The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel
for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasability of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.
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 o
rders 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.
One of the most fundamental results in combinatorial optimization is the polynomial-time 3/2-approximation algorithm for the metric traveling salesman problem. It was presented by Christofides in 1976 and is well known as the Christofides algorithm.
Recently, some authors started calling it Christofides-Serdyukov algorithm, pointing out that it was published independently in the USSR in 1978. We provide some historic background on Serdyukovs findings and a translation of his article from Russian into English.
The known linear-time kernelizations for $d$-Hitting Set guarantee linear worst-case running times using a quadratic-size data structure (that is not fully initialized). Getting rid of this data structure, we show that problem kernels of asymptotical
ly optimal size $O(k^d)$ for $d$-Hitting Set are computable in linear time and space. Additionally, we experimentally compare the linear-time kernelizations for $d$-Hitting Set to each other and to a classical data reduction algorithm due to Weihe.
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.
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 a
nd 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 $c$.
We show algorithms for computing representative families for matroid intersections and use them in fixed-parameter algorithms for set packing, set covering, and facility location problems with multiple matroid constraints. We complement our tractability results by hardness results.
Given a graph $G=(V,E)$, two vertices $s,tin V$, and two integers $k,ell$, the Short Secluded Path problem is to find a simple $s$-$t$-path with at most $k$ vertices and $ell$ neighbors. We study the parameterized complexity of the problem with respe
ct to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with $k$ and $ell$. We also obtain a $2^{O(w)}cdot ell^2cdot n$-time algorithm for graphs of treewidth $w$, which yields subexponential-time algorithms in several graph classes.
Inductive $k$-independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced $c$-colorable subgraphs, whic
h is a generalization of finding maximum independent sets, naturally occurs when selecting $c$ sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive $k$-independent graphs. We show that the Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes---a subclass of inductive 2-independent graphs. In contrast, we prove that the more general Maximum $c$-Colorable Subgraph problem is fixed-parameter tractable on edge-wise unions of cluster and chordal graphs, which are 2-simplicial. In both cases, the parameter is the solution size. Aside from this, we survey other graph classes between inductive 1-inductive and inductive 2-inductive graphs with applications in scheduling.
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.
