Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Deterministic Algorithm for Maximizing Submodular Functions
174
0
0.0
(
0
)
Added by
Shahar Dobzinski
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The problem of maximizing a non-negative submodular function was introduced by Feige, Mirrokni, and Vondrak [FOCS07] who provided a deterministic local-search based algorithm that guarantees an approximation ratio of $frac 1 3$, as well as a randomized $frac 2 5$-approximation algorithm. An extensive line of research followed and various algorithms with improving approximation ratios were developed, all of them are randomized. Finally, Buchbinder et al. [FOCS12] presented a randomized $frac 1 2$-approximation algorithm, which is the best possible. This paper gives the first deterministic algorithm for maximizing a non-negative submodular function that achieves an approximation ratio better than $frac 1 3$. The approximation ratio of our algorithm is $frac 2 5$. Our algorithm is based on recursive composition of solutions obtained by the local search algorithm of Feige et al. We show that the $frac 2 5$ approximation ratio can be guaranteed when the recursion depth is $2$, and leave open the question of whether the approximation ratio improves as the recursion depth increases.
rate research
Read More
We design new serial and parallel approximation algorithms for computing a maximum weight $b$-matching in an edge-weighted graph with a submodular objective function. This problem is NP-hard; the new algorithms have approximation ratio $1/3$, and are relaxations of the Greedy algorithm that rely only on local information in the graph, making them parallelizable. We have designed and implemented Local Lazy Greedy algorithms for both serial and parallel computers. We have applied the approximate submodular $b$-matching algorithm to assign tasks to processors in the computation of Fock matrices in quantum chemistry on parallel computers. The assignment seeks to reduce the run time by balancing the computational load on the processors and bounding the number of messages that each processor sends. We show that the new assignment of tasks to processors provides a four fold speedup over the currently used assignment in the NWChemEx software on $8000$ processors on the Summit supercomputer at Oak Ridge National Lab.
In a minimum cost submodular cover problem (MinSMC), given a monotone non-decreasing submodular function $fcolon 2^V rightarrow mathbb{Z}^+$, a cost function $c: Vrightarrow mathbb R^{+}$, an integer $kleq f(V)$, the goal is to find a subset $Asubseteq V$ with the minimum cost such that $f(A)geq k$. MinSMC has a lot of applications in machine learning and data mining. In this paper, we design a parallel algorithm for MinSMC which obtains a solution with approximation ratio at most $frac{H(min{Delta,k})}{1-5varepsilon}$ with probability $1-3varepsilon$ in $O(frac{log mlog nlog^2 mn}{varepsilon^4})$ rounds, where $Delta=max_{vin V}f(v)$, $H(cdot)$ is the Hamornic number, $n=f(V)$, $m=|V|$ and $varepsilon$ is a constant in $(0,frac{1}{5})$. This is the first paper obtaining a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to $H(min{Delta,k})$.
We study the problem of maximizing a monotone $k$-submodular function $f$ under a knapsack constraint, where a $k$-submodular function is a natural generalization of a submodular function to $k$ dimensions. We present a deterministic $(frac12-frac{1}{2e})$-approximation algorithm that evaluates $f$ $O(n^5k^4)$ times.
We show a deterministic algorithm for computing edge connectivity of a simple graph with $m$ edges in $m^{1+o(1)}$ time. Although the fastest deterministic algorithm by Henzinger, Rao, and Wang [SODA17] has a faster running time of $O(mlog^{2}mloglog m)$, we believe that our algorithm is conceptually simpler. The key tool for this simplication is the expander decomposition. We exploit it in a very straightforward way compared to how it has been previously used in the literature.
In this work, we give a new parallel algorithm for the problem of maximizing a non-monotone diminishing returns submodular function subject to a cardinality constraint. For any desired accuracy $epsilon$, our algorithm achieves a $1/e - epsilon$ approximation using $O(log{n} log(1/epsilon) / epsilon^3)$ parallel rounds of function evaluations. The approximation guarantee nearly matches the best approximation guarantee known for the problem in the sequential setting and the number of parallel rounds is nearly-optimal for any constant $epsilon$. Previous algorithms achieve worse approximation guarantees using $Omega(log^2{n})$ parallel rounds. Our experimental evaluation suggests that our algorithm obtains solutions whose objective value nearly matches the value obtained by the state of the art sequential algorithms, and it outperforms previous parallel algorithms in number of parallel rounds, iterations, and solution quality.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
