Online hypergraph coloring

In this paper we investigate the online hypergraph coloring problem. In this online problem the algorithm receives the vertices of the hypergraph in some order v₁,…,vn and it must color vi by only looking at the subhypergraph Hi=(Vi,Ei) where Vi={v₁,…,vi} and Ei contains the edges of the hypergraph which are subsets of Vi. We show that there exists no online hypergraph coloring algorithm with sublinear competitive ratio. Furthermore we investigate some particular classes of hypergraphs (k-uniform hypergraphs, hypergraphs with bounded matching number, projective planes), we analyse the online algorithm FF and give matching lower bounds for these classes.     

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

Given an undirected multigraph G=(V,E), a family $\mathcal{W}Given an undirected multigraph G=(V,E), a family W\mathcal{W} of areas W⊆V, and a target connectivity k≥1, we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least k edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W ? WW\in \mathcal{W} . So far this problem was shown to be NP-complete in the case of k=1 and polynomially solvable in the case of k=2. In this paper, we show that the problem for k≥3 can be solved in O(m+n(k ³+n ²)(p+kn+nlog n)log k+pkn ³log (n/k)) time, where n=|V|, m=|{{u,v}|(u,v)∈E}|, and p=|W|p=|\mathcal{W}| .     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ⋅s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

Small Space Representations for Metric Min-sum k-Clustering and Their Applications

The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters C ₁,…,C _k ?P such that $\sum_{i=1}^{k}\sum_{p,q\in C_{i}}d(p,q)The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters C ₁,…,C _k ⊆P such that ?_i=1^k?_{p,q ? Ci}d(p,q)\sum_{i=1}^{k}\sum_{p,q\in C_{i}}d(p,q) is minimized. We show the first efficient construction of a coreset for this problem. Our coreset construction is based on a new adaptive sampling algorithm. With our construction of coresets we obtain two main algorithmic results.     

寻找无向图中回路的并行算法

对无向简单图Ｇ＝（Ｖ，Ｅ），｜Ｖ｜＝ｎ，｜Ｅ｜＝ｍ，给出对下述问题的ＮＣ算法：（１）寻找Ｇ中最短回路；（２）寻找Ｇ中最短偶（奇）长度回路；（３）求解Ｃ_ｋ，ｋ＝３，４，这里Ｃ_ｋ表示Ｇ中长度为ｋ的回路．     

On Rooted Node-Connectivity Problems

Let G be a graph which is k -outconnected from a specified root node r , that is, G has k openly disjoint paths between r and v for every node v . We give necessary and sufficient conditions for the existence of a pair rv,rw of edges for which replacing these edges by a new edge vw gives a graph that is k -outconnected from r . This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k -node-connectivity. We also prove that if C is a cycle in G such that each edge in C is critical with respect to k -outconnectivity from r , then C has a node v , distinct from r , which has degree k . This result is the rooted counterpart of a theorem due to Mader. We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements c _u for each node u , find a minimum-weight subgraph that contains max {c _u ,c _v } openly disjoint paths between every pair of nodes u,v . For metric weights, our approximation guarantee is 3 . For uniform weights, our approximation guarantee is \min{ 2, (k+2q-1)/k} . Here k is the maximum node requirement, and q is the number of positive node requirements. Received September 15, 1998; revised March 10, 2000, and April 17, 2000.     

The Solution of Linear Probabilistic Recurrence Relations

Abstract. Linear probabilistic divide-and-conquer recurrence relations arise when analyzing the running time of divide-and-conquer randomized algorithms. We consider first the problem of finding the expected value of the random process T(x) , described as the output of a randomized recursive algorithm T . On input x , T generates a sample (h ₁ ,···,h _k ) from a given probability distribution on [0,1] ^k and recurses by returning g(x) + Σ _i=1 ^k c _i T(h _i x) until a constant is returned when x becomes less than a given number. Under some minor assumptions on the problem parameters, we present a closed-form asymptotic solution of the expected value of T(x) . We show that E[T(x)] = Θ( x ^p + x ^p ∈t ₁ ^x (g(u)/ u ^p+1 ) du) , where p is the nonnegative unique solution of the equation Σ _i=1 ^k c _i E[h _i ^p ] = 1 . This generalizes the result in [1] where we considered the deterministic version of the recurrence. Then, following [2], we argue that the solution holds under a broad class of perturbations including floors and ceilings that usually accompany the recurrences that arise when analyzing randomized divide-and-conquer algorithms.     

