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 \(L\) - \(\mathbf{Top}_{0}\) .     

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 \(h\) .     

Fault tolerance of hypercubes and folded hypercubes

Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\) . We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.     

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.     

Some kinds of primitive and non-primitive words

If the length of a primitive word \(p\) is equal to the length of another primitive word \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . This was obtained separately by Tetsuo Moriya in 2008 and Shyr and Yu in 1994. In this paper, we prove that if the length of \(p\) is divisible by the length of \(q\) and the length of \(p\) is less than or equal to \(m\) times the length of \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . Then we show that if \(uv,u\) are non-primitive words and the length of \(u\) is divisible by the length \(v\) or one of the length of \(u\) and \(uv\) is odd for any two nonempty words \(u\) and \(v\) , then \(u\) is a power of \(v\) .     

Thermal entanglement of the mixed spin-1/2 and spin-5/2 Heisenberg model under an external magnetic field

We use the concept of negativity to study the entanglement of spin-1/2 and spin-5/2 antiferromagnetic Heisenberg model with an inhomogeneous magnetic field. Analytical conclusions of the model are acquired. It is found that the critical temperature \(T_\mathrm{c}\) goes up, as the increase of anisotropy parameter \(k\) . The temperature \(T_\mathrm{c}\) becomes bigger than the results of spin-1/2 and spin-3/2 Heisenberg XXZ chain for the same value of \(k\) . And we can gain more entanglement at higher temperature by coordinating the value of inhomogeneity \(b\) .     

Strong Conflict-Free Coloring for Intervals

We consider the \(k\) -strong conflict-free ( \(k\) -SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval \(I\) of the family there are at least \(k\) colors each appearing exactly once in \(I\) . We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when \(k=1\) and \(5-\frac{2}{k}\) when \(k\ge 2\) . In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any \(k \ge 1\) . We also provide, in case \(k=O({{\mathrm{polylog}}}(n))\) , a quasipolynomial time algorithm to decide the existence of a \(k\) -SCF coloring that uses at most \(q\) colors.     

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.     

Growth rates for persistently excited linear systems

We consider a family of linear control systems \(\dot{x}=Ax+\alpha Bu\) on \(\mathbb {R}^d\) , where \(\alpha \) belongs to a given class of persistently exciting signals. We seek maximal \(\alpha \) -uniform stabilization and destabilization by means of linear feedbacks \(u=Kx\) . We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if there exists at least one \(K\) such that the Lie algebra generated by \(A\) and \(BK\) is equal to the set of all \(d\times d\) matrices, then the maximal rate of convergence of \((A,B)\) is equal to the maximal rate of divergence of \((-A,-B)\) . We also provide more precise results in the general single-input case, where the above result is obtained under the simpler assumption of controllability of the pair \((A,B)\) .     

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals \(I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx\) and \(I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx\) . These integrals are not defined in the regular sense; \(I^{(1)}[g]\) is defined in the sense of Cauchy Principal Value while \(I^{(2)}[g]\) is defined in the sense of Hadamard Finite Part. With \(h=(b-a)/n, \,n=1,2,\ldots \) , and \(t=a+kh\) for some \(k\in \{1,\ldots ,n-1\}, \,t\) being fixed, the numerical quadrature formulas \({Q}^{(1)}_n[g]\) for \(I^{(1)}[g]\) and \(Q^{(2)}_n[g]\) for \(I^{(2)}[g]\) are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that \(g\in C^\infty [a,b]\) . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants \(c^{(k)}_i\) being independent of \(h\) . In this work, we apply the Richardson extrapolation to \({Q}^{(k)}_n[g]\) to obtain approximations of very high accuracy to \(I^{(k)}[g]\) . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function \(g(x)\) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.     

