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Register a new userThe Complexity of Mixed-Connectivity
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Added by
Sergio Cabello
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We investigate the parameterized complexity in $a$ and $b$ of determining whether a graph~$G$ has a subset of $a$ vertices and $b$ edges whose removal disconnects $G$, or disconnects two prescribed vertices $s, t in V(G)$.
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Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum $s$-$t$ cut, the minimum multiway cut, and the minimum $k$-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $r$-dimensional simplices of minimum cardinality so that $mathcal{S}$ meets every cycle homologous to $zeta$. Our main result is that this problem admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size $k$. On the positive side, we show that THS admits an FPT algorithm with respect to $k+d$, where $d$ is the maximum degree of the Hasse graph of the complex $mathsf{K}$. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $(r+1)$-dimensional simplices of minimum cardinality so that the removal of $mathcal{S}$ from $mathsf{K}$ makes $zeta$ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an $O(log n)$-approximation FPT algorithm for $(r+1)$-dimensional complexes with the $(r+1)$-th Betti number $beta_{r+1}$ as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.
We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$, one can query a vector $x in {0,1}^n$ and receive the answer $Ax$. We give a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after $O(log^4(n))$ matrix-vector multiplication queries to the adjacency matrix. This complements a result of Sun et al. (ICALP 2019) that gives a randomized algorithm that can output a spanning forest of a graph after $O(log^4(n))$ matrix-vector multiplication queries to the signed vertex-edge incidence matrix of the graph. As an application, we show that a quantum algorithm can output a spanning forest of an unweighted graph after $O(log^5(n))$ cut queries, improving and simplifying a result of Lee, Santha, and Zhang (SODA 2021), which gave the bound $O(log^8(n))$. In the second part of the paper, we turn to showing lower bounds on the linear query complexity of determining if a graph is connected. If $w$ is the weight vector of a graph (viewed as an $binom{n}{2}$ dimensional vector), in a linear query one can query any vector $z in mathbb{R}^{n choose 2}$ and receive the answer $langle z, wrangle$. We show that a zero-error randomized algorithm must make $Omega(n)$ linear queries in expectation to solve connectivity. As far as we are aware, this is the first lower bound of any kind on the unrestricted linear query complexity of connectivity. We show this lower bound by looking at the linear query emph{certificate complexity} of connectivity, and characterize this certificate complexity in a linear algebraic fashion.
We study the space complexity of solving the bias-regularized SVM problem in the streaming model. This is a classic supervised learning problem that has drawn lots of attention, including for developing fast algorithms for solving the problem approximately. One of the most widely used algorithms for approximately optimizing the SVM objective is Stochastic Gradient Descent (SGD), which requires only $O(frac{1}{lambdaepsilon})$ random samples, and which immediately yields a streaming algorithm that uses $O(frac{d}{lambdaepsilon})$ space. For related problems, better streaming algorithms are only known for smooth functions, unlike the SVM objective that we focus on in this work. We initiate an investigation of the space complexity for both finding an approximate optimum of this objective, and for the related ``point estimation problem of sketching the data set to evaluate the function value $F_lambda$ on any query $(theta, b)$. We show that, for both problems, for dimensions $d=1,2$, one can obtain streaming algorithms with space polynomially smaller than $frac{1}{lambdaepsilon}$, which is the complexity of SGD for strongly convex functions like the bias-regularized SVM, and which is known to be tight in general, even for $d=1$. We also prove polynomial lower bounds for both point estimation and optimization. In particular, for point estimation we obtain a tight bound of $Theta(1/sqrt{epsilon})$ for $d=1$ and a nearly tight lower bound of $widetilde{Omega}(d/{epsilon}^2)$ for $d = Omega( log(1/epsilon))$. Finally, for optimization, we prove a $Omega(1/sqrt{epsilon})$ lower bound for $d = Omega( log(1/epsilon))$, and show similar bounds when $d$ is constant.
A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such that the largest strong connectivity component in $D-X$ has at most $ell$ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $ell=1.$ We study parametered complexity of DCOC for general and semicomplete digraphs with the following parameters: $k, ell,ell+k$ and $n-ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$ unless the Exponential Time Hypothesis (ETH) fails. gutin{The upper bound $O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-ell)})$ for the parameter $n-ell.$ We complement the latter by showing that there is no algorithm of time complexity $O^*(2^{o({n-ell})})$ unless ETH fails.} Finally, we improve viol{(in dependency on $ell$)} the upper bound of G{{o}}ke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $ell+k$ on general digraphs from $O^*(2^{O(kelllog (kell))})$ to $O^*(2^{O(klog (kell))}).$ Note that Drange, Dregi and van t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $O^*(2^{o(klog ell)})$ unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $O^*(2^{o(klog k)}).$
We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(log log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($n log n$ random bits are used) and time optimal ($log log n$ rounds are used). This improves the time complexity of previous results in the same setting by a $log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes its time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach it as needed otherwise).
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