Approximation algorithms for the Weighted t-Uniform Sparsest Cut and some other graph partitioning problems

We study the Weighted t-Uniform Sparsest Cut (Weighted t-USC) and other related problems. In an instance of the Weighted t-USC problem, a parameter t and an undirected graph $G=(V,E)$ with edge-weights $w:E\to {\mathbb{R}}_{\ge 0}$ and vertex-weights $\eta :V\to {\mathbb{R}}_{+}$ are given. The goal is to find a vertex set $S\subseteq V$ with $|S|\le t$ while minimizing $w(S,V\backslash S)/\eta (S)$, where $w(S,V\backslash S)$ is the total weight of the edges with exactly one endpoint in S and $\eta (S)={\sum }_{v\in S}\eta (v)$. For this problem, we present a $(O(\log t),1+ϵ)$ factor bicriteria approximation algorithm. Our algorithm outperforms the current best algorithm when $t=n^{o(1)}$. We also present better approximation algorithms for Weighted ρ-Unbalanced Cut and Min–Max k-Partitioning problems.     

A shortest cycle for each vertex of a graph

We present an algorithm that finds, for each vertex of an undirected graph, a shortest cycle containing it. While for directed graphs this problem reduces to the All-Pairs Shortest Paths problem, this is not known to be the case for undirected graphs.We present a truly sub-cubic randomized algorithm for the undirected case. Given an undirected graph with n vertices and integer weights in $1,\dots ,M$, it runs in $\stackrel{?}{O}(\sqrt{M}{n}^{(\omega +3)/2})$ time where $\omega <2.376$ is the exponent of matrix multiplication. As a by-product, our algorithm can be used to determine which vertices lie on cycles of length at most t in $\stackrel{?}{O}(M{n}^{\omega }t)$ time. For the case of bounded real edge weights, a variant of our algorithm solves the problem up to an additive error of ? in $\stackrel{?}{O}(n^{(\omega +6)/3})$ time.     

On multiple-instance learning of halfspaces

In multiple-instance learning the learner receives bags, i.e., sets of instances. A bag is labeled positive if it contains a positive example of the target. An $\Omega (d\log r)$ lower bound is given for the VC-dimension of bags of size r for d-dimensional halfspaces and it is shown that the same lower bound holds for halfspaces over any large point set in general position. This lower bound improves an $\Omega (\log r)$ lower bound of Sabato and Tishby, and it is sharp in order of magnitude. We also show that the hypothesis finding problem is NP-complete and formulate several open problems.     

Efficient algorithms for the round-trip 1-center and 1-median problems

This paper presents improved algorithms for the round-trip single-facility location problem on a general graph, in which a set A of collection depots is given and the service distance of a customer is defined to be the distance from the server, to the customer, then to a depot, and back to the server. Each customer i is associated with a subset $A^i\subseteq A$ of depots that i can potentially select from and use. When $A^i=A$ for each customer i, the problem is unrestricted; otherwise it is restricted. For the restricted round-trip 1-center problem, we give an $O(mn\lg n)$-time algorithm. For the restricted 1-median problem, we give an $O(mn\lg (|A|/m)+n^2\lg n)$-time algorithm. For the unrestricted 1-median problem, we give an $O(mn+n^2\lg \lg n)$-time algorithm.     

Edge-fault-tolerant pancyclicity and bipancyclicity of Cartesian product graphs with faulty edges

Let r≥ 4 be an even integer. Graph G is r-bipancyclic if it contains a cycle of every even length from r to $2\lfloor \frac{n(G)}{2}\rfloor$, where $n(G)$ is the number of vertices in G. A graph G is r-pancyclic if it contains a cycle of every length from r to $n(G)$, where $r\ge 3$. A graph is k-edge-fault Hamiltonian if, after deleting arbitrary k edges from the graph, the resulting graph remains Hamiltonian. The terms k-edge-fault r-bipancyclic and k-edge-fault r-pancyclic can be defined similarly. Given two graphs G and H, where $n(G)$, $n(H)\ge$ 9, let $k_1$, $k_2\ge 5$ be the minimum degrees of G and H, respectively. This study determined the edge-fault r-bipancyclic and edge-fault r-pancyclic of Cartesian product graph $G\times H$ with some conditions. These results were then used to evaluate the edge-fault pancyclicity (bipancyclicity) of ${\mathit{NQ}}_{m_r,\dots ,m_1}$ and ${\mathit{GQ}}_{m_r,\dots ,m_1}$.     

