Parameterized complexity of three edge contraction problems with degree constraints

For any graph class \(\mathcal{H}\) , the \(\mathcal{H}\) -Contraction problem takes as input a graph \(G\) and an integer \(k\) , and asks whether there exists a graph \(H\in \mathcal{H}\) such that \(G\) can be modified into \(H\) using at most \(k\) edge contractions. We study the parameterized complexity of \(\mathcal{H}\) -Contraction for three different classes \(\mathcal{H}\) : the class \(\mathcal{H}_{\le d}\) of graphs with maximum degree at most  \(d\) , the class \(\mathcal{H}_{=d}\) of \(d\) -regular graphs, and the class of \(d\) -degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters \(k\) , \(d\) , and \(d+k\) . Moreover, we show that \(\mathcal{H}\) -Contraction admits an \(O(k)\) vertex kernel on connected graphs when \(\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}\) , while the problem is \(\mathsf{W}[2]\) -hard when \(\mathcal{H}\) is the class of \(2\) -degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that \(\mathcal{H}\) -Contraction admits a linear vertex kernel when \(\mathcal{H}\) is the class of cycles.     

The categories L-\mathbf{Top}_{0} and L-\mathbf{Sob} as epireflective hulls

We show that the category \(L\) - \(\mathbf{Top}_{0}\) of \(T_{0}\) - \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}\) of \(L\) -topological spaces and the category \(L\) - \(\mathbf{Sob}\) of sober \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}_{0}\) .     

Nonmonotone Barzilai–Borwein Gradient Algorithm for \ell _{1}-Regularized Nonsmooth Minimization in Compressive Sensing

This study aims to minimize the sum of a smooth function and a nonsmooth \(\ell _{1}\) -regularized term. This problem as a special case includes the \(\ell _{1}\) -regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, and so on. However, the non-differentiability of the \(\ell _{1}\) -norm causes more challenges especially in large problems encountered in many practical applications. This study proposes, analyzes, and tests a Barzilai–Borwein gradient algorithm. At each iteration, the generated search direction demonstrates descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the \(\ell _{1}\) -norm. A nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where each iteration requiring the values of the objective function and the gradient of the smooth term. Under some conditions, the proposed algorithm appears globally convergent. The limited experiments using some nonconvex unconstrained problems from the CUTEr library with additive \(\ell _{1}\) -regularization illustrate that the proposed algorithm performs quite satisfactorily. Extensive experiments for \(\ell _{1}\) -regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms that have been specifically designed in recent years.     

Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\) , where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\) , inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\) , the derivative of the time- \(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\) . Our main result states that generically in the space of such systems, for sufficiently large \(N\) , all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\) -open and \(C^\infty \) -dense set of maps \(A\) , and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\) . We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.     

Join-Reachability Problems in Directed Graphs

For a given collection \(\mathcal{G}\) of directed graphs we define the join-reachability graph of \(\mathcal{G}\) , denoted by \(\mathcal{J}(\mathcal{G})\) , as the directed graph that, for any pair of vertices u and v, contains a path from u to v if and only if such a path exists in all graphs of  \(\mathcal{G}\) . Our goal is to compute an efficient representation of  \(\mathcal{J}(\mathcal{G})\) . In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for  \(\mathcal{G}\) . In the implicit version we wish to build an efficient data structure, in terms of space and query time, such that we can report fast the set of vertices that reach a query vertex in all graphs of  \(\mathcal{G}\) . This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problems. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.     

Quantum entropy-typical subspace and universal data compression

The quantum entropy-typical subspace theory is specified. It is shown that any \(\rho ^{\otimes n}\) with von Neumann \(\hbox {entropy}\le h\) can be preserved approximately by the entropy-typical subspace with \(\hbox {entropy}=h\) . This result implies an universal compression scheme for the case that the von Neumann entropy of the source does not exceed \(h\) .     

