We present a new algorithm that solves the minimum Steiner tree problem on an undirected graph with n nodes, k terminals, and m edges with bounded integer weights in time Õ((2− ε)n + nm). Our algorithm beats the aesthetically appealing and seemingly inherent 2 bound of the current best algorithm for Steiner tree problem from Björklund, Husfeldt, Kaski, and Koivisto. Our algorithm is based on fa...

We study a bottleneck Steiner tree problem: given a set P = {p1,p2, . . . , pn} of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the pr...

Let A be a nonnegative real matrix which is expanding i e with all eigenvalues j j and suppose that jdet A j is an integer Let D consist of exactly jdet A j nonnegativevectors inR We classify all pairs A D such that every x in the orthant Rn has at least one radix expansion in base A using digits inD The matrixAmust be a diagonalmatrix times a permutation matrix In addition A must be similar to...

Branching in mixed-integer (or integer) linear programming requires choosing both the branching variable and the branching direction. This paper develops a number of new methods for making those two decisions either independently or together with the goal of reaching the first integer-feasible solution quickly. These new methods are based on estimating the probability of satisfying a constraint...

In this paper we construct trees having only integer eigenvalues with arbitrarily large diameters. In fact, we prove that for every set S of positive integers there exists a tree whose positive eigenvalues are exactly the elements of S. If the set S is different from the set {1} then the constructed tree will have diameter 2|S|.

Linear Relation Analysis [CH78] suffers from the cost of operations on convex polyhedra, which can be exponential with the number of involved variables. In order to reduce this cost, we propose to detect when a polyhedron is a Cartesian product of polyhedra of lower dimensions, i.e., when groups of variables are unrelated with each other. Classical operations are adapted to work on such factore...

We study the representation number for some special sparse graphs. For graphs with a single edge and for complete binary trees we give an exact formula, and for hypercubes we improve the known lower bound. We also study the prime factorization of the representation number of graphs with one edge.

Let D:= (X,B,I) be a 2-design. For each xeX set Bx:= {jeB:xIj}, and define Xh dually for each h e B. We call D reduced if Xh = Xt implies that h = i for all h,ieB. Assume that D is a reduced 2-design with parameters v, b, k, r, X. Let G be a flag transitive automorphism group of D. By F*(G) we mean (as usual) the generalized Fitting subgroup of G which is defined to be the inverse image in G of...

We present modifications to Recursive Auto-Associative Memory which increase its robustness and storage capacity. This is done by introducing an extra layer to the compressor and reconstructor networks, employing integer rather than realvalued representations, pre-conditioning the weights and presetting the representations to be compatible with them, and using a quick-prop modification. Initial...