arXiv subjects
Publications and source records attributed to Andreas Emil Feldmann.
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.
We consider the classic Facility Location, $k$-Median, and $k$-Means problems in metric spaces of doubling dimension $d$. We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is $2^{(\log(1/\eps)/\eps)^{O(d^2)}} n \log^4 n + 2^{O(d)} n \log^9 n$, making a significant improvement over the state-of-the-art algorithms which run in time $n^{(d/\eps)^{O(d)}}$. Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting $k$-Medians and $k$-Means, and efficient bicriteria approximation schemes for $k$-Medians with outliers, $k$-Means with outliers and $k$-Center.
We develop an FPT algorithm and a bi-kernel for the Weighted Edge Clique Partition (WECP) problem, where a graph with $n$ vertices and integer edge weights is given together with an integer $k$, and the aim is to find $k$ cliques, such that every edge appears in exactly as many cliques as its weight. The problem has been previously only studied in the unweighted version called Edge Clique Partition (ECP), where the edges need to be partitioned into $k$ cliques. It was shown that ECP admits a kernel with~$k^2$ vertices [Mujuni and Rosamond, 2008], but this kernel does not extend to WECP. The previously fastest algorithm known for ECP has a runtime of $2^{\mathcal{O}(k^2)}n^{O(1)}$ [Issac, 2019]. For WECP we develop a bi-kernel with $4^k$ vertices, and an algorithm with runtime $2^{\mathcal{O}(k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}$, where $w$ is the maximum edge weight. The latter in particular improves the runtime for ECP to~$2^{\mathcal{O}(k^{3/2}\log k)}n^{O(1)}$.
In this paper we study the hardness of the $k$-Center problem on inputs that model transportation networks. For the problem, a graph $G=(V,E)$ with edge lengths and an integer $k$ are given and a center set $C\subseteq V$ needs to be chosen such that $|C|\leq k$. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the $k$-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers $k$, the highway dimension $h$, and the pathwidth $p$. Moreover, under the Exponential Time Hypothesis there is no $f(k,p,h)\cdot n^{o(p+\sqrt{k+h})}$ time algorithm for any computable function $f$. Thus it is unlikely that the optimum solution to $k$-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized $(1+\varepsilon)$-approximation algorithm for inputs of doubling dimension $d$ with runtime $(k^k/\varepsilon^{O(kd)})\cdot n^{O(1)}$. This generalizes a previous result, which considered inputs in $D$-dimensional $L_q$ metrics.
(see paper for full abstract) Given a vertex-weighted directed graph $G=(V,E)$ and a set $T=\{t_1, t_2, \ldots t_k\}$ of $k$ terminals, the objective of the SCSS problem is to find a vertex set $H\subseteq V$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $i\neq j$. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel $n^{O(k)}$ algorithm for the SCSS problem, where $n$ is the number of vertices in the graph and $k$ is the number of terminals. We explore how much easier the problem becomes on planar directed graphs: - Our main algorithmic result is a $2^{O(k)}\cdot n^{O(\sqrt{k})}$ algorithm for planar SCSS, which is an improvement of a factor of $O(\sqrt{k})$ in the exponent over the algorithm of Feldman and Ruhl. - Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an $f(k)\cdot n^{o(\sqrt{k})}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. The following additional results put our upper and lower bounds in context: - In general graphs, we cannot hope for such a dramatic improvement over the $n^{O(k)}$ algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. - Feldman and Ruhl generalized their $n^{O(k)}$ algorithm to the more general Directed Steiner Network (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source $s_i$ there is a path to the corresponding terminal $t_i$. We show that, assuming ETH, there is no $f(k)\cdot n^{o(k)}$ time algorithm for DSN on acyclic planar graphs.
(see paper for full abstract) Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number $k$ of terminals and the size $p$ of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for several directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gap-instances" under stronger hypotheses such as the (Gap-)Exponential Time Hypothesis (ETH).
We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015].
Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a reduction framework which identifies some necessary conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite 2D grid without holes. First we use the framework to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfying ratio. Then we consider solutions in which the sets may deviate from being equal-sized. Our framework is used on grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase the more stringent the limit on the set sizes is. These are the first bicriteria inapproximability results for the problem.
We consider the $k$-Center problem and some generalizations. For $k$-Center a set of $k$ center vertices needs to be found in a graph $G$ with edge lengths, such that the distance from any vertex of $G$ to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any $\varepsilon>0$ computing a $(2-\varepsilon)$-approximation is W[2]-hard for parameter $k$ and NP-hard for graphs with highway dimension $O(\log^2 n)$. The latter does not rule out fixed-parameter $(2-\varepsilon)$-approximations for the highway dimension parameter $h$, but implies that such an algorithm must have at least doubly exponential running time in $h$ if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of $2$ by combining the parameters $k$ and $h$: we develop a fixed-parameter $3/2$-approximation with running time $2^{O(kh\log h)}\cdot n^{O(1)}$. Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter $(2-\varepsilon)$-approximations for only the parameter $h$. We also provide similar fixed-parameter approximations for the weighted $k$-Center and $(k,\mathcal{F})$-Partition problems, which generalize $k$-Center.
In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(δ(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+ε)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).
A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the Vertex Bisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n^{O(d)}. We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.
In this paper we show that the price of stability of Shapley network design games on undirected graphs with k players is at most (k^3(k+1)/2-k^2) / (1+k^3(k+1)/2-k^2) H_k = (1 - Θ(1/k^4)) H_k, where H_k denotes the k-th harmonic number. This improves on the known upper bound of H_k, which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected Shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to Nash equilibria that minimize the potential function of the game. We also present a game with k=3 players in which such a restricted price of stability is 1.634. This shows that the analysis of Bilò and Bove (Journal of Interconnection Networks, Volume 12, 2011) is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to 1.571.