The region approach for computing relative neighbourhood graphs in the Lp metric

The following geometrical proximity concepts are discussed: relative closeness and geographic closeness. Consider a setV={v ₁,v ₂, ...,v _n } of distinct points in atwo-dimensional space. The pointv _j is said to be arelative neighbour ofv _i ifd _p (v _i ,v _j )≤max{d _p (v _j ,v _k ),d _p (v _j ,v _k )} for allv _k ∈V, whered _p denotes the distance in theL _p metric, 1≤p≤∞. After dividing the space around the pointv _i into eight sectors (regions) of equal size, a closest point tov _i in some region is called anoctant (region, orgeographic) neighbour ofv _i. For anyL _p metric, a relative neighbour ofv _i is always an octant neighbour in some region atv _i. This gives a direct method for computing all relative neighbours, i.e. for establishing therelative neighbourhood graph ofV. For every pointv _i ofV, first search for the octant neighbours ofv _i in each region, and then for each octant neighbourv _j found check whether the pointv _j is also a relative neighbour ofv _i. In theL _p metric, 1<p<∞, the total number of octant neighbours is shown to be θ(n) for any set ofn points; hence, even a straightforward implementation of the above method runs in θn ²) time. In theL ₁ andL _∞ metrics the method can be refined to a θ(n logn+m) algorithm, wherem is the number of relative neighbours in the output,n-1≤m≤n(n-1). TheL ₁ (L _∞) algorithm is optimal within a constant factor.     

On covering problems of codes

LetC be a binary code of lengthn (i.e., a subset of {0, 1} ⁿ ). TheCovering Radius of C is the smallest integerr such that each vector in {0, 1} ⁿ is at a distance at mostr from some code word. Our main result is that the decision problem associated with the Covering Radius of arbitrary binary codes is NP-complete. This result is established as follows. TheRadius of a binary codeC is the smallest integerr such thatC is contained in a radius-r ball of the Hamming metric space 〈{0, 1} ⁿ ,d〉. It is known [K] that the problems of computing the Radius and the Covering Radius are equivalent. We show that the 3SAT problem is polynomially reducible to the Radius decision problem. A central tool in our reduction is a metrical characterization of the set ofdoubled vectors of length 2n: {v=(v ₁ v ₂ …v _2n ) | ∀i:v _2i =v _2i−1}. We show that there is a setY ⊂ {0, 1}²ⁿ such that for everyv ε {0, 1}²ⁿ :v is doubled iffY is contained in the radius-n ball centered atv; moreover,Y can be constructed in time polynomial inn.     

Well-Separated Clusters and Optimal Fuzzy Partitions

Abstract

Two separation indices are considered for partitions P = {X₁, …, X_k} of a finite data set X in a general inner product space. Both indices increase as the pairwise distances between the subsets X_i become large compared to the diameters of X_i Maximally separated partitions p' are defined and it is shown that as the indices of p' increase without bound, the characteristic functions of X_i' in P' are approximated more and more closely by the membership functions in fuzzy partitions which minimize certain fuzzy extensions of the k-means squared error criterion function.     

A Neural Algorithm for the Maximum Clique Problem: Analysis,Experiments, and Circuit Implementation

{In this paper we design and analyze a neural approximation algorithm for the Maximum Clique problem. This algorithm, having as input an arbitrary undirected graph G = \langle V, E\rangle , constructs a finite sequence of Hopfield networks such that the attractor of the last network in the sequence represents a maximal clique of G . We prove that D(G) ⋅ |E ^{\rm c} | , where D(G) = max _{{i,j}\notin E} \min{d _i , d _j } , d _i is the degree of the vertex i of G , and |E ^{\rm c} | denotes the cardinality of the set of edges in the complement graph, is an upper bound to the number of the networks in the sequence. Some experiments made on the second DIMACS benchmark and on random graphs show that: 1. The quality of the solutions found by the algorithm is satisfactory. 2. The theoretical upper bound D(G) ⋅ |E ^{\rm c} | is quite pessimistic. For random graphs we propose an empirical formula that gives a better estimate of the number of networks in the sequence. Moreover, thanks to the simplicity of the algorithm, we are able to design a uniform family of circuits of small size (\approx 10n ² log ₂ n ) that implements it. The circuit, which solves the problems for graphs of at most 32 vertices, has then been programmed on FPGAs (Field Programmable Gate Arrays). An analysis in terms of size and time complexity is given. Received November 10, 1998; revised December 2000.     

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

A Characterization of Planar Graphs by Pseudo-Line Arrangements

Let Γ be an arrangement of pseudo-lines, i.e., a collection of unbounded x -monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (Γ, E) is a graph for which the vertices are the pseudo-lines of Γ and the edges are some unordered pairs of pseudo-lines of Γ . A diamond of a pseudo-line graph (Γ, E) is a pair of edges {p,q} , {p',q'}∈ E , {p,q} ∩ {p',q'}= , such that the crossing point of the pseudo-lines p and q lies vertically between p' and q' and the crossing point of p' and q' lies vertically between p and q . We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(k ^1/3 n) upper bound on the k -level complexity of an arrangement of straight lines, which was very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.     

