Isolation, Matching, and Counting Uniform and Nonuniform Upper Bounds

We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2 O(k)), an improvement over previous algorithms for some of these problems running in time O(n+k O(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent.

https://www.cs.mcgill.ca/~sue/papers/fpt-puzzles.pdf

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

We introduce divide-and-color, a new technique for the solution of hard graph problems. It is a combination of the well-known divide-and-conquer paradigm and color-coding [2]. Our approach first randomly colors all edges or nodes of a graph black and white, and then solves the problem recursively on the two induced parts.

We demonstrate this technique by giving new randomized algorithms for the solution of two important problems. These yield runtime bounds of O*(4k) for finding a simple path of length k and O*(4( h−−1) k) for finding k edge-disjoint (resp. vertex-disjoint) copies of a graph H with h edges (resp. h nodes) in a given graph. Derandomization gives deterministic algorithms for these problems with running times O*(24 k) and O*(24hk), respectively.

All these results significantly improve over the currently known best bounds. In particular, our generic algorithms beat specialized ones that have been designed to find k triangles or paths of length two.

Divide-and-color

Given a set of values x1, x2, ... x n , of which k are nonzero, the compaction problem is the problem of moving the nonzero elements into the first k consecutive memory locations. The chaining problem asks that the nonzero elements be put into a linked list. One can in addition require that the elements remain in the same order, leading to the problems of ordered compaction and ordered chaining respectively. Parallel algorithms for these problems were considered by Hagerup and Nowak (ICALP 1989). This paper improves their results by introducing a technique involving perfect hash functions. This leads to a deterministic algorithm for ordered compaction running on a CRCW PRAM in time O(log k/log log n) using n processors. A matching lower bound for unordered compaction is given. In addition, the ordered chaining problem is shown to be solvable in time O(α(k)) with n processors (where α is a functional inverse of Ackermann's function) and unordered chaining can be solved in constant time with n processors when k<n1/4−e.

The parallel simplicity of compaction and chaining

Efficient algorithms based upon Balinski's signature method are described for solving then × n assignment problem. These algorithms are special variants of the dual simplex method and are shown to have computational bounds of O(n3). Variants for solving sparse assignment problems withm arcs that require O(m) space and O(mn + n2 logn) time in the worst case are also presented.

/pdf/efficient-dual-simplex-algorithms-for-the-assignment-problem-4qk8kswf50.pdf

Szemeredi

Papers

Storing a sparse table with O(1) worst case access time