Let $H$ be a fixed undirected graph on $k$ vertices. The $H$-hitting set problem asks for deleting a minimum number of vertices from a given graph $G$ in such a way that the resulting graph has no copies of $H$ as a subgraph. This problem is a special case of the hypergraph vertex cover problem on $k$-uniform hypergraphs, and thus admits an efficient $k$-factor approximation algorithm. The purpose of this article is to investigate the question that for which graphs $H$ this trivial approximation factor $k$ can be improved.
We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph $H$ and an input graph $G$, the task is to find a set $X \subseteq V(G)$ of minimum size that hits all subgraphs of $G$ isomorphic to $H$. In the colorful variant of the problem, each vertex of $G$ is precolored with some color from $V(H)$ and we require to hit only $H$-subgraphs with matching colors. Standard techniques shows that for every fixed $H$, the problem is fixed-parameter tractable parameterized by the treewidth of $G$; however, it is not clear how exactly the running time shou…
For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph $H$, the smallest function $f_H(t)$ such that $H$-IS-Deletion can be solved in time $f_H(t) \cdot n^{O(1)}$ assuming the Exponential Time Hypothesis (ETH), where $t$ and $n$ denote the treewidth and the number of vertices of the input graph, respec…
We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster vertex deletion in the literature and admits a straightforward 3-approximation algorithm since it is a special case of the vertex cover problem on a 3-uniform hypergraph. Recently, You, Wang, and Cao described an efficient 5/2-approximation algorithm for the unweighted version of the problem. Our main result is a 9/4-approximation algorithm for arbitrary weights, using the local rati…
A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family $\mathcal F$. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs $G$, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals w…
Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vert…
Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $Ω(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs.
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…
We consider problems of the following type: given a graph $G$, how many edges are needed in the worst case for a sparse subgraph $H$ that approximately preserves distances between a given set of node pairs $P$? Examples include pairwise spanners, distance preservers, reachability preservers, etc. There has been a trend in the area of simple constructions based on the hitting set technique, followed by somewhat more complicated constructions that improve over the bounds obtained from hitting sets by roughly a $\log$ factor. In this note, we point out that the simpler constructions based on hitt…
We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system $(V,\mathcal{F},\mathcal{B})$ with two families $\mathcal{F},\mathcal{B}$ of subsets of $V$. The task is to find a hitting set for $\mathcal{F}$ that minimizes the maximum number of elements in any of the sets of $\mathcal{B}$. Our focus is on determining the complexity of some special cases of Sparse-HS with respect to the sparseness $k$, which is the optimum number of hitting set elements in any set of $\mathcal{B}$. For the Sparse Vertex Cover (Sparse-VC) problem, $V$ is given by the vertex set of a grap…
Hitting Axis-Parallel Segments with Weighted Points
Indraprastha Institute of Information Technology Delhi（德里印度理工学院）；Indian Institute of Science Bengaluru（班加罗尔印度科学研究所）；Indian Institute of Technology Delhi（德里印度理工学院）
We present the first polynomial-time approximation schemes, i.e., (1 + ε)-approximation algorithm for any constant ε > 0, for the minimum three-edge connected spanning subgraph problem and the minimum three-vertex connected spanning subgraph problem in undirected planar graphs. Both the approximation schemes run in linear time.
Dept of Computer Science, Univ of British Columbia（不列颠哥伦比亚大学计算机科学系）；Systems Biology Dept, Harvard Medical School（哈佛医学院系统生物学系）；Dept of Computer Science, MIT（麻省理工学院计算机科学系）；Google（谷歌）
This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an un-supervised grammar induction problem from computational linguistics. We present a reduction to some known problems and algorithms from graph theory, provide computational complexity results, and describe an approximation algorithm.
Computational Learning and Computational Linguistics Research Group（计算学习与计算语言学研究组）；University of Geneva（日内瓦大学）
Previous work has shown the effectiveness of random walk hitting times as a measure of dissimilarity in a variety of graph-based learning problems such as collaborative filtering, query suggestion or finding paraphrases. However, application of hitting times has been limited to small datasets because of computational restrictions. This paper develops a new approximation algorithm with which hitting times can be computed on very large, disk-resident graphs, making their application possible to problems which were previously out of reach. This will potentially benefit a range of large-scale prob…
Jaberi [7] presented approximation algorithms for the problem of computing a minimum size 2-vertex strongly biconnected subgraph in directed graphs. We have implemented approximation algorithms presented in [7] and we have tested the implementation on some graphs. The experimental results show that these algorithms work well in practice.
We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönni…
We obtain a polynomial-time 17/12-approximation algorithm for the minimum-cost 2-vertex-connected spanning subgraph problem, restricted to graphs of minimum degree at least 3. Our algorithm uses the framework of ear-decompositions for approximating connectivity problems, which was previously used in algorithms for finding the smallest 2-edge-connected spanning subgraph by Cheriyan, Sebő and Szigeti (SIAM J.Discrete Math. 2001) who gave a 17/12-approximation algorithm for this problem, and by Sebő and Vygen (Combinatorica 2014), who improved the approximation ratio to 4/3.
