For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family $\mathcal{H}$ ($\textbf{mod}_{\mathcal{H}}$), elimination distance to $\mathcal{H}$ ($\textbf{ed}_{\mathcal{H}}$), and $\mathcal{H}$-treewidth ($\textbf{tw}_{\mathcal{H}}$). While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of latter two) in the context of approximation algorithms is mostly unexplored.
The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index, then the existence of an FPT Approximation Scheme (FPT-AS, i.e., ($1\pm ε$)-approximation) parameterized these three parameters is in fact equivalent. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by these three parameters.
A recent trend in the design of FPT algorithms is exploiting the half-integrality of LP relaxations. In other words, starting with a half-integral optimal solution to an LP relaxation, we assign integral values to variables one-by-one by branch and bound. This technique is general and the resulting time complexity has a low dependency on the parameter. However, the time complexity often becomes a large polynomial in the input size because we need to compute half-integral optimal LP solutions. In this paper, we address this issue by providing an $O(km)$-time algorithm for solving the LPs arisin…
For a hereditary graph class $\mathcal{H}$, the $\mathcal{H}$-elimination distance of a graph $G$ is the minimum number of rounds needed to reduce $G$ to a member of $\mathcal{H}$ by removing one vertex from each connected component in each round. The $\mathcal{H}$-treewidth of a graph $G$ is the minimum, taken over all vertex sets $X$ for which each connected component of $G - X$ belongs to $\mathcal{H}$, of the treewidth of the graph obtained from $G$ by replacing the neighborhood of each component of $G-X$ by a clique and then removing $V(G) \setminus X$. These parameterizations recently at…
The treedepth of a graph $G$ is the least possible depth of an elimination forest of $G$: a rooted forest on the same vertex set where every pair of vertices adjacent in $G$ is bound by the ancestor/descendant relation. We propose an algorithm that given a graph $G$ and an integer $d$, either finds an elimination forest of $G$ of depth at most $d$ or concludes that no such forest exists; thus the algorithm decides whether the treedepth of $G$ is at most $d$. The running time is $2^{O(d^2)}\cdot n^{O(1)}$ and the space usage is polynomial in $n$. Further, by allowing randomization, the time and…
We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing …
We generalize the fractional packing framework of Garg and Koenemann to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $\max\{c^T x : Ax \leq b, x \in C \}$, where the matrix $A$ contains no negative entries and $C$ is a cone that is generated by a finite set $S$ of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approxi…
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size $k$ in time $f(k) \cdot n^{O(1)}$, or width parameterizations, finding arbitrarily large optimal solutions in time $f(w) \cdot n^{O(1)}$ for some width measure $w$ like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simult…
Understanding the interactions between different combinatorial optimisation problems in real-world applications is a challenging task. Recently, the traveling thief problem (TTP), as a combination of the classical traveling salesperson problem and the knapsack problem, has been introduced to study these interactions in a systematic way. We investigate the underlying non-linear packing while traveling (PWT) problem of the TTP where items have to be selected along a fixed route. We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS…
We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least …
We show that for various classes C of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to C is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph G to a minor-closed class C is FPT.
Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk where every vertex occurs at most $r$ times and the total number of vertex occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial $P$ on $n$ variables and integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing pr…
For a graph class ${\cal H}$, the graph parameters elimination distance to ${\cal H}$ (denoted by ${\bf ed}_{\cal H}$) [Bulian and Dawar, Algorithmica, 2016], and ${\cal H}$-treewidth (denoted by ${\bf tw}_{\cal H}$) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class ${\cal H}$. Here, the torso of a vertex set $S$ in a graph $G$ is the graph with vertex set $S$ and an edge between two vertices $u, v \in S$ if there is a path between $u$ and $v$ in $G$ whose internal vertices all lie outside …
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, wh…
A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al. 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canon…
Diagonal Packing for Efficient Homomorphic Sparse Matrix-Vector Multiplication
Faculty of Engineering and Natural Sciences, Sabancı University（萨班哲大学工程与自然科学学院）；VERIM, Center of Excellence in Data Analytics, Sabancı University（萨班哲大学VERIM数据分析卓越中心）
We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $κ$ distinct lengths, and another interval $B$, can we pack the intervals in $\mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $\mathcal{I}$) by moving $τ$ intervals and keeping the other $σ= n - τ$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $κ$, $τ$, and $σ$ as single parameter, but are FPT with combined parameters $κ$ and $τ$. We also obtain improved polynomial-time algorithms for packing and covering, inc…
Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is particularly hard -- it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present …
We present a data structure that in a dynamic graph of treedepth at most $d$, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time $2^{{\cal O}(d^2)}$, which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time $f(d)$ for some non-elementary (i.e. tower-exponential) function $f$. As a by-product, we improve known up…
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of struct…
We give an algebraic, determinant-based algorithm for the K-Cycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the $O^*(2^{|K|})$ running time of the algorithm of Björklund et al. (SODA, 2012). Furthermore, our approach is open for treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to a polynomial compression of the problem, i.e., a polynomial-time reduction of the $K$-Cycle problem into an algebraic problem with coding size $O(|K|^3)$. This is s…
We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite $(3,3)$-hypergraphs. The proof technique of this paper uses the general correlation decay technique and a new combinatorial analysis of the underlying structures of the intersection graphs. The proof method could be also of independent…
In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use $2^{\frac1ε}$ passes or $n^{\frac1ε}$ time to compute a $(1-ε)$ approximation. In this paper we provide the first FPTAS (polynomial in $n,\frac1ε$) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the …
The component size of a graph is the maximum number of edges in any connected component of the graph. Given a graph $G$ and two integers $k$ and $c$, $(k,c)$-Decomposition is the problem of deciding whether $G$ admits an edge partition into $k$ subgraphs with component size at most $c$. We prove that for any fixed $k \ge 2$ and $c \ge 2$, $(k,c)$-Decomposition is NP-complete in bipartite graphs. Also, when both $k$ and $c$ are part of the input, $(k,c)$-Decomposition is NP-complete even in trees. Moreover, $(k,c)$-Decomposition in trees is W[1]-hard with parameter $k$, and is FPT with paramete…
We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an $\ell_{\infty}$-minimal element in a Minkowski sum $\mathcal{X}= \sum_{C \in \mathcal{C}} X_C$ of non-empty closed convex sets $X_C \subseteq \mathbb{R}^{\mathcal{R}}_{\geq 0}$, where $\mathcal{C}$ and $\mathcal{R}$ are finite sets. We assume that an oracle for approximate linear minimization over $X_C$ is given. In this setting, the currently fastest known FPTAS is due to Müller, Radke, and Vygen. For $δ\…
For the vast majority of local graph problems standard dynamic programming techniques give c^tw V^O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw^O(tw) V^O(1) time. We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Feedback Vertex Set and Connected Dominating Set, consequently answerin…
In this paper, we propose an algorithm that, given an undirected graph $G$ of $m$ edges and an integer $k$, computes a graph $G'$ and an integer $k'$ in $O(k^4 m)$ time such that (1) the size of the graph $G'$ is $O(k^2)$, (2) $k'\leq k$, and (3) $G$ has a feedback vertex set of size at most $k$ if and only if $G'$ has a feedback vertex set of size at most $k'$. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is $2k^2+k$ vertices and $4k^2$ edges, which is smaller than the previous best of $4k^2$ vertices and $8k^2$ edges. Thus, we improve t…
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex …
A Framework for the Design of Efficient Diversification Algorithms for Packing and Covering Problems
Universidad de Concepción（康塞普西翁大学）；Queens College of CUNY（纽约市立大学皇后学院）；University of Chile（智利大学）；CUNY Graduate Center（纽约市立大学研究生院）；Academia Sinica（中央研究院）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）
We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid $M$, a weight function $ω:E(M)\to\mathbb{N}$, and integers $k\geq 1, d\geq 0$. The task is to decide if there is a collection of $k$ bases $B_{1}, \dotsc, B_{k}$ of $M$ such that the weight of the symmetric difference of any pair of these bases is at least $d$. This is a diverse variant of the classical matroid base packing problem. The …
