ,n\}$, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most $k$ elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most $\ell$ times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by $R{U}_{\ell }^k\left(n\right)$. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of $R{U}_{\ell }^k\left(n\right)$ and an upper bound which differs from the lower one by at most $2\ell +1$. We also give its exact value when $n$ is sufficiently large compared to $k$. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case $\ell =1$.

We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length $n$ with bounded variation, 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 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 $O(nlogn)$, which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.

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).

Došlić et al. defined the Mostar index of a graph $G$ as $\sum _{uv\in E\left(G\right)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}{n}^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}{n}^3$.

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.

Došlić et al. defined the Mostar index of a graph $G$ as $\sum _{uv\in E\left(G\right)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}{n}^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}{n}^3$.

We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set $\{1,\dots ,n\}$, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most $k$ elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most $\ell$ times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by $R{U}_{\ell }^k\left(n\right)$. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of $R{U}_{\ell }^k\left(n\right)$ and an upper bound which differs from the lower one by at most $2\ell +1$. We also give its exact value when $n$ is sufficiently large compared to $k$. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case $\ell =1$.

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.

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 $D=(V,A)$$2T$-connected for some $T\subseteq V$ if $D$ is 2-arc-connected and $D-v$ is strongly connected for all $v\in T$. We deduce a characterization of the graphs admitting a $2T$-connected orientation from the theorem of Thomassen.

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