We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomial-time $(2-\frac{2}{Δ+1})$-approximation algorithm for the problem, where $Δ$ is the maximum degree of the graph. Finally, we s…
In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: -- Matching: First, there exists an $\tilde{O}(k^2)$ space algorithm that returns an exact maximum matching on the assumption the cardinality is at most $k$. The best previous algorithm used $\tilde{O}(kn)$ spa…
The minimum-cost subset $k$-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph $G=(V,E)$ with costs on edges and a set of terminals $T$. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by $k$ openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset $k$-connected subgraph problem which improves on the previous best approximation guarantee of $O(k^2\log{k})$ by Nutov (FOCS 2009). Our approximation guaran…
We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.
We obtain approximation ratio $2(2+\frac{1}{\ell})$ for the (undirected) $k$-Connected Subgraph problem, where $\ell \approx \frac{1}{2} (\log_k n-1)$ is the largest integer such that $2^{\ell-1} k^{2\ell+1} \leq n$. For large values of $n$ this improves the $6$-approximation of Cheriyan and Végh when $n =Ω(k^3)$, which is the case $\ell=1$. For $k$ bounded by a constant we obtain ratio $4+ε$. For large values of $n$ our ratio matches the best known ratio $4$ for the augmentation version of the problem, as well as the best known ratios for $k=6,7$. Similar results are shown for the problem of …
A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put on the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. Some approaches aim at finding disjoint subgraphs, while in many real-world networks communities are often overlapping. An approach introduced to find possible overlapping subgraphs is the Top-k Overlapping Densest Subgraphs problem. For a given integer k >= 1,…
Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to compute a set $S \subset V$ of cardinality $\left\lvert S\right\rvert \leq k$ such that the weight of edges inside $S$ is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a $\mathcal{O}\left({n^{1/4 + ε}}\right)$ approximation in time $n^{\mathcal{O}\left(1/ε\right)}$, for any $ε> 0$. We ask what are some "easier" instances of this problem? We propose …
Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bha…
In the Directed Long Cycle Hitting Set} problem we are given a directed graph $G$, and the task is to find a set $S$ of at most $k$ vertices/arcs such that $G-S$ has no cycle of length longer than $\ell$. We show that the problem can be solved in time $2^{\mathcal O(\ell k^3\log k + k^5\log k\log\ell)}\cdot n^{\mathcal O(1)}$, that is, it is fixed-parameter tractable (FPT) parameterized by $k$ and $\ell$. This algorithm can be seen as a far-reaching generalization of the fixed-parameter tractability of {\sc Mixed Graph Feedback Vertex Set} [Bonsma and Lokshtanov WADS 2011], which is already a …
In the classical Maximum Acyclic Subgraph problem (MAS), given a directed-edge weighted graph, we are required to find an ordering of the nodes that maximizes the total weight of forward-directed edges. MAS admits a 2 approximation, and this approximation is optimal under the Unique Game Conjecture. In this paper we consider a generalization of MAS, the Restricted Maximum Acyclic Subgraph problem (RMAS), where each node is associated with a list of integer labels, and we have to find a labeling of the nodes so as to maximize the weight of edges whose head label is larger than the tail label. T…
We study the problem of identity testing of markov chains. In this setting, we are given access to a single trajectory from a markov chain with unknown transition matrix $Q$ and the goal is to determine whether $Q = P$ for some known matrix $P$ or $\text{Dist}(P, Q) \geq ε$ where $\text{Dist}$ is suitably defined. In recent work by Daskalakis, Dikkala and Gravin, 2018, it was shown that it is possible to distinguish between the two cases provided the length of the observed trajectory is at least super-linear in the hitting time of $P$ which may be arbitrarily large. In this paper, we propose a…
The claw number of a graph $G$ is the largest number $v$ such that $K_{1,v}$ is an induced subgraph of $G$. Interval graphs with claw number at most $v$ are cluster graphs when $v = 1$, and are proper interval graphs when $v = 2$. Let $κ(n,v)$ be the smallest number $k$ such that every interval graph with $n$ vertices admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. Let $\checkκ(w,v)$ be the smallest number $k$ such that every interval graph with claw number $w$ admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. We show that $κ(…
Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known re…
Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$ sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cov…
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for $k \geq 2$ less is known in the distributed setting. In this paper, we present fast randomized distributed approximation algorithms for $k$-ECSS in the CONGEST model. Our first contribution is an $\widetilde{O}(D + \sqrt{n})$-ro…
Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network…
We propose a simple and natural approximation algorithm for the problem of finding a 2-edge-connected spanning subgraph of minimum total edge cost in a graph. The algorithm maintains a spanning forest starting with an empty edge set. In each iteration, a new edge incident to a leaf is selected in a natural greedy manner and added to the forest. If this produces a cycle, this cycle is contracted. This growing phase ends when the graph has been contracted into a single node and a subsequent cleanup step removes redundant edges in reverse order. We analyze the algorithm using the primal-dual meth…
We study the recently introduced problem of finding dense common subgraphs: Given a sequence of graphs that share the same vertex set, the goal is to find a subset of vertices $S$ that maximizes some aggregate measure of the density of the subgraphs induced by $S$ in each of the given graphs. Different choices for the aggregation function give rise to variants of the problem that were studied recently. We settle many of the questions left open by previous works, showing NP-hardness, hardness of approximation, non-trivial approximation algorithms, and an integrality gap for a natural relaxation…
In the $k$-edge-connected spanning subgraph ($k$ECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to $k$ link failures: Given an $n$-node $m$-edge graph with a cost function on the edges, our goal is to compute a minimum-cost $k$-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a sp…
