Quantum and Classical Query Complexities of Local Search Are Polynomially Related

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).     

(<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-Distance Antimagic Labeling of Some Types of Graphs

We analyze the necessary existence conditions for (a, d)-distance antimagic labeling of a graph G = (V, E) of order n. We obtain theorems that expand the family of not (a, d) -distance antimagic graphs. In particular, we prove that the crown P _n ○ P ₁ does not admit an (a, 1)-distance antimagic labeling for n ≥ 2 if a ≥ 2. We determine the values of a at which path P _n can be an (a, 1)-distance antimagic graph. Among regular graphs, we investigate the case of a circulant graph.     

On Fixed Cost k-Flow Problems

In the Fixed Cost k-Flow problem, we are given a graph G = (V, E) with edge-capacities {u _e ∣e ∈ E} and edge-costs {c _e ∣e ∈ E}, source-sink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. By an approximation-preserving reduction from Group Steiner Tree problem to Fixed Cost k-Flow, we obtain the first polylogarithmic lower bound for the problem; this also implies the first non-constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k-Flow. In the Bipartite Fixed-Cost k-Flow problem, we are given a bipartite graph G = (A ∪ B, E) and an integer k > 0. The goal is to find a node subset S ? A ∪ B of minimum size |S| such G has k pairwise edge-disjoint paths between S ∩ A and S ∩ B. We give an \(O(\sqrt {k\log k})\) approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = |A| + |B|. In the Generalized-P2P problem we are given an undirected graph G = (V, E) with edge-costs and integer charges {b _v : v ∈ V}. The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [11]. Besides that, it generalizes many problems such as Steiner Forest, k-Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log^3+?? n approximation scheme for it using Group Steiner Tree techniques.     

Time versus cost tradeoffs for deterministic rendezvous in networks

Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds. Each agent has a distinct integer label from the set \(\{1,\ldots ,L\}\). Two main efficiency measures of rendezvous are its time (the number of rounds until the meeting) and its cost (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound E on the time of exploration (where E and a corresponding exploration procedure are known to both agents) and of the size L of the label space. We present two natural rendezvous algorithms. Algorithm Cheap has cost O(E) (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly E) and time O(EL). Algorithm Fast has both time and cost \(O(E\log L)\). Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically E (i.e., of cost \(E+o(E)\)) must have time \(\varOmega (EL)\). On the other hand, we show that any deterministic rendezvous algorithm with time complexity \(O(E\log L)\) must have cost \(\varOmega (E\log L)\).     

The weight-constrained maximum-density subtree problem and related problems in trees

Given a tree T=(V,E) of n nodes such that each node v is associated with a value-weight pair (val _v ,w _v ), where value val _v is a real number and weight w _v is a non-negative integer, the density of T is defined as \(\frac{\sum_{v\in V}{\mathit{val}}_{v}}{\sum_{v\in V}w_{v}}\). A subtree of T is a connected subgraph (V′,E′) of T, where V′?V and E′?E. Given two integers w _min? and w _max?, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying w _min?≤∑_v∈V′ w _v ≤w _max?. In this paper, we first present an O(w _max? n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S?V, we also present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.     

A Priori knowledge based secure payload estimation

Many contemporary steganographic schemes aim to embed fixed-length secret message in the cover while minimizing the stego distortion. However, in some cases, the secret message sender requires to embed a variable-length secret payload within his expected stego security. This kind of problem is named as secure payload estimation (SPE). In this paper, we propose a practical SPE approach for individual cover. The stego security metric we adopt here is the detection error rate of steganalyzer (P _E ). Our method is based on a priori knowledge functions, which are two kinds of functions to be determined before the estimation. The first function is the relation function of detection error rate and stego distortion (P _E ? D function). The second function reflects the relationship between stego distortion and payload rate (D ? α) of the chosen cover. The P _E ? D is the general knowledge, which is calculated from image library. On the other hand, D ? α is for specific cover, which is needed to be determined on site. The estimating procedure is as follows: firstly, the sender solves the distortion D under his expected P _E via P _E ? D, and then calculates the corresponding secure payload α via D ? α of the cover. For on-site operations, the most time-consuming part is calculating D ? α function for cover image, which costs 1 time of STC coding. Besides this, the rest on-site operations are solving single-variable formulas, which can be easily tackled. Our approach is an efficient and practical solution for SPE problem.     

Mathematical models and routing algorithms for CAD technological preparation of cutting processes

Resource-conscious technologies for cutting sheet material include the ICP and ECP technologies that allow for aligning fragments of the contours of cutouts. In this work, we show the mathematical model for the problem of cutting out parts with these technologies and algorithms for finding cutting tool routes that satisfy technological constraints. We give a solution for the problem of representing a cutting plan as a plane graph G = (V,F,E), which is a homeomorphic image of the cutting plan. This has let us formalize technological constraints on the trajectory of cutting the parts according to the cutting plan and propose a series of algorithms for constructing a route in the graph G = (V,F,E), which is an image of an admissible trajectory. Using known coordinates of the preimages of vertices of graph G = (V,F,E) and the locations of fragments of the cutting plan that are preimages of edges of graph G = (V,F,E), the resulting route in the graph G = (V,E) can be interpreted as the cutting tool’s trajectory.The proposed algorithms for finding routes in a connected graph G have polynomial computational complexity. To find the optimal route in an unconnected graph G, we need to solve, for every dividing face f of graph G, a travelling salesman problem on the set of faces incident to f.     

