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Register a new userOn approximate data reduction for the Rural Postman Problem: Theory and experiments
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Added by
Ren\\'e van Bevern
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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 $c$.
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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.
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Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and Vegh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.
Recent work has shown that many problems of satisfiability and resiliency in workflows may be viewed as special cases of the authorization policy existence problem (APEP), which returns an authorization policy if one exists and No otherwise. However, in many practical settings it would be more useful to obtain a least bad policy than just a No, where least bad is characterized by some numerical value indicating the extent to which the policy violates the base authorization relation and constraints. Accordingly, we introduce the Valued APEP, which returns an authorization policy of minimum weight, where the (non-negative) weight is determined by the constraints violated by the returned solution. We then establish a number of results concerning the parameterized complexity of Valued APEP. We prove that the problem is fixed-parameter tractable (FPT) if the set of constraints satisfies two restrictions, but is intractable if only one of these restrictions holds. (Most constraints known to be of practical use satisfy both restrictions.) We also introduce a new type of resiliency for workflow satisfiability problem, show how it can be addressed using Valued APEP and use this to build a set of benchmark instances for Valued APEP. Following a set of computational experiments with two mixed integer programming (MIP) formulations, we demonstrate that the Valued APEP formulation based on the user profile concept has FPT-like running time and usually significantly outperforms a naive formulation.
The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms, also known as polynomial kernelizations. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0. In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.
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