Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Hierarchical Chinese Postman Problem: the slightest disorder makes it hard, yet disconnectedness is manageable
72
0
0.0
(
0
)
Added by
Ren\\'e van Bevern
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges of a graph respecting precedence constraints given by a partial order on classes of edges. We show that the special case with connected classes is NP-hard even on orders decomposable into a chain and an incomparable class. For the case with linearly ordered (possibly disconnected) classes, we get 5/3-approximations and fixed-parameter algorithms by transferring results from the Rural Postman Problem.
rate research
Read More
Given an undirected graph with edge weights and a subset $R$ of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of $R$. We prove that RPP is WK[1]-complete parameterized by the number and cost $d$ of edges traversed additionally to the required ones. Thus, in particular, RPP instances cannot be polynomial-time compressed to instances of size polynomial in $d$ unless the polynomial-time hierarchy collapses. In contrast, denoting by $bleq 2d$ the number of vertices incident to an odd number of edges of $R$ and by $cleq d$ the number of connected components formed by the edges in $R$, we show how to reduce any RPP instance $I$ to an RPP instance $I$ with $2b+O(c/varepsilon)$ vertices in $O(n^3)$ time so that any $alpha$-approximate solution for $I$ gives an $alpha(1+varepsilon)$-approximate solution for $I$, for any $alphageq 1$ and $varepsilon>0$. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on wide-spread benchmark data sets as well as on two real snow plowing instances from Berlin. On instances with few connected components, the number of vertices and required edges is reduced to about $50,%$ at a $1,%$ solution quality loss. We also make first steps towards a PSAKS for the parameter $c$.
The recent availability of quantum annealers has fueled a new area of information technology where such devices are applied to address practically motivated and computationally difficult problems with hardware that exploits quantum mechanical phenomena. D-Wave annealers are promising platforms to solve these problems in the form of quadratic unconstrained binary optimization. Here we provide a formulation of the Chinese postman problem that can be used as a tool for probing the local connectivity of graphs and networks. We treat the problem classically with a tabu algorithm and using a D-Wave device. We systematically analyze computational parameters associated with the specific hardware. Our results clarify how the interplay between the embedding due to limited connectivity of the Chimera graph, the definition of logical qubits, and the role of spin-reversal controls the probability of reaching the expected solution.
We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine) who performs the tasks. A typical workers objective is to minimize the amount of work he does (he is ``lazy), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse scheduling problems, which we denote ``Lazy Bureaucrat Problems, gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.
The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k = 3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achive on two restrictions of the problem,namely (i) when the records values are over a binary alphabet and k = 3, and (ii) when the records have length at most 8 and k = 4, showing that these restrictions of the problem are APX-hard.
We investigate the parameterized complexity of the following edge coloring problem motivated by the problem of channel assignment in wireless networks. For an integer q>1 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. This problem is NP-hard for q>1, and has been well-studied from the point of view of approximation. Our main focus is the case when q=2, which is already theoretically intricate and practically relevant. We show fixed-parameter tractable algorithms for both the standard and the dual parameter, and for the latter problem, the result is based on a linear vertex kernel.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