Constant-Degree Graph Expansions that Preserve Treewidth

Many hard algorithmic problems dealing with graphs, circuits, formulas and constraints admit polynomial-time upper bounds if the underlying graph has small treewidth. The same problems often encourage reducing the maximal degree of vertices to simplify theoretical arguments or address practical concerns. Such degree reduction can be performed through a sequence of splittings of vertices, resulting in an expansion of the original graph. We observe that the treewidth of a graph may increase dramatically if the splittings are not performed carefully. In this context we address the following natural question: is it possible to reduce the maximum degree to a constant without substantially increasing the treewidth?We answer the above question affirmatively. We prove that any simple undirected graph G=(V,E) admits an expansion G′=(V′,E′) with the maximum degree ≤3 and tw(G′)≤tw(G)+1, where tw(?) is the treewidth of a graph. Furthermore, such an expansion will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed efficiently from a tree-decomposition of G. We also construct a family of examples for which the increase by 1 in treewidth cannot be avoided.     

Position-based coding and convex splitting for private communication over quantum channels

The classical-input quantum-output (cq) wiretap channel is a communication model involving a classical sender X, a legitimate quantum receiver B, and a quantum eavesdropper E. The goal of a private communication protocol that uses such a channel is for the sender X to transmit a message in such a way that the legitimate receiver B can decode it reliably, while the eavesdropper E learns essentially nothing about which message was transmitted. The \(\varepsilon \)-one-shot private capacity of a cq wiretap channel is equal to the maximum number of bits that can be transmitted over the channel, such that the privacy error is no larger than \(\varepsilon \in (0,1)\). The present paper provides a lower bound on the \(\varepsilon \)-one-shot private classical capacity, by exploiting the recently developed techniques of Anshu, Devabathini, Jain, and Warsi, called position-based coding and convex splitting. The lower bound is equal to a difference of the hypothesis testing mutual information between X and B and the “alternate” smooth max-information between X and E. The one-shot lower bound then leads to a non-trivial lower bound on the second-order coding rate for private classical communication over a memoryless cq wiretap channel.     

Chain sequences with ordered enclosing

To enhance the efficiency of numerical control machines for cutting of sheet material, a method of searching for the cutter trajectory as a path with ordered enclosing in the flat graph G = (V, E) corresponding to the cutting design is proposed. It is shown that this path can be represented as a sequence of no more than n/2 + 1 chains with ordered enclosing, where n is the number of odd-order vertices in G. An algorithm for finding the sought sequence of chains, with a complexity of O(|E|) is developed.     

Determining Steady-State Characteristics of Some Queuing Systems with Erlangian Distributions

This article proposes a method to study M / E _s / 1 / m, E _r E _s /1 / m, and E _r / M / n / m queuing systems including the case when m = ∞. Recurrence relations are obtained to compute the stationary distribution of the number of customers in a system and its steady-state characteristics. The developed algorithms are tested on examples using simulation models constructed with the help of the GPSS World tools.     

On transitive partitions of an <Emphasis Type="Italic">n</Emphasis>-cube into codes

We present methods to construct transitive partitions of the set E ⁿ of all binary vectors of length n into codes. In particular, we show that for all n = 2 ^k ? 1, k ≥ 3, there exist transitive partitions of E ⁿ into perfect transitive codes of length n.     

Optimisation of a non linear fuzzy function

 We present a first study concerning the optimization of a non linear fuzzy function f depending both on a crisp variable and a fuzzy number: therefore the function value is a fuzzy number. More specifically, given a real fuzzy number ?∈F and the function f(a,x):R ²→R, we consider the fuzzy extension induced by f, f˜ : F × R → F, f˜(?,x) = Y˜. If K is a convex subset of R, the problem we consider is “maximizing”f˜(?,x), xˉ∈ K. The first problem is the meaning of the word “maximizing”: in fact it is well-known that ranking fuzzy numbers is a complex matter. Following a general method, we introduce a real function (evaluation function) on real fuzzy numbers, in order to get a crisp rating, induced by the order of the real line. In such a way, the optimization problem on fuzzy numbers can be written in terms of an optimization problem for the real-valued function obtained by composition of f with a suitable evaluation function. This approach allows us to state a necessary and sufficient condition in order that ∈K is the maximum for f˜ in K, when f(a,x) is convex-concave (Theorem 4.1).     