Online scheduling of equal-length jobs with incompatible families on multiple batch machines to maximize the weighted number of early jobs

This paper studies the online scheduling of equal-length jobs with incompatible families on multiple batch machines which can process the jobs from a common family in batches, where each batch has a capacity b with $b=\infty$ in the unbounded batching and $b<\infty$ in the bounded batching. Each job J has an equal-length integral processing time $p>0$, an integral release time $r(J)?0$, an integral deadline $d(J)?0$ and a real weight $w(J)?0$. The goal is to determine a preemptive schedule with restart which maximizes the weighted number of early jobs. When $p=1$, we show that a simple greedy online algorithm has a competitive ratio 2, and establish the lower bound $2?1/b$. This means that the greedy algorithm is of the best possible for $b=\infty$. When p is any positive integer, we provide an online algorithm of competitive ratio $3+2\sqrt{2}$ for both $b=\infty$ and $b<\infty$. This is the first online algorithm for the problem with a constant competitive ratio.     

Proximity and average eccentricity of a graph

Let $G=(V,E)$ be a connected graph on n vertices. The proximity $\pi (G)$ of G is the minimum average distance from a vertex of G to all others. The eccentricity $e(v)$ of a vertex v in G is the largest distance from v to another vertex, and the average eccentricity $\mathit{ecc}(G)$ of the graph G is $\frac{1}{n}{\sum }_{v\in V(G)}e(v)$. Recently, it was conjectured by Aouchiche and Hansen (2011) [3] that for any connected graph G on $n?3$ vertices, $\mathit{ecc}(G)?\pi (G)?\mathit{ecc}(P_n)?\pi (P_n)$, with equality if and only if $G?P_n$. In this paper, we show that this conjecture is true.     

Deterministic extractors for small-space sources

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space s sources on ${\{0,1\}}^n$ as sources generated by width $2^s$ branching programs. Specifically, there is a constant $\eta >0$ such that for any $\zeta >n^{?\eta }$, our algorithm extracts $m=(\delta ?\zeta )n$ bits that are exponentially close to uniform (in variation distance) from space s sources with min-entropy δn, where $s=\Omega ({\zeta }^{3}n)$. Previously, nothing was known for $\delta ?1/2$, even for space 0. Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of r independent smaller sources over ${\{0,1\}}^{?}$, where the total min-entropy over all the smaller sources is k. We give deterministic extractors for such sources when k is as small as $\mathsf{polylog}(r)$, for small enough ?.     

Equations over sets of integers with addition only

Systems of equations of the form $X=Y+Z$ and $X=C$ are considered, in which the unknowns are sets of integers, the plus operator denotes element-wise sum of sets, and C is an ultimately periodic constant. For natural numbers, such equations are equivalent to language equations over a one-symbol alphabet using concatenation and regular constants. It is shown that such systems are computationally universal: for every recursive (r.e., co-r.e.) set $S\subseteq \mathbb{N}$ there exists a system with a unique (least, greatest) solution containing a component T with $S=\{n|16n+13\in T\}$. Testing solution existence for these equations is ${\mathrm{\Pi }}_1^0$-complete, solution uniqueness is ${\mathrm{\Pi }}_2^0$-complete, and finiteness of the set of solutions is ${\mathrm{\Sigma }}_3^0$-complete. A similar construction for integers represents any hyper-arithmetical set $S\subseteq \mathbb{Z}$ by a set $T\subseteq \mathbb{Z}$ satisfying $S=\{n|16n\in T\}$, whereas testing solution existence is ${\mathrm{\Sigma }}_1^1$-complete.