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We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimum-weight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dual-change � for each tree. As a benchmark of the algorithm’s performance, solving a 100,000-node geometric instance on a 200 Mhz Pentium-Pro computer takes approximately 3 minutes.

. We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms ...

Dominance constraints are logical descriptions of trees that are widely used in computational linguistics. Their general

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm+n 2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3

Recently, algorithms for computing game-theoretic solutions have been deployed in real-world security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strategy, a so-called Stackelberg model. As pointed out by Kiekintveld et al. (Kiekintveld et al. 2009), in these applications, generally, multiple resources need to be assigned to multiple targets, resulting in an exponential number of pure strategies for the defender. In this paper, we study how to compute optimal Stackelberg strategies in such games, showing that this can be done in polynomial time in some cases, and is NP-hard in others.

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial-time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm + n² log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore, there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear-time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/1. This improves the previously best performance ratio of 3/2. Our algorithm is not only of theoretical interest, but because it is easy to implement and the constants involved are quite small it is also useful in practice.

The sum coloring problem asks to find a vertex coloring of a given graph G, using natural numbers, such that the total sum of the colors of vertices is minimized amongst all proper vertex colorings of G. This minimum total sum is the chromatic sum of the graph, \Sigma(G), and a coloring which achieves this total sum is called an optimum coloring. There are some graphs for which the optimum coloring needs more colors than indicated by the chromatic number. The minimum number of colors needed in any optimum coloring of a graph is called the strength of the graph, which we denote by s(G). Trivially (G) s(G). In this thesis we present various results about the sum coloring problem. We prove the NP-Hardness of finding the vertex strength for graphs with \Delta = 6 and also give some logarithmic upper bounds for the vertex strength of graphs with small chromatic number. We also prove that the sum coloring problem is NP-complete for split graphs. Polynomial time algorithms are presented for the sum coloring of k-split graphs, P 4 -reducible graphs, chain bipartite graphs, and cobipartite graphs. We can

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Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants ɛ(H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 − ɛ(H))|G|, then the H-packing number of G and the H-covering number of G can be computed in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given. 1