Nonparametric estimation of signal amplitude in white Gaussian noise

We assume that a transmitted signal is of the form S(t)f(t), where f(t) is a known function vanishing at some points of the observation interval and S(t) is a function of a known smoothness class. The signal is transmitted over a communication channel with additive white Gaussian noise of small intensity ?. For this model, we construct an estimator for S(t) which is optimal with respect to the rate of convergence of the risk to zero as ? → 0.     

Shadow Prices in Territory Division

We consider a geographic optimization problem in which we are given a region R, a probability density function f(?) defined on R, and a collection of n utility density functions u _i (?) defined on R. Our objective is to divide R into n sub-regions R _i so as to “balance” the overall utilities on the regions, which are given by the integrals \(\iint _{R_{i}}f(x)u_{i}(x)\, dA\). Using a simple complementary slackness argument, we show that (depending on what we mean precisely by “balancing” the utility functions) the boundary curves between optimal sub-regions are level curves of either the difference function u _i (x) ? u _j (x) or the ratio u _i (x)/u _j (x). This allows us to solve the problem of optimally partitioning the region efficiently by reducing it to a low-dimensional convex optimization problem. This result generalizes, and gives very short and constructive proofs of, several existing results in the literature on equitable partitioning for particular forms of f(?) and u _i (?). We next give two economic applications of our results in which we show how to compute a market-clearing price vector in an aggregate demand system or a variation of the classical Fisher exchange market. Finally, we consider a dynamic problem in which the density function f(?) varies over time (simulating population migration or transport of a resource, for example) and derive a set of partial differential equations that describe the evolution of the optimal sub-regions over time. Numerical simulations for both static and dynamic problems confirm that such partitioning problems become tractable when using our methods.     

Clique Numbers of Random Subgraphs of Some Distance Graphs

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

     

Occurrence indices of elements in linear recurrence sequences over primary residue rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring \(\mathbb{Z}_{p^n } \). We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over \(\mathbb{Z}_{p^n } \), we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over ?₄ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.     

On ruin probability minimization under excess reinsurance

The problem of ruin probability minimization in the Cramer-Lundberg risk model under excess reinsurance is studied. Together with traditional maximization of the Lundberg characteristic coefficient R is considered the problem of direct calculation of insurer’s ruin probability ? _r (x) as an initial-capital function x under the prescribed level of net-retention r. To solve this problem, we propose the excess variant of the Cramer integral equation which is an equivalent to the Hamilton-Jacobi-Bellman equation. The continuation method is used for solving this equation; by means of it is found the analytical solution to the Markov risk model. We demonstrated on a series of standard examples that with any admissible value of x the ruin probability ? _x (r): = ? _r (x) is usually a unimodal function r. A comparison of the analytic representation of ruin probability ? _r(x) with its asymptotic approximation with x → ∞ was conducted.     

Reverse Furthest Neighbors Query in Road Networks

Given a road network G = (V,E), where V (E) denotes the set of vertices(edges) in G, a set of points of interest P and a query point q residing in G, the reverse furthest neighbors (Rfn _R ) query in road networks fetches a set of points p ∈ P that take q as their furthest neighbor compared with all points in P ∪ {q}. This is the monochromatic Rfn _R (Mrfn _R ) query. Another interesting version of Rfn _R query is the bichromatic reverse furthest neighbor (Brfn _R ) query. Given two sets of points P and Q, and a query point q ∈ Q, a Brfn _R query fetches a set of points p ∈ P that take q as their furthest neighbor compared with all points in Q. This paper presents efficient algorithms for both Mrfn _R and Brfn _R queries, which utilize landmarks and partitioning-based techniques. Experiments on real datasets confirm the efficiency and scalability of proposed algorithms.     

Guaranteed strategies for escaping encirclement

This paper considers a conflict situation on the plane as follows. A fast evader E has to break out the encirclement of slow pursuers P _{j1,...,j n} = {P _j1,..., P _jn }, n ≥ 3, with a miss distance not smaller than r ≥ 0. First, we estimate the minimum guaranteed miss distance from E to a pursuer P _a , a ∈ {j ₁,..., j _n }, when the former moves along a given straight line. Then the obtained results are used to calculate the guaranteed estimates to a group of two pursuers P _b,c = {P _b , P _c }, b, c ∈ {j ₁,..., j _n }, b ≠ c, when E maneuvers by crossing the rectilinear segment P _b P _c , and the state passes to the domain of the game space where E applies a strategy under which the miss distance to any of the pursuers is not decreased. In addition, we describe an approach to the games with a group of pursuers P _{j1,... jn} , n ≥ 3, in which E seeks to break out the encirclement by passing between two pursuers P _b and P _c , entering the domain of the game space where E can increase the miss distance to all pursuers by straight motion. By comparing the guaranteed miss distances with r for all alternatives b, c ∈ {j ₁,..., j _n }, b ≠ c, and a ? {b, c}, it is possible to choose the best alternative and also to extract the histories of the game in which the designed evasion strategies guarantee a safe break out from the encirclement.