Isomorphisms and functors of fuzzy sets and cut systems

Any fuzzy set \(X\) in a classical set \(A\) with values in a complete (residuated) lattice \( Q\) can be identified with a system of \(\alpha \) -cuts \(X_{\alpha }\) , \(\alpha \in Q\) . Analogical results were proved for sets with similarity relations with values in \( Q\) (e.g. \( Q\) -sets), which are objects of two special categories \({\mathbf K}={Set}( Q)\) or \({SetR}( Q)\) of \( Q\) -sets, and for fuzzy sets defined as morphisms from an \( Q\) -set into a special \(Q\) -set \(( Q,\leftrightarrow )\) . These fuzzy sets can be defined equivalently as special cut systems \((C_{\alpha })_{\alpha }\) , called f-cuts. This equivalence then represents a natural isomorphism between covariant functor of fuzzy sets \(\mathcal{F}_{\mathbf K}\) and covariant functor of f-cuts \(\mathcal{C}_{\mathbf K}\) . In this paper, we prove that analogical natural isomorphism exists also between contravariant versions of these functors. We are also interested in relationships between sets of fuzzy sets and sets of f-cuts in an \(Q\) -set \((A,\delta )\) in the corresponding categories \({Set}( Q)\) and \({SetR}( Q)\) , which are endowed with binary operations extended either from binary operations in the lattice \(Q\) , or from binary operations defined in a set \(A\) by the generalized Zadeh’s extension principle. We prove that the resulting binary structures are (under some conditions) isomorphic.     

Informational correlation between two parties of a quantum system: spin-1/2 chains

We introduce the informational correlation \(E^{AB}\) between two interacting quantum subsystems \(A\) and \(B\) of a quantum system as the number of arbitrary parameters \(\varphi _i\) of a unitary transformation \(U^A\) (locally performed on the subsystem \(A\) ) which may be detected in the subsystem \(B\) by the local measurements. This quantity indicates whether the state of the subsystem \(B\) may be effected by means of the unitary transformation applied to the subsystem \(A\) . Emphasize that \(E^{AB}\ne E^{BA}\) in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem \(B\) . However, there is some non-reducible part of \(E^{AB}(t)\) which may not be decreased by any unitary transformation of the subsystem \(B\) at a fixed time instant \(t\) . Two examples of the informational correlations between two parties of the four-node spin-1/2 chain with mixed initial states are studied. The long chains with a single initially excited spin (the pure initial state) are considered as well.     

The Equivalence of Sampling and Searching

In a sampling problem, we are given an input x∈{0,1} ⁿ , and asked to sample approximately from a probability distribution \(\mathcal{D}_{x}\) over \(\operatorname{poly} ( n ) \) -bit strings. In a search problem, we are given an input x∈{0,1} ⁿ , and asked to find a member of a nonempty set A _x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity to show that sampling and search problems are “essentially equivalent.” More precisely, for any sampling problem S, there exists a search problem R _S such that, if \(\mathcal{C}\) is any “reasonable” complexity class, then R _S is in the search version of \(\mathcal{C}\) if and only if S is in the sampling version. What makes this nontrivial is that the same R _S works for every  \(\mathcal{C}\) . As an application, we prove the surprising result that SampP=SampBQP if and only if FBPP=FBQP. In other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve.     

Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis

Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen’s (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .     

On requirement verification for evolving Statecharts specifications

Software development processes have been evolving from rigid, pre-specified, and sequential to incremental, and iterative. This evolution has been dictated by the need to accommodate evolving user requirements and reduce the delay between design decision and feedback from users. Formal verification techniques, however, have largely ignored this evolution and even when they made enormous improvements and found significant uses in practice, like in the case of model checking, they remained confined into the niches of safety-critical systems. Model checking verifies if a system’s model \(\mathcal{M}\) satisfies a set of requirements, formalized as a set of logic properties \(\Phi\) . Current model-checking approaches, however, implicitly rely on the assumption that both the complete model \(\mathcal{M}\) and the whole set of properties \(\Phi\) are fully specified when verification takes place. Very often, however, \(\mathcal{M}\) is subject to change because its development is iterative and its definition evolves through stages of incompleteness, where alternative design decisions are explored, typically to evaluate some quality trade-offs. Evolving systems specifications of this kind ask for novel verification approaches that tolerate incompleteness and support incremental analysis of alternative designs for certain functionalities. This is exactly the focus of this paper, which develops an incremental model-checking approach for evolving Statecharts. Statecharts have been chosen both because they are increasingly used in practice natively support model refinements.     

Design patterns percolating to parallel programming framework implementation

Structured parallel programming is recognised as a viable and effective means of tackling parallel programming problems. Recently, a set of simple and powerful parallel building blocks ( \(\mathsf{RISC\text{- }pb^2l}\) ) has been proposed to support modelling and implementation of parallel frameworks. In this work we demonstrate how that same parallel building block set may be used to model both general purpose parallel programming abstractions, not usually listed in classical skeleton sets, and more specialized domain specific parallel patterns. We show how an implementation of \(\mathsf{RISC\text{- }pb^2l}\) can be realised via the FastFlow framework and present experimental evidence of the feasibility and efficiency of the approach.     

