The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input, assuming the SETH. Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.
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The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the information-theoretic method, Gilmer \cite{gilmer2022constant} recently showed that there exists an element contained in at least a proportion $0.01$ of the sets of such $\mathcal{F}$. He conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}\approx0.38197$ which was subsequently confirmed by several researchers \cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}. Furthermore, Sawin \cite{sawin2022improved} showed that Gilmer's technique can be improved to obtain a bound better than $\frac{3-\sqrt{5}}{2}$. This paper further improves Gilmer's technique to derive new bounds in the optimization form for the union-closed sets conjecture. These bounds include Sawin's improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin's improvement computable, and then evaluate it numerically which yields a bound around $0.38234$, slightly better than $\frac{3-\sqrt{5}}{2}$.
We study the fine-grained complexity of graph connectivity problems in unweighted undirected graphs. Recent development shows that all variants of edge connectivity problems, including single-source-single-sink, global, Steiner, single-source, and all-pairs connectivity, are solvable in $m^{1+o(1)}$ time, collapsing the complexity of these problems into the almost-linear-time regime. While, historically, vertex connectivity has been much harder, the recent results showed that both single-source-single-sink and global vertex connectivity can be solved in $m^{1+o(1)}$ time, raising the hope of putting all variants of vertex connectivity problems into the almost-linear-time regime too. We show that this hope is impossible, assuming conjectures on finding 4-cliques. Moreover, we essentially settle the complexity landscape by giving tight bounds for combinatorial algorithms in dense graphs. There are three separate regimes: (1) all-pairs and Steiner vertex connectivity have complexity $\hat{\Theta}(n^{4})$, (2) single-source vertex connectivity has complexity $\hat{\Theta}(n^{3})$, and (3) single-source-single-sink and global vertex connectivity have complexity $\hat{\Theta}(n^{2})$. For graphs with general density, we obtain tight bounds of $\hat{\Theta}(m^{2})$, $\hat{\Theta}(m^{1.5})$, $\hat{\Theta}(m)$, respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time.
We consider a simple one-way averaging protocol on graphs. Initially, every node of the graph has a value. A node $u$ is chosen uniformly at random and $u$ samples $k$ neighbours $v_1,v_2,\cdots, v_k \in N(u)$ uniformly at random. Then, $u$ averages its value with $v$ as follows: $\xi_u(t+1) = \alpha \xi_u(t) + \frac{(1-\alpha)}{k} \sum_{i=1}^k \xi_{v_i}(t)$ for some $\alpha \in (0,1)$, where $\xi_u(t)$ is the value of node $u$ at time $t$. Note that, in contrast to neighbourhood value balancing, only $u$ changes its value. Hence, the sum (and the average) of the values of all nodes changes over time. Our results are two-fold. First, we show a bound on the convergence time (the time it takes until all values are roughly the same) that is asymptotically tight for some initial assignments of values to the nodes. Our second set of results concerns the ability of this protocol to approximate well the initial average of all values: we bound the probability that the final outcome is significantly away from the initial average. Interestingly, the variance of the outcome does not depend on the graph structure. The proof introduces an interesting generalisation of the duality between coalescing random walks and the voter model.
Graph neural networks (GNNs) are widely used for modeling complex interactions between entities represented as vertices of a graph. Despite recent efforts to theoretically analyze the expressive power of GNNs, a formal characterization of their ability to model interactions is lacking. The current paper aims to address this gap. Formalizing strength of interactions through an established measure known as separation rank, we quantify the ability of certain GNNs to model interaction between a given subset of vertices and its complement, i.e. between sides of a given partition of input vertices. Our results reveal that the ability to model interaction is primarily determined by the partition's walk index -- a graph-theoretical characteristic that we define by the number of walks originating from the boundary of the partition. Experiments with common GNN architectures corroborate this finding. As a practical application of our theory, we design an edge sparsification algorithm named Walk Index Sparsification (WIS), which preserves the ability of a GNN to model interactions when input edges are removed. WIS is simple, computationally efficient, and markedly outperforms alternative methods in terms of induced prediction accuracy. More broadly, it showcases the potential of improving GNNs by theoretically analyzing the interactions they can model.
We consider the problem of testing whether a single coefficient is equal to zero in high-dimensional fixed-design linear models. In the high-dimensional setting where the dimension of covariates $p$ is allowed to be in the same order of magnitude as sample size $n$, to achieve finite-population validity, existing methods usually require strong distributional assumptions on the noise vector (such as Gaussian or rotationally invariant), which limits their applications in practice. In this paper, we propose a new method, called \emph{residual permutation test} (RPT), which is constructed by projecting the regression residuals onto the space orthogonal to the union of the column spaces of the original and permuted design matrices. RPT can be proved to achieve finite-population size validity under fixed design with just exchangeable noises, whenever $p < n / 2$. Moreover, RPT is shown to be asymptotically powerful for heavy tailed noises with bounded $(1+t)$-th order moment when the true coefficient is at least of order $n^{-t/(1+t)}$ for $t \in [0,1]$. We further proved that this signal size requirement is essentially optimal in the minimax sense. Numerical studies confirm that RPT performs well in a wide range of simulation settings with normal and heavy-tailed noise distributions.
