Subexponential Algorithms for Partial Cover Problems

Partial Cover problems are optimization versions of fundamental and well studied problems like {\sc Vertex Cover} and {\sc Dominating Set}. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number $k$ of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by $k$. It was recently shown by Amini et. al. [{\em FSTTCS 08}\,] that {\sc Partial Vertex Cover} and {\sc Partial Dominating Set} are fixed parameter tractable on large classes of sparse graphs, namely $H$.minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2^{\cO(k)}n^{\cO(1)}.publishedVersio

(Total) Vector Domination for Graphs with Bounded Branchwidth

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) has at least $d(v)$ neighbors in $S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the $k$-tuple dominating set problem (this $k$ is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto $k$, where $k$ is the size of solution.Comment: 16 page

Bidimensionality and EPTAS

Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. Two of the most widely used approaches to obtain PTASs on planar graphs are the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained EPTASs for a multitude of problems. We unify the two strenghtened approaches to combine the best of both worlds. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n X is f(e). Here, OPT is the objective function value of the problem in question and f is a function depending only on e. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including cycle packing, vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no EPTASs on planar graphs were previously known

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

Linear kernels for outbranching problems in sparse digraphs

In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are given a directed graph $D$ with a designated root $r$ and a nonnegative integer $k$. The question is to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $O(k^2)$ vertices are known on general graphs. In this work we show that $k$-Leaf Out-Branching admits a kernel with $O(k)$ vertices on $\mathcal{H}$-minor-free graphs, for any fixed family of graphs $\mathcal{H}$, whereas $k$-Internal Out-Branching admits a kernel with $O(k)$ vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page