No Arabic abstract
Many load balancing problems that arise in scientific computing applications ask to partition a graph with weights on the vertices and costs on the edges into a given number of almost equally-weighted parts such that the maximum boundary cost over all parts is small. Here, this partitioning problem is considered for bounded-degree graphs G=(V,E) with edge costs c: E->R+ that have a p-separator theorem for some p>1, i.e., any (arbitrarily weighted) subgraph of G can be separated into two parts of roughly the same weight by removing a vertex set S such that the edges incident to S in the subgraph have total cost at most proportional to (SUM_e c^p_e)^(1/p), where the sum is over all edges e in the subgraph. We show for all positive integers k and weights w that the vertices of G can be partitioned into k parts such that the weight of each part differs from the average weight by less than MAX{w_v; v in V}, and the boundary edges of each part have cost at most proportional to (SUM_e c_e^p/k)^(1/p) + MAX_e c_e. The partition can be computed in time nearly proportional to the time for computing a separator S of G. Our upper bound on the boundary costs is shown to be tight up to a constant factor for infinitely many instances with a broad range of parameters. Previous results achieved this bound only if one has c=1, w=1, and one allows parts with weight exceeding the average by a constant fraction.
We consider cost constrain
In this paper we present a new data structure for double ended priority queue, called min-max fine heap, which combines the techniques used in fine heap and traditional min-max heap. The standard operations on this proposed structure are also presented, and their analysis indicates that the new structure outperforms the traditional one.
We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $Delta$ and an integer $k > 3Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is emph{perfectly} uniform over the set of all proper $k$-colorings; the expected running time of the algorithm is $mathrm{poly}(k,n)=widetilde{O}(nDelta^2cdot log(k))$. This improves upon a result of Huber~(STOC 1998) who obtained a polynomial time perfect sampling algorithm for $k>Delta^2+2Delta$. Prior to our work, no algorithm with expected running time $mathrm{poly}(k,n)$ was known to guarantee perfectly sampling with sub-quadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Hubers) is based on the Coupling from the Past method. Inspired by the emph{bounding chain} approach, pioneered independently by Huber~(STOC 1998) and Haggstrom & Nelander~(Scand.{} J.{} Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem.
The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: roughly 4 for estimation and roughly $6 + 2sqrt{10}$ for construction. We propose an algorithm that constructs an allocation with value within a factor of $6 + delta$ from the optimum for any constant $delta > 0$. The running time is polynomial in the input size for any constant $delta$ chosen.
In a bipartite max-min LP, we are given a bipartite graph $myG = (V cup I cup K, E)$, where each agent $v in V$ is adjacent to exactly one constraint $i in I$ and exactly one objective $k in K$. Each agent $v$ controls a variable $x_v$. For each $i in I$ we have a nonnegative linear constraint on the variables of adjacent agents. For each $k in K$ we have a nonnegative linear objective function of the variables of adjacent agents. The task is to maximise the minimum of the objective functions. We study local algorithms where each agent $v$ must choose $x_v$ based on input within its constant-radius neighbourhood in $myG$. We show that for every $epsilon>0$ there exists a local algorithm achieving the approximation ratio ${Delta_I (1 - 1/Delta_K)} + epsilon$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${Delta_I (1 - 1/Delta_K)}$. Here $Delta_I$ is the maximum degree of a vertex $i in I$, and $Delta_K$ is the maximum degree of a vertex $k in K$. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.