This paper pushes further the intrinsic capabilities of the GFEM$^{gl}$ global-local approach introduced initially in [1]. We develop a distributed computing approach using MPI (Message Passing Interface) both for the global and local problems. Regarding local problems, a specific scheduling strategy is introduced. Then, to measure correctly the convergence of the iterative process, we introduce a reference solution that revisits the product of classical and enriched functions. As a consequence, we are able to propose a purely matrix-based implementation of the global-local problem. The distributed approach is then compared to other parallel solvers either direct or iterative with domain decomposition. The comparison addresses the scalability as well as the elapsed time. Numerical examples deal with linear elastic problems: a polynomial exact solution problem, a complex micro-structure, and, finally, a pull-out test (with different crack extent). 1: C. A. Duarte, D.-J. Kim, and I. Babu\v{s}ka. A global-local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks. In Advances in Meshfree Techniques, Dordrecht, 2007. Springer
Sublinear time algorithms for approximating maximum matching size have long been studied. Much of the progress over the last two decades on this problem has been on the algorithmic side. For instance, an algorithm of Behnezhad [FOCS'21] obtains a 1/2-approximation in $\tilde{O}(n)$ time for $n$-vertex graphs. A more recent algorithm by Behnezhad, Roghani, Rubinstein, and Saberi [SODA'23] obtains a slightly-better-than-1/2 approximation in $O(n^{1+\epsilon})$ time. On the lower bound side, Parnas and Ron [TCS'07] showed 15 years ago that obtaining any constant approximation of maximum matching size requires $\Omega(n)$ time. Proving any super-linear in $n$ lower bound, even for $(1-\epsilon)$-approximations, has remained elusive since then. In this paper, we prove the first super-linear in $n$ lower bound for this problem. We show that at least $n^{1.2 - o(1)}$ queries in the adjacency list model are needed for obtaining a $(\frac{2}{3} + \Omega(1))$-approximation of maximum matching size. This holds even if the graph is bipartite and is promised to have a matching of size $\Theta(n)$. Our lower bound argument builds on techniques such as correlation decay that to our knowledge have not been used before in proving sublinear time lower bounds. We complement our lower bound by presenting two algorithms that run in strongly sublinear time of $n^{2-\Omega(1)}$. The first algorithm achieves a $(\frac{2}{3}-\epsilon)$-approximation; this significantly improves prior close-to-1/2 approximations. Our second algorithm obtains an even better approximation factor of $(\frac{2}{3}+\Omega(1))$ for bipartite graphs. This breaks the prevalent $2/3$-approximation barrier and importantly shows that our $n^{1.2-o(1)}$ time lower bound for $(\frac{2}{3}+\Omega(1))$-approximations cannot be improved all the way to $n^{2-o(1)}$.
We study the question of when we can provide direct access to the k-th answer to a Conjunctive Query (CQ) according to a specified order over the answers in time logarithmic in the size of the database, following a preprocessing step that constructs a data structure in time quasilinear in database size. Specifically, we embark on the challenge of identifying the tractable answer orderings, that is, those orders that allow for such complexity guarantees. To better understand the computational challenge at hand, we also investigate the more modest task of providing access to only a single answer (i.e., finding the answer at a given position), a task that we refer to as the selection problem, and ask when it can be performed in quasilinear time. We also explore the question of when selection is indeed easier than ranked direct access. We begin with lexicographic orders. For each of the two problems, we give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orders for every CQ without self-joins. We then continue to the more general orders by the sum of attribute weights and establish the corresponding decidable characterizations, for each of the two problems, of the tractable CQs without self-joins. Finally, we explore the question of when the satisfaction of Functional Dependencies (FDs) can be utilized for tractability, and establish the corresponding generalizations of our characterizations for every set of unary FDs.
Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.
The aim of this work is to develop a fully-distributed algorithmic framework for training graph convolutional networks (GCNs). The proposed method is able to exploit the meaningful relational structure of the input data, which are collected by a set of agents that communicate over a sparse network topology. After formulating the centralized GCN training problem, we first show how to make inference in a distributed scenario where the underlying data graph is split among different agents. Then, we propose a distributed gradient descent procedure to solve the GCN training problem. The resulting model distributes computation along three lines: during inference, during back-propagation, and during optimization. Convergence to stationary solutions of the GCN training problem is also established under mild conditions. Finally, we propose an optimization criterion to design the communication topology between agents in order to match with the graph describing data relationships. A wide set of numerical results validate our proposal. To the best of our knowledge, this is the first work combining graph convolutional neural networks with distributed optimization.
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