A $k$-bit Hamming prefix code is a binary code with the following property: for any codeword $x$ and any prefix $y$ of another codeword, both $x$ and $y$ having the same length, the Hamming distance between $x$ and $y$ is at least $k$. Given an alphabet $A = [a_1,\ldots,a_n]$ with corresponding probabilities $[p_1,\ldots,p_n]$, the $k$-bit … Read more
We consider a class of two-stage linear stochastic programming problems, introduced by Shmoys and Swamy (2004), motivated by a relaxation of a stochastic set cover problem. We show that the sample size required to solve this problem by the sample average approximation (SAA) method with a relative accuracy $\kappa>0$ and confidence $1-\alpha$ is polynomial in … Read more
We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers $n, d, k$ where $k \le d$, and a set $P$ of $n$ points in $R^d$. The goal is to compute the {\em outer $k$-radius} of $P$, denoted by $\kflatr(P)$, which is the minimum, over all $(d-k)$-dimensional … Read more
The paper describes various combinatorial and algorithmic applications of hyperbolic (multivariate) polynomials . Section 2.2 introduces a new class of polynomials , which include as hyperbolic polynomials as well volume polynomials $Vol(x_1C_1+…+x_nC_n)$ , where $C_i$ are convex compact subsets of $R^n$. This extension leads to randomized poly-time algorithm to approximate $M(C_1,…,C_n)$ (the mixed volume) within … Read more
Local optimizations introduced to obtain improved tours for Traveling Salesman Problem have a great impact on the final solution. That is way we introduce a new ant system algorithm with a new local updating pheromone rule, and the tours are improved using k-opt techniques. The tests use different parameters, in order to obtain solutions close … Read more
We consider 3-partitioning the vertices of a graph into sets $S_1, S_2$ and $S_3$ of specified cardinalities, such that the total weight of all edges joining $S_1$ and $S_2$ is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that … Read more
In this paper we consider the following two types of complex quadratic maximization problems: (i) maximize $z^{\HH} Q z$, subject to $z_k^m=1$, $k=1,…,n$, where $Q$ is a Hermitian matrix with $\tr Q=0$ and $z\in \C^n$ is the decision vector; (ii) maximize $\re y^{\HH}Az$, subject to $y_k^m=1$, $k=1,…,p$, and $z_l^m=1$, $l=1,…,q$, where $A\in \C^{p\times q}$ and … Read more
One of the fundamental clustering problems is to assign $n$ points into $k$ clusters based on the minimal sum-of-squares(MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP). We show that our 0-1 SDP model provides an unified framework for … Read more
For a given schedule of a round-robin tournament and a matrix of distances between homes of teams, an optimal home-away assignment problem is to find a home-away assignment that minimizes the total traveling distance. We propose a technique to transform the problem to MIN RES CUT. We apply Goemans and Williamson’s 0.878-approximation algorithm for MAX … Read more
We consider the problem of computing the minimum value $p_{\min}$ taken by a polynomial $p(x)$ of degree $d$ over the standard simplex $\Delta$. This is an NP-hard problem already for degree $d=2$. For any integer $k\ge 1$, by minimizing $p(x)$ over the set of rational points in $\Delta$ with denominator $k$, one obtains a hierarchy … Read more