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Register a new userLinear Time Parameterized Algorithms via Skew-Symmetric Multicuts
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Added by
Ramanujan M. S.
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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A skew-symmetric graph $(D=(V,A),sigma)$ is a directed graph $D$ with an involution $sigma$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $Xsubseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $sigma(v)$ are in different connected components of $D=(V,Asetminus (Xcup sigma(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4ell)$ and we obtain algorithms for {sc Odd Cycle Transversal} and {sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $ell$ being the length of the input formula.
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In the Subset Sum problem we are given a set of $n$ positive integers $X$ and a target $t$ and are asked whether some subset of $X$ sums to $t$. Natural parameters for this problem that have been studied in the literature are $n$ and $t$ as well as the maximum input number $rm{mx}_X$ and the sum of all input numbers $Sigma_X$. In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in $n$. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time $n^{O(1)}$. Our main question is: When can dense Subset Sum be solved in near-linear time $tilde{O}(n)$? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters $n,t,rm{mx}_X,Sigma_X$ for which dense Subset Sum is in time $tilde{O}(n)$. For notational convenience we assume without loss of generality that $t ge rm{mx}_X$ (as larger numbers can be ignored) and $t le Sigma_X/2$ (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP91], we show that Subset Sum is in near-linear time $tilde{O}(n)$ if $t gg rm{mx}_X Sigma_X/n^2$. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with $t ll rm{mx}_X Sigma_X/n^2$, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.
We study the algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. We identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^{O(sqrt{k}log k)}cdot n^{O(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.
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 respect 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.
It has long been known that Feedback Vertex Set can be solved in time $2^{mathcal{O}(wlog w)}n^{mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{mathcal{O}(w)}n^{mathcal{O}(1)}$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class $mathcal{P}$ of graphs, the Bounded $mathcal{P}$-Block Vertex Deletion problem asks, given a graph~$G$ on $n$ vertices and positive integers~$k$ and~$d$, whether $G$ contains a set~$S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $mathcal{P}$. Assuming that $mathcal{P}$ is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of $d$: if $mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{mathcal{O}(wd^2)} n^{mathcal{O}(1)}$, and if $mathcal{P}$ contains a graph with an induced cycle of length $ellge 4$, then the problem is not solvable in time $2^{o(wlog w)} n^{mathcal{O}(1)}$ even for fixed $d=ell$, unless the ETH fails. We also study a similar problem, called Bounded $mathcal{P}$-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if $d$ is part of the input and $mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.
In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain faster algorithms for variety of linear algebraic problems. Key results include: $bullet$ A nearly linear time algorithm for solving the inverse of symmetric $M$-matrices, a strict superset of Laplacians and SDD matrices. $bullet$ An $tilde{O}(n^2)$ time algorithm for solving $n times n$ linear systems that are constant spectral approximations of Laplacians or more generally, SDD matrices. $bullet$ An $tilde{O}(n^2)$ algorithm to recover a spectral approximation of a $n$-vertex graph using only $tilde{O}(1)$ matrix-vector multiplies with its Laplacian matrix. The previous best results for each problem either used a trivial number of queries to exactly recover the matrix or a trivial $O(n^omega)$ running time, where $omega$ is the matrix multiplication constant. We achieve these results by generalizing recent semidefinite programming based linear sized sparsifier results of Lee and Sun (2017) and providing iterative methods inspired by the semistreaming sparsification results of Kapralov, Lee, Musco, Musco and Sidford (2014) and input sparsity time linear system solving results of Li, Miller, and Peng (2013). We hope that by initiating study of these natural problems, expanding the robustness and scope of recent nearly linear time linear system solving research, and providing general matrix recovery machinery this work may serve as a stepping stone for faster algorithms.
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