Approximation Algorithms For Problems Research Articles

On computing the 2-vertex-connected components of directed graphs

In this paper we consider the problem of computing the 2-vertex-connected components of directed graphs. We present a new algorithm for solving this problem in O(nm) time. Furthermore, we show that the old algorithm of Erusalimskii and Svetlov runs in O(nm2) time. In this paper, we investigate the relationship between 2-vertex-connected components and dominator trees. We also consider three applications of our new algorithm, which are approximation algorithms for problems that are generalization of the problem of approximating the smallest 2-vertex-connected spanning subgraph of 2-vertex-connected directed graph.

Improved approximations for buy-at-bulk and shallow-light $$k$$ k -Steiner trees and $$(k,2)$$ ( k , 2 ) -subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light $$k$$k-Steiner tree problem (SL$$k$$kST), we are given an undirected graph $$G=(V,E)$$G=(V,E) with terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, a cost function $$c:E\rightarrow \mathbb {R}^+$$c:E?R+, a length function $$\ell :E\rightarrow \mathbb {R}^+$$l:E?R+, a bound $$L>0$$L>0 and an integer $$k\ge 1$$k?1. The goal is to find a minimum $$c$$c-cost $$r$$r-rooted Steiner tree containing at least $$k$$k terminals whose diameter under $$\ell $$l metric is at most $$L$$L. The input to the buy-at-bulk $$k$$k-Steiner tree problem (BB$$k$$kST) is similar: graph $$G=(V,E)$$G=(V,E), terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, cost and length functions $$c,\ell :E\rightarrow \mathbb {R}^+$$c,l:E?R+, and an integer $$k\ge 1$$k?1. The goal is to find a minimum total cost $$r$$r-rooted Steiner tree $$H$$H containing at least $$k$$k terminals, where the cost of each edge $$e$$e is $$c(e)+\ell (e)\cdot f(e)$$c(e)+l(e)·f(e) where $$f(e)$$f(e) denotes the number of terminals whose path to root in $$H$$H contains edge $$e$$e. We present a bicriteria $$(O(\log ^2 n),O(\log n))$$(O(log2n),O(logn))-approximation for SL$$k$$kST: the algorithm finds a $$k$$k-Steiner tree with cost at most $$O(\log ^2 n\cdot \text{ opt }^*)$$O(log2n·opt?) where $$\text{ opt }^*$$opt? is the cost of an LP relaxation of the problem and diameter at most $$O(L\cdot \log n)$$O(L·logn). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX'06/Algorithmica'09) which had ratio $$(O(\log ^4 n), O(\log ^2 n))$$(O(log4n),O(log2n)). Using this, we obtain an $$O(\log ^3 n)$$O(log3n)-approximation for BB$$k$$kST, which improves upon the $$O(\log ^4 n)$$O(log4n)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost $$2$$2-edge-connected subgraph with at least $$k$$k vertices, which is introduced as the $$(k,2)$$(k,2)-subgraph problem in Lau et al. (2009) (STOC'07/SICOMP09). This generalizes some well-studied classical problems such as the $$k$$k-MST and the minimum cost $$2$$2-edge-connected subgraph problems. We give an $$O(\log n)$$O(logn)-approximation algorithm for this problem which improves upon the $$O(\log ^2 n)$$O(log2n)-approximation algorithm of Lau et al. (2009).

Approximation algorithms for graph approximation problems

Several versions of the graph approximation problem are under study. Approximation algorithms for these problems are proposed, and performance guarantees of the algorithms are obtained. In particular, it is shown that the problem of approximation by graphs with a bounded number of connected components belongs to the class APX.

Randomized coalition structure generation

Randomization can be employed to achieve constant factor approximations to the coalition structure generation problem in less time than all previous approximation algorithms. In particular, this manuscript presents a new randomized algorithm that can generate a 2 3 approximate solution in O ( n 2.587 n ) time, improving upon the previous algorithm that required O ( n 2.83 n ) time to guarantee the same performance. Also, the presented new techniques allow a 1 4 approximate solution to be generated in the optimal time of O ( 2 n ) and improves on the previous best approximation ratio obtainable in O ( 2 n ) time of 1 8 . The presented algorithms are based upon a careful analysis of the sizes and numbers of coalitions in the smallest optimal coalition structures. An empirical analysis of the new randomized algorithms compared to their deterministic counterparts is provided. We find that the presented randomized algorithms generate solutions with utility comparable to what is returned by their deterministic counterparts (in some cases producing better results on average). Moreover, a significant speedup was found for most approximation ratios for the randomized algorithms over the deterministic algorithms. In particular, the randomized 1 2 approximate algorithm runs in approximately 22.4 % of the time required for the deterministic 1 2 approximation algorithm for problems with between 20 and 27 agents.

Passing Messages to Lonely Numbers

Message-passing methods provide powerful approximation algorithms for problems that can be formulated in terms of (probabilistic) graphical models. These methods find applications in statistical physics, inference, and combinatorial optimization. Sudoku, a popular number puzzle, is a simple optimization problem that message-passing algorithms can help solve. Therefore, Sudoku is an ideal vehicle to demonstrate these methods' strengths and limitations.

Approximation algorithms for problems in scheduling with set-ups

In this paper, we present new approximation results for the offline problem of single machine scheduling with sequence-independent set-ups and item availability, where the jobs to be scheduled are independent (i.e., have no precedence constraints) and have a common release time. We present polynomial-time approximation algorithms for two versions of this problem. In the first version, the input includes a weight for each job, and the goal is to minimize the total weighted completion time. On any input, our algorithm produces a schedule whose total weighted completion time is within a factor 2 of optimal for that input. In the second version, the input includes a due date for each job, and the goal is to minimize the maximum lateness of any job. On any input, our algorithm produces a schedule with the following performance guarantee: the maximum lateness of a job is at most the maximum lateness of the optimal schedule on a machine that runs at half the speed of our machine.

Approximation algorithms for knapsack problems with cardinality constraints

We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting-plane approaches to integer linear programs. We focus our attention on approximation algorithms for the problem, describing a linear-storage Polynomial Time Approximation Scheme (PTAS) and a dynamic-programming based Fully Polynomial Time Approximation Scheme (FPTAS). The main ideas contained in our PTAS are used to derive PTAS's for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTAS's. We finally illustrate better PTAS's and FPTAS's for the subset sum case of the problem in which profits and weights coincide.