Time-series decomposition and forecasting

Any stationary time-series can be decomposed by means of an optimization operator, called the ζ-optimator, into several components (the time-series){Y _t ⁱ}, i =1,2,…, p, such that the first component {V _t ⁱ} t = 1,2,…,v is a smooth process having a larger autocorrelation in comparison with the original process {Y _t}, i.e. ρ_vi > ρ_y. Usually only a few such components are sufficient for approximating the time-series with good accuracy. The ζ-optimator involves a shape parameter a, so the decomposition is unique provided that a. is fixed. Since the component {V _t ¹} involves much of the useful information it can be used for computing predictors for control purposes. Thus, given the observations Y_v, Y_v-1, Y_v-2,…, a predictor of Y_v+1 is ρ_vi V _v ¹ (q) where, V_v ¹(q) = qY^v + q(1-q)² Y_v-2, …, the weights q(1-q)^r, r=0,1,2,…, decreasing rapidly as q = q(α) ε (0,1) Further, one may choose q rather than choosing α, since q(α) is a one-one mapping. Once q is fixed, the predictor ρ_v1 V _v ¹(q) is obtained in a straightforward way by using the formula above. It is shown that ρ_v1 V _v ¹(q) converges to the best predictor as α → 0. Some examples are worked out, illustrating both the decomposition and the forecasting procedures.     

Generation of robust root loci for linear systems with parametric uncertainties in an ellipsoid

Given a parametric polynomial family p(s; Q) := {ⁿ _k=0 a_k (q)s^k : q ] Q}, Q R ^m , the robust root locus of p(s; Q) is defined as the two-dimensional zero set _p,Q := {s ] C:p(s; q) = 0 for some q ] Q}. In this paper we are concerned with the problem of generating robust root loci for the parametric polynomial family p(s; E) whose polynomial coefficients depend polynomially on elements of the parameter vector q ] E which lies in an m-dimensional ellipsoid E. More precisely, we present a computational technique for testing the zero inclusion/exclusion of the value set p(z; E) for a fixed point z in C, and then apply an integer-labelled pivoting procedure to generate the boundary of each subregion of the robust root locus _p,E . The proposed zero inclusion/exclusion test algorithm is based on using some simple sufficient conditions for the zero inclusion and exclusion of the value set p(z,E) and subdividing the domain E iteratively. Furthermore, an interval method is incorporated in the algorithm to speed up the process of zero inclusion/exclusion test by reducing the number of zero inclusion test operations. To illustrate the effectiveness of the proposed algorithm for the generation of robust root locus, an example is provided.     

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form , y=Cx with feedback of the form u(x)=v(x)Cx is considered, where v(x) is real-valued and has domain which is all of . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x){k₁,k₂} for all x and for some k₁,k₂), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)^-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.     

Average-Case Analysis of Dynamic Graph Algorithms

We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., not random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2-edge connectivity, k -edge connectivity, k -vertex connectivity, and bipartiteness. Given a random graph G with m ₀ edges and n vertices and a sequence of l update operations such that the graph contains m _i edges after operation i , the expected time for performing the updates for any l is in the case of minimum spanning forests, connectivity, 2-edge connectivity, and bipartiteness. The expected time per update operation is O(n) in the case of maximum matching. We also give improved bounds for k -edge and k -vertex connectivity. Additionally we give an insertions-only algorithm for maximum cardinality matching with worst-case O(n) amortized time per insertion. Received June 11, 1995; revised March 8, 1996.     

On efficient implementation of an approximation algorithm for the Steiner tree problem

Summary This paper studies the design and implementation of an approximation algorithm for the Steiner tree problem. Given any undirected distance graph G and a set of Steiner points S, the algorithm produces a Steiner tree with total weight on its edges no more than 2(1–1/L) times the total weight on the optimal Steiner tree, where L is the number of leaves in the optimal Steiner tree. Our implementation of the algorithm, in the worst case, makes it run in 0(¦E _g¦+¦V _g–S¦log¦V _g–S¦+¦S¦log ¦S¦) time for general graph G and in 0(¦S¦ log¦S¦+M log (M,¦V _g–S¦)) time for sparse graph G, where E _g is the set of edges in G, V_g is the set of vertices in G, M = min {¦E _g, (¦V _g–S¦–1)²/2} and (x,y) = min {i¦log⁽ⁱ⁾ y x/y}.The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦V _g¦ vertices and ¦E _g¦ edges and the minimum spanning tree of a graph with ¦V _g–S¦ vertices and M edges.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983