\frac{13}{9}-Approximation for Graphic TSP

The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of \(\frac{3}{2}\) , even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only \(\frac{4}{3}\) . Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of \(\frac{13}{9}\) on the approximation factor, as well as a bound of \(\frac{19}{12}+\varepsilon\) for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.     

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

The Induced Graph Matching problem asks to find \(k\) disjoint induced subgraphs isomorphic to a given graph  \(H\) in a given graph \(G\) such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in \(k\) on claw-free graphs when \(H\) is a fixed connected graph, and even admits a polynomial kernel when  \(H\) is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove \(\mathsf {W}[1]\) -hardness of Induced Graph Matching on graphs excluding \(K_{1,4}\) as an induced subgraph, for any fixed complete graph \(H\) . In particular, we show that Independent Set is \(\mathsf {W}[1]\) -hard on \(K_{1,4}\) -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or \(\mathsf {NP}\) -complete, depending on the connectivity of \(H\) and the structure of \(G\) .     

Greedy routing in small-world networks with power-law degrees

In this paper we study decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variant of J. Kleinberg’s grid-based small-world model in which (1) the number of long-range edges of each node is not fixed, but is drawn from a power-law probability distribution with exponent parameter \(\alpha \ge 0\) and constant mean, and (2) the long-range edges are considered to be bidirectional for the purposes of routing. This model is motivated by empirical observations indicating that several real networks have degrees that follow a power-law distribution. The measured power-law exponent \(\alpha \) for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when \(2 < \alpha < 3\) the standard greedy routing algorithm, in which a node forwards the message to its neighbor that is closest to the target in the grid, finishes in an expected number of \(O(\log ^{\alpha -1} n\cdot \log \log n)\) steps, for any source–target pair. This is asymptotically smaller than the \(O(\log ^2 n)\) steps needed in Kleinberg’s original model with the same average degree, and approaches \(O(\log n)\) as \(\alpha \) approaches 2. Further, we show that when \(0\le \alpha < 2\) or \(\alpha \ge 3\) the expected number of steps is \(O(\log ^2 n)\) , while for \(\alpha = 2\) it is \(O(\log ^{4/3} n)\) . We complement these results with lower bounds that match the upper bounds within at most a \(\log \log n\) factor.     

Łukasiewicz logic and Riesz spaces

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from \([0,1]\) . Extending Mundici’s equivalence between MV-algebras and \(\ell \) -groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C \(^*\) -algebras. The propositional calculus \({\mathbb R}{\mathcal L}\) that has Riesz MV-algebras as models is a conservative extension of ?ukasiewicz \(\infty \) -valued propositional calculus and is complete with respect to evaluations in the standard model \([0,1]\) . We prove a normal form theorem for this logic, extending McNaughton theorem for ? ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in \({\mathbb R}{\mathcal L},\) and relate them with the analogue of de Finetti’s coherence criterion for \({\mathbb R}{\mathcal L}\) .     

Direct proof of security of Wegman–Carter authentication with partially known key

Information-theoretically secure (ITS) authentication is needed in quantum key distribution (QKD). In this paper, we study security of an ITS authentication scheme proposed by Wegman & Carter, in the case of partially known authentication key. This scheme uses a new authentication key in each authentication attempt, to select a hash function from an Almost Strongly Universal \(_2\) hash function family. The partial knowledge of the attacker is measured as the trace distance between the authentication key distribution and the uniform distribution; this is the usual measure in QKD. We provide direct proofs of security of the scheme, when using partially known key, first in the information-theoretic setting and then in terms of witness indistinguishability as used in the universal composability (UC) framework. We find that if the authentication procedure has a failure probability \(\varepsilon \) and the authentication key has an \(\varepsilon ^{\prime }\) trace distance to the uniform, then under ITS, the adversary’s success probability conditioned on an authentic message-tag pair is only bounded by \(\varepsilon +|\mathcal T |\varepsilon ^{\prime }\) , where \(|\mathcal T |\) is the size of the set of tags. Furthermore, the trace distance between the authentication key distribution and the uniform increases to \(|\mathcal T |\varepsilon ^{\prime }\) after having seen an authentic message-tag pair. Despite this, we are able to prove directly that the authenticated channel is indistinguishable from an (ideal) authentic channel (the desired functionality), except with probability less than \(\varepsilon +\varepsilon ^{\prime }\) . This proves that the scheme is ( \(\varepsilon +\varepsilon ^{\prime }\) )-UC-secure, without using the composability theorem.