Computer Science

Linear algebra algorithms often require some sort of iteration or recursion as is illustrated by standard algorithms for Gaussian elimination, matrix inversion, and transitive closure. A key characteristic shared by these algorithms is that they allow looping for a number of steps that is bounded by the matrix dimension. In this paper we extend the matrix query language MATLANG with this type of recursion, and show that this suffices to express classical linear algebra algorithms. We study the expressive power of this language and show that it naturally corresponds to arithmetic circuit families, which are often said to capture linear algebra. Furthermore, we analyze several sub-fragments of our language, and show that their expressive power is closely tied to logical formalisms on semiring-annotated relations.

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces R n . These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X . We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x) and Dim(x) to individual points x∈X --to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages E 0 , E 1 , E 2 ,… used to construct a self-similar fractal E in the plane are elements of the hyperspace of the plane, and they converge to E in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets E that are analytic, i.e., Σ 1 1 , a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.

Is Fully Polynomial-time Randomized Approximation Scheme (FPRAS) for a problem via an MCMC algorithm possible when it is known that rapid mixing provably fails? We introduce several weight-preserving maps for the eight-vertex model on planar and on bipartite graphs, respectively. Some are one-to-one, while others are holographic which map superpositions of exponentially many states from one setting to another, in a quantum-like many-to-many fashion. In fact we introduce a set of such mappings that forms a group in each case. Using some holographic maps and their compositions we obtain FPRAS for the eight-vertex model at parameter settings where it is known that rapid mixing provably fails due to an intrinsic barrier. This FPRAS is indeed the same MCMC algorithm, except its state space corresponds to superpositions of the given states, where rapid mixing holds. FPRAS is also given for torus graphs for parameter settings where natural Markov chains are known to mix torpidly. Our results show that the eight-vertex model is the first problem with the provable property that while NP-hard to approximate on general graphs (even #P-hard for planar graphs in exact complexity), it possesses FPRAS on both bipartite graphs and planar graphs in substantial regions of its parameter space.

A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates is not larger than 1. The Hamiltonian path (cycle) problem is to determine whether a graph contains a simple path (cycle) in which each vertex of the graph appears exactly once. This problem is NP-complete for general graphs and it is also NP-complete for general supergrid graphs. Despite the many applications of the problem, it is still open for many classes, including solid supergrid graphs and supergrid graphs with some holes. A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In this paper, first we will study the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of rectangular supergrid graphs with a rectangular hole. Next, we will show that C-shaped supergrid graphs are Hamiltonian connected except few conditions. Finally, we will compute a longest path between two distinct vertices in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory's core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.

We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph of size n and one wants to recover this planted clique. For k=Ω( n − − √ ) , polynomial time algorithms can find the planted clique. The fastest such algorithms run in time linear O( n 2 ) (or nearly linear) in the size of the input [FR10,DGGP14,DM15a]. In this work, we initiate the development of sublinear time algorithms that find the planted clique when k=ω( nloglogn − − − − − − − − √ ) . Our algorithms can recover the clique in time O ˜ (n+( n k ) 3 )= O ˜ ( n 3 2 ) when k=Ω( nlogn − − − − − − √ ) , and in time O ˜ ( n 2 /exp( k 2 24n )) for ω( nloglogn − − − − − − − − √ )=k=o( nlogn − − − − − − √ ) . An Ω(n) running time lower bound for the planted clique recovery problem follows easily from the results of [RS19] and therefore our recovery algorithms are optimal whenever k=Ω( n 2 3 ) . As the lower bound of [RS19] builds on purely information theoretic arguments, it cannot provide a detection lower bound stronger than Ω ˜ ( n 2 k 2 ) . Since our algorithms for k=Ω( nlogn − − − − − − √ ) run in time O ˜ ( n 3 k 3 +n) , we show stronger lower bounds based on computational hardness assumptions. With a slightly different notion of the planted clique problem we show that the Planted Clique Conjecture implies the following. A natural family of non-adaptive algorithms---which includes our algorithms for clique detection---cannot reliably solve the planted clique detection problem in time O( n 3−δ k 3 ) for any constant δ>0 . Thus we provide evidence that if detecting small cliques is hard, it is also likely that detecting large cliques is not \textit{too} easy.

The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of bounded depth in this work. For them a classification splitting the types in easy (strongly sub-quadratic) and hard (essentially quadratic time under SETH) is known. We take a very fine-grained look at the hard pattern types from this classification and show a dichotomy: few types allow super-poly-logarithmic improvements while the algorithms for the other pattern types can only be improved by a constant number of log-factors, assuming the Formula-SAT Hypothesis.

For graphs G and H , a \emph{homomorphism} from G to H is an edge-preserving mapping from the vertex set of G to the vertex set of H . For a fixed graph H , by \textsc{Hom( H )} we denote the computational problem which asks whether a given graph G admits a homomorphism to H . If H is a complete graph with k vertices, then \textsc{Hom( H )} is equivalent to the k -\textsc{Coloring} problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that \textsc{Hom( H )} is polynomial-time solvable if H is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Nešetřil, JCTB 1990]. In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph G . If G has n vertices and is given along with its tree decomposition of width tw(G) , then the problem can be solved in time |V(H) | tw(G) ⋅ n O(1) , using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if H is a \emph{projective core}, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the \textsc{Hom( H )} problem cannot be solved in time (|V(H)|−ϵ ) tw(G) ⋅ n O(1) , for any ϵ>0 . This result provides a full complexity characterization for a large class of graphs H , as almost all graphs are projective cores. We also notice that the naive algorithm can be improved for some graphs H , and show a complexity classification for all graphs H , assuming two conjectures from algebraic graph theory. In particular, there are no known graphs H which are not covered by our result. In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.

For graphs G,H , a homomorphism from G to H is an edge-preserving mapping from V(G) to V(H) . In the list homomorphism problem, denoted by \textsc{LHom}( H ), we are given a graph G and lists L:V(G)→ 2 V(H) , and we ask for a homomorphism from G to H which additionally respects the lists L . Very recently Okrasa, Piecyk, and Rzążewski [ESA 2020] defined an invariant i ∗ (H) and proved that under the SETH O ∗ ( i ∗ (H ) tw(G) ) is the tight complexity bound for \textsc{LHom}( H ), parameterized by the treewidth tw(G) of the instance graph G . We study the complexity of the problem under dirretent parameterizations. As the first result, we show that i ∗ (H) is also the right complexity base if the parameter is the size of a minimum feedback vertex set of G . Then we turn our attention to a parameterization by the cutwidth ctw(G) of G . Jansen and Nederlof~[ESA 2018] showed that \textsc{List k -Coloring} (i.e., \textsc{LHom}( K k )) can be solved in time O ∗ ( c ctw(G) ) where c does not depend on k . Jansen asked if this behavior extends to graph homomorphisms. As the main result of the paper, we answer the question in the negative. We define a new graph invariant mi m ∗ (H) and prove that \textsc{LHom}( H ) problem cannot be solved in time O ∗ ((mi m ∗ (H)−ε ) ctw(G) ) for any ε>0 , unless the SETH fails. This implies that there is no c , such that for every odd cycle the non-list version of the problem can be solved in time O ∗ ( c ctw(G) ) . Finally, we generalize the algorithm of Jansen and Nederlof, so that it can be used to solve \textsc{LHom}( H ) for every graph H ; its complexity depends on ctw(G) and another invariant of H , which is constant for cliques.

The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].