Asymptotic spectral distributions of Manhattan products of C_{n}\sharp P_{m}

The Manhattan product of directed cycles \(C_{n}\) and directed paths \(P_{m}\) is a diagraph. Recently, in quantum probability theory, several authors have studied the spectrum of graph, as mentioned also by A. Hora and N. Obata. In the paper, we study asymptotic spectral distribution of the Manhattan products of simple digraphs- \(C_{n}\sharp P_{m}\) . The limit of the spectral distribution of \(C_{n}\sharp P_{2}\) as \(n\rightarrow \infty \) exists in the sense of weak convergence, and its concrete form is obtained. We insist on the fact that this note does not contain any new results, which is only some parallel results with Obata (Interdiscip Inf Sci 18(1):43–54, 2012) or Obata (Ann Funct Anal 3:136–144, 2012). But, we have only been written to convey the information from quantum probability to spectral analysis of graph.     

Fault tolerance in distributed systems using fused state machines

Replication is a standard technique for fault tolerance in distributed systems modeled as deterministic finite state machines (DFSMs or machines). To correct \(f\) crash or \(\lfloor f/2 \rfloor \) Byzantine faults among \(n\) different machines, replication requires \(nf\) backup machines. We present a solution called fusion that requires just \(f\) backup machines. First, we build a framework for fault tolerance in DFSMs based on the notion of Hamming distances. We introduce the concept of an ( \(f\) , \(m\) )-fusion, which is a set of \(m\) backup machines that can correct \(f\) crash faults or \(\lfloor f/2 \rfloor \) Byzantine faults among a given set of machines. Second, we present an algorithm to generate an ( \(f\) , \(f\) )-fusion for a given set of machines. We ensure that our backups are efficient in terms of the size of their state and event sets. Third, we use locality sensitive hashing for the detection and correction of faults that incurs almost the same overhead as that for replication. We detect Byzantine faults with time complexity \(O(n f)\) on average while we correct crash and Byzantine faults with time complexity \(O(n \rho f)\) with high probability, where \(\rho \) is the average state reduction achieved by fusion. Finally, our evaluation of fusion on the widely used MCNC’91 benchmarks for DFSMs shows that the average state space savings in fusion (over replication) is 38 % (range 0–99 %). To demonstrate the practical use of fusion, we describe its potential application to two areas: sensor networks and the MapReduce framework. In the case of sensor networks a fusion-based solution can lead to significantly fewer sensor-nodes than a replication-based solution. For the MapReduce framework, fusion can reduce the number of map-tasks compared to replication. Hence, fusion results in considerable savings in state space and other resources such as the power needed to run the backups.     

Deterministic function computation with chemical reaction networks

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. CRNs have been shown to be efficiently Turing-universal (i.e., able to simulate arbitrary algorithms) when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates (a multi-dimensional generalization of “eventually periodic” sets). We introduce the notion of function, rather than predicate, computation by representing the output of a function \({f:{\mathbb{N}}^k\to{\mathbb{N}}^l}\) by a count of some molecular species, i.e., if the CRN starts with \(x_1,\ldots,x_k\) molecules of some “input” species \(X_1,\ldots,X_k, \) the CRN is guaranteed to converge to having \(f(x_1,\ldots,x_k)\) molecules of the “output” species \(Y_1,\ldots,Y_l\) . We show that a function \({f:{\mathbb{N}}^k \to {\mathbb{N}}^l}\) is deterministically computed by a CRN if and only if its graph \({\{({\bf x, y}) \in {\mathbb{N}}^k \times {\mathbb{N}}^l | f({\bf x}) = {\bf y}\}}\) is a semilinear set. Finally, we show that each semilinear function f (a function whose graph is a semilinear set) can be computed by a CRN on input x in expected time \(O(\hbox{polylog} \|{\bf x}\|_1)\) .     

A note on the growth factor in Gaussian elimination for accretive-dissipative matrices

This short note proves that if \(A\) is accretive-dissipative, then the growth factor for such \(A\) in Gaussian elimination is less than \(4\) . If \(A\) is a Higham matrix, i.e., the accretive-dissipative matrix \(A\) is complex symmetric, then the growth factor is less than \(2\sqrt{2}\) . The result obtained improves those of George et al. in [Numer. Linear Algebra Appl. 9, 107–114 (2002)] and is one step closer to the final solution of Higham’s conjecture.     

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 \(G\) .