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Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem -- leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd-order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.

A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii) When the theory is empty, in which case every valid sequent admits a rational proof tree, where the total number of pairwise distinct sequents occurring in the proof tree is doubly exponential w.r.t.\ the size of the end-sequent. We also show that the validity problem is undecidable for a wide class of theories, even with a very low expressive power.

We study the problem of computing the vitality with respect to max flow of edges and vertices in undirected planar graphs, where the vitality of an edge/vertex in a graph with respect to max flow between two fixed vertices $s,t$ is defined as the max flow decrease when the edge/vertex is removed from the graph. We show that the vitality of any $k$ selected edges can be computed in $O(kn + n\log\log n)$ worst-case time, and that a $\delta$ additive approximation of the vitality of all edges with capacity at most $c$ can be computed in $O(\frac{c}{\delta}n +n \log \log n)$ worst-case time, where $n$ is the size of the graph. Similar results are given for the vitality of vertices. All our algorithms work in $O(n)$ space.

The expressive power of interval temporal logics (ITLs) makes them one of the most natural choices in a number of application domains, ranging from the specification and verification of complex reactive systems to automated planning. However, for a long time, because of their high computational complexity, they were considered not suitable for practical purposes. The recent discovery of several computationally well-behaved ITLs has finally changed the scenario. In this paper, we investigate the finite satisfiability and model checking problems for the ITL D, that has a single modality for the sub-interval relation, under the homogeneity assumption (that constrains a proposition letter to hold over an interval if and only if it holds over all its points). We first prove that the satisfiability problem for D, over finite linear orders, is PSPACE-complete, and then we show that the same holds for its model checking problem, over finite Kripke structures. In such a way, we enrich the set of tractable interval temporal logics with a new meaningful representative.

We contribute to fulfil the long-lasting gap in the understanding of the spatial search with multiple marked vertices. The theoretical framework is that of discrete-time quantum walks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. QW based search algorithms are well understood when they have to tackle the fundamental problem of finding only one marked element in a $d-$dimensional grid and it has been proven they provide a quadratic advantage over classical searching protocols. However, once we consider to search more than one element, the behaviour of the algorithm may be affected by the spatial configuration of the marked elements, due to the quantum interference among themselves and even the quantum advantage is no longer granted. Here our main contribution is twofold : (i) we provide strong numerical evidence that spatial configurations are almost all optimal; and (ii) we analytically prove that the quantum advantage with respect to the classical counterpart is not always granted and it does depend on the proportion of searched elements over the total number of grid points $\tau$. We finally providing a clear phase diagram for the QW search advantage with respect to the classical random algorithm.

Zero-free based algorithm is a major technique for deterministic approximate counting. In Barvinok's original framework[Bar17], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts[PR17] later gave a refinement of Barvinok's framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs. In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.

A family of sets is $(p,q)$-intersecting if every nonempty subfamily of $p$ or fewer sets has at least $q$ elements in its total intersection. A family of sets has the $(p,q)$-Helly property if every nonempty $(p,q)$-intersecting subfamily has total intersection of cardinality at least $q$. The $(2,1)$-Helly property is the usual Helly property. A hypergraph is $(p,q)$-Helly if its edge family has the $(p,q)$-Helly property and hereditary $(p,q)$-Helly if each of its subhypergraphs has the $(p,q)$-Helly property. A graph is $(p,q)$-clique-Helly if the family of its maximal cliques has the $(p,q)$-the Helly property and hereditary $(p,q)$-clique-Helly if each of its induced subgraphs is $(p,q)$-clique-Helly. The classes of $(p,q)$-biclique-Helly and hereditary $(p,q)$-biclique-Helly graphs are defined analogously. We prove several characterizations of hereditary $(p,q)$-Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. We give an improved time bound for the recognition of $(p,q)$-Helly hypergraphs for each fixed $q$ and show that the recognition of hereditary $(p,q)$-Helly hypergraphs can be solved in polynomial time if $p$ and $q$ are fixed but co-NP-complete if $p$ is part of the input. In addition, we generalize to $(p,q)$-clique-Helly graphs the characterization of $p$-clique-Helly graphs in terms of expansions and give different characterizations of hereditary $(p,q)$-clique-Helly graphs, including one by forbidden induced subgraphs. We give an improvement on the time bound for the recognition of $(p,q)$-clique-Helly graphs and prove that the recognition problem of hereditary $(p,q)$-clique-Helly graphs is polynomial-time solvable for $p$ and $q$ fixed but NP-hard if $p$ or $q$ is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (hereditary) $(p,q)$-biclique-Helly graphs.

In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees yet is challenging because variances are often not known a priori. Recently, considerable progress has been made by Zhang et al. (2021) where they obtain a variance-adaptive regret bound for linear bandits without knowledge of the variances and a horizon-free regret bound for linear mixture Markov decision processes (MDPs). In this paper, we present novel analyses that improve their regret bounds significantly. For linear bandits, we achieve $\tilde O(d^{1.5}\sqrt{\sum_{k}^K \sigma_k^2} + d^2)$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k^2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d^3$ improvement. For linear mixture MDPs with the assumption of maximum cumulative reward in an episode being in $[0,1]$, we achieve a horizon-free regret bound of $\tilde O(d \sqrt{K} + d^2)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d^{3.5}$ improvement in the leading term and $d^7$ in the lower order term. Our analysis critically relies on a novel elliptical potential `count' lemma. This lemma allows a novel regret analysis in conjunction with the peeling trick, which is of independent interest.

The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the $p$th power $G^p$ of $G$. This extends some characterizations of graphs with connected medians (case $p=1$) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any $p$. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $G^2$-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

The application of deep learning to symbolic domains remains an active research endeavour. Graph neural networks (GNN), consisting of trained neural modules which can be arranged in different topologies at run time, are sound alternatives to tackle relational problems which lend themselves to graph representations. In this paper, we show that GNNs are capable of multitask learning, which can be naturally enforced by training the model to refine a single set of multidimensional embeddings $\in \mathbb{R}^d$ and decode them into multiple outputs by connecting MLPs at the end of the pipeline. We demonstrate the multitask learning capability of the model in the relevant relational problem of estimating network centrality measures, i.e. is vertex $v_1$ more central than vertex $v_2$ given centrality $c$?. We then show that a GNN can be trained to develop a $lingua$ $franca$ of vertex embeddings from which all relevant information about any of the trained centrality measures can be decoded. The proposed model achieves $89\%$ accuracy on a test dataset of random instances with up to 128 vertices and is shown to generalise to larger problem sizes. The model is also shown to obtain reasonable accuracy on a dataset of real world instances with up to 4k vertices, vastly surpassing the sizes of the largest instances with which the model was trained ($n=128$). Finally, we believe that our contributions attest to the potential of GNNs in symbolic domains in general and in relational learning in particular.