Branchwidth determines how graphs, and more generally, arbitrary connectivity (basically symmetric and submodular) functions could be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable (FPT) 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations could decrease the width of a branch decomposition or that the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework. $\bullet$ An algorithm that for a given $n$-vertex graph $G$ and integer $k$ in time $2^{2^{O(k)}} n^2$ either constructs a rank decomposition of $G$ of width at most $2k$ or concludes that the rankwidth of $G$ is more than $k$. It also yields a $(2^{2k+1}-1)$-approximation algorithm for cliquewidth within the same time complexity, which in turn, improves to $f(k)n^2$ the running times of various algorithms on graphs of cliquewidth $k$. Breaking the "cubic barrier" for rankwidth and cliquewidth was an open problem in the area. $\bullet$ An algorithm that for a given $n$-vertex graph $G$ and integer $k$ in time $2^{O(k)} n$ either constructs a branch decomposition of $G$ of width at most $2k$ or concludes that the branchwidth of $G$ is more than $k$. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].
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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 simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is $k$, we show how to find a decomposition of cost $k^{O(1)}$ in time $f(k) \cdot n^{O(1)}$. This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.
Let $\Pi$ be a hereditary graph class. The problem of deletion to $\Pi$, takes as input a graph $G$ and asks for a minimum number (or a fixed integer $k$) of vertices to be deleted from $G$ so that the resulting graph belongs to $\Pi$. This is a well-studied problem in paradigms including approximation and parameterized complexity. Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether $k$ vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set. In this paper, we do a deep dive on pairs of specific graph classes ($\Pi_1, \Pi_2$) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. We design a general FPT algorithm and approximation algorithm for pairs of graph classes (possibly having infinite forbidden sets) satisfying certain conditions. These algorithms cover several pairs of popular graph classes. Our algorithm makes non-trivial use of the branching technique and as a black box, FPT algorithms for deletion to individual graph classes.
We consider Broyden's method and some accelerated schemes for nonlinear equations having a strongly regular singularity of first order with a one-dimensional nullspace. Our two main results are as follows. First, we show that the use of a preceding Newton-like step ensures convergence for starting points in a starlike domain with density 1. This extends the domain of convergence of these methods significantly. Second, we establish that the matrix updates of Broyden's method converge q-linearly with the same asymptotic factor as the iterates. This contributes to the long-standing question whether the Broyden matrices converge by showing that this is indeed the case for the setting at hand. Furthermore, we prove that the Broyden directions violate uniform linear independence, which implies that existing results for convergence of the Broyden matrices cannot be applied. Numerical experiments of high precision confirm the enlarged domain of convergence, the q-linear convergence of the matrix updates, and the lack of uniform linear independence. In addition, they suggest that these results can be extended to singularities of higher order and that Broyden's method can converge r-linearly without converging q-linearly. The underlying code is freely available.
There are many applications of max flow with capacities that depend on one or more parameters. Many of these applications fall into the "Source-Sink Monotone" framework, a special case of Topkis's monotonic optimization framework, which implies that the parametric min cuts are nested. When there is a single parameter, this property implies that the number of distinct min cuts is linear in the number of nodes, which is quite useful for constructing algorithms to identify all possible min cuts. When there are multiple Source-Sink Monotone parameters and the vector of parameters are ordered in the usual vector sense, the resulting min cuts are still nested. However, the number of distinct min cuts was an open question. We show that even with only two parameters, the number of distinct min cuts can be exponential in the number of nodes.
In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters $d$-pathwidth and clique preserving $d$-pathwidth denoted by $d-pw(G)$ and $d-cpw(G)$ where $G$ is a graph. We show that $2-cpw(G) \leq O(tw(G) \Delta(G))$ where $tw(G)$ and $\Delta(G)$ are, respectively the treewidth and maximal degree of $G$. Using this upper bound, we demonstrate that each CNF $\psi$ can be represented as a conjunction of two OBDDs of size $2^{O(\Delta(\psi)*tw(\psi)^2)}$ where $tw(\psi)$ is the treewidth of the primal graph of $\psi$ and each variable occurs in $\psi$ at most $\Delta(\psi)$ times. Next we use $d$-pathwdith to obtain lower bounds for monotone branching programs. In particular, we consider the monotone version of syntactic nondeterministic read $d$ times branching programs (just forbidding negative literals as edge labels) and introduce a further restriction that each computational path can be partitioned into at most $d$ read-once subpaths. We call the resulting model separable monotone read $d$ times branching programs and abbreviate them $d$-SMNBPs. For each graph $G$ without isolated vertices, we introduce a CNF $\psi(G)$ whsose clauses are $(u \vee e \vee v)$ for each edge $e=\{u,v\}$ of $G$. We prove that a $d$-SMNBP representing $\psi(G)$ is of size at least $\Omega(c^{d-pw(G)})$ where $c=(8/7)^{1/12}$. We use this 'generic' lower bound to obtain an exponential lower bound for a 'concrete' class of CNFs $\psi(K_n)$. In particular, we demonstrate that for each $0<a<1$, the size of $n^{a}$-SMNBP representing $\psi(K_n)$ is at least $c^{n^b}$ where $b$ is an arbitrary constant such that $a+b<1$. This lower bound is tight in the sense $\psi(K_n)$ can be represented by a poly-sized $n$-SMNBP.
We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices which are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the matrix pencil method is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the matrix pencil method but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the matrix pencil method for noisy data and for signal approximation by short exponential sums.
Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.
The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.
In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve strictly a wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms to theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios and that no GNN can perform better than with these ratios. This paper is the first to elucidate approximation ratios of GNNs for combinatorial problems. Furthermore, we prove that adding coloring or weak-coloring to each node feature improves these approximation ratios. This indicates that preprocessing and feature engineering theoretically strengthen model capabilities.
Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.
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