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Register a new userParameterized complexity of machine scheduling: 15 open problems
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Added by
Ren\\'e van Bevern
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.
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We present three new quantum algorithms in the quantum query model for textsc{graph-collision} problem: begin{itemize} item an algorithm based on tree decomposition that uses $Oleft(sqrt{n}t^{sfrac{1}{6}}right)$ queries where $t$ is the treewidth of the graph; item an algorithm constructed on a span program that improves a result by Gavinsky and Ito. The algorithm uses $O(sqrt{n}+sqrt{alpha^{**}})$ queries, where $alpha^{**}(G)$ is a graph parameter defined by [alpha^{**}(G):=min_{VCtext{-- vertex cover of}G}{max_{substack{Isubseteq VCItext{-- independent set}}}{sum_{vin I}{deg{v}}}};] item an algorithm for a subclass of circulant graphs that uses $O(sqrt{n})$ queries. end{itemize} We also present an example of a possibly difficult graph $G$ for which all the known graphs fail to solve graph collision in $O(sqrt{n} log^c n)$ queries.
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.
Production and inventory planning have become crucial and challenging in nowadays competitive industrial and commercial sectors, especially when multiple plants or warehouses are involved. In this context, this paper addresses the complexity of uncapacitated multi-plant lot-sizing problems. We consider a multi-item uncapacitated multi-plant lot-sizing problem with fixed transfer costs and show that two of its very restricted special cases are already NP-hard. Namely, we show that the single-item uncapacitated multi-plant lot-sizing problem with a single period and the multi-item uncapacitated two-plant lot-sizing problem with fixed transfer costs are NP-hard. Furthermore, as a direct implication of the proven results, we also show that a two-echelon multi-item lot-sizing with joint setup costs on transportation is NP-hard.
A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size $k$ by showing that DOCT does not admit an algorithm with running time $f(k)n^{O(1)}$ unless FPT = W[1]. On the positive side, we give a factor $2$ fixed parameter tractable (FPT) approximation algorithm for the problem. More precisely, our algorithm takes as input $D$ and $k$, runs in time $2^{O(k^2)}n^{O(1)}$, and either concludes that $D$ does not have a directed odd cycle transversal of size at most $k$, or produces a solution of size at most $2k$. Finally, we provide evidence that there exists $epsilon > 0$ such that DOCT does not admit a factor $(1+epsilon)$ FPT-approximation algorithm.
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problems fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: for every graph property P expressible by a first order-logic formula phiin Sigma_3, that is, of the form phi=exists x_1exists x_2cdots exists x_r forall y_1forall y_2cdots forall y_s exists z_1exists z_2cdots exists z_t psi, where psi is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Sigma_3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: there are formulas phiin Pi_3, for which computing elimination distance is W[2]-hard.
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