New complexity results for the k-covers problem

The k-covers problem (kCP) asks us to compute a minimum cardinality set of stringsof given length k > 1 that covers a given string. It was shown in a recent paper, by reduction to 3-SAT, that the k-covers problem is NP-complete. In this paper we introduce a new problem, that we call the k-Bounded Relaxed Vertex Cover Problem (RVCPk), which we show is equivalent to k-Bounded Set Cover (SCPk). We show further that kCP is a special case of RVCPk restricted to certain classes Gx,k of graphs that represent all strings x. Thus a minimum k-cover can be approximated to within a factor k in polynomial time. We discuss approximate solutions of kCP, and we state a number of conjectures and open problems related to kCP and Gx,k

Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs)

Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables

We study the structure and learnability of sums of independent integer random variables (SIIRVs). For $k \in \mathbb{Z}_{+}$, a $k$-SIIRV of order $n \in \mathbb{Z}_{+}$ is the probability distribution of the sum of $n$ independent random variables each supported on $\{0, 1, \dots, k-1\}$. We denote by ${\cal S}_{n,k}$ the set of all $k$-SIIRVs of order $n$. In this paper, we tightly characterize the sample and computational complexity of learning $k$-SIIRVs. More precisely, we design a computationally efficient algorithm that uses $\widetilde{O}(k/\epsilon^2)$ samples, and learns an arbitrary $k$-SIIRV within error $\epsilon,$ in total variation distance. Moreover, we show that the {\em optimal} sample complexity of this learning problem is $\Theta((k/\epsilon^2)\sqrt{\log(1/\epsilon)}).$ Our algorithm proceeds by learning the Fourier transform of the target $k$-SIIRV in its effective support. Its correctness relies on the {\em approximate sparsity} of the Fourier transform of $k$-SIIRVs -- a structural property that we establish, roughly stating that the Fourier transform of $k$-SIIRVs has small magnitude outside a small set. Along the way we prove several new structural results about $k$-SIIRVs. As one of our main structural contributions, we give an efficient algorithm to construct a sparse {\em proper} $\epsilon$-cover for ${\cal S}_{n,k},$ in total variation distance. We also obtain a novel geometric characterization of the space of $k$-SIIRVs. Our characterization allows us to prove a tight lower bound on the size of $\epsilon$-covers for ${\cal S}_{n,k}$, and is the key ingredient in our tight sample complexity lower bound. Our approach of exploiting the sparsity of the Fourier transform in distribution learning is general, and has recently found additional applications.Comment: Main differences from v1: Changed title and restructured introduction. Added new sample optimal algorithm. Generalized sample lower bound for any value of

From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP

Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idem potent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semi complete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its Pi2-restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Message passing for vertex covers

Constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints. In this paper, we develop and analyze message passing techniques, namely warning and survey propagation, which serve as efficient heuristic algorithms for solving these computational hard problems. We show also, how previously obtained results on the typical-case behavior of vertex covers of random graphs can be recovered starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR

Almost Settling the Hardness of Noncommutative Determinant

In this paper, we study the complexity of computing the determinant of a matrix over a non-commutative algebra. In particular, we ask the question, "over which algebras, is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following hardness and easiness of noncommutative determinant computation. * [Hardness] Computing the determinant of an n \times n matrix whose entries are themselves 2 \times 2 matrices over a field is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, who proved a similar result which however required the entries to be of linear dimension. * [Easiness] Determinant of an n \times n matrix whose entries are themselves d \times d upper triangular matrices can be computed in poly(n^d) time. Combining the above with the decomposition theorem of finite dimensional algebras (in particular exploiting the simple structure of 2 \times 2 matrix algebras), we can extend the above hardness and easiness statements to more general algebras as follows. Let A be a finite dimensional algebra over a finite field with radical R(A). * [Hardness] If the quotient A/R(A) is non-commutative, then computing the determinant over the algebra A is as hard as computing the permanent. * [Easiness] If the quotient A/R(A) is commutative and furthermore, R(A) has nilpotency index d (i.e., the smallest d such that R(A)d = 0), then there exists a poly(n^d)-time algorithm that computes determinants over the algebra A. In particular, for any constant dimensional algebra A over a finite field, since the nilpotency index of R(A) is at most a constant, we have the following dichotomy theorem: if A/R(A) is commutative, then efficient determinant computation is feasible and otherwise determinant is as hard as permanent.Comment: 20 pages, 3 figure