Raphael Yuster

Color-coding

We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, in particular, the following new results: • For every fixed k, if a graph G = (V,E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V ω log V ) worst-case time, where ω < 2.376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of |V | and |E| whenever no confusion may arise.) • For every fixed k, if a planar graph G = (V,E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V log V ) worst-case time. The same algorithm applies, in fact, not only to planar graphs, but to any minor closed family of graphs which is not the family of all graphs. • If a graph G = (V,E) contains a subgraph isomorphic to a bounded tree-width graph H = (VH , EH) where |VH | = O(log V ), then such a copy of H can be found in polynomial time. This was not previously known even if H were just a path of length O(log V ). These results improve upon previous results of many authors. The third result resolves in the affirmative a conjecture of Papadimitriou and Yannakakis that the LOG PATH problem is in P. We can show that it is even in NC.

The algorithmic aspects of the regularity lemma

Abstract The regularity lemma of Szemeredi asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an n -vertex graph, can be found in time O ( M ( n )), where M ( n ) = O ( n 2.376 ) is the time needed to multiply two n by n matrices with 0, 1-entries over the integers. The algorithm can be parallelized and implemented in NC 1 . Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the regularity lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.

Fast sparse matrix multiplication

Let <i>A</i> and <i>B</i> two <i>n</i>×<i>n</i> matrices over a ring <i>R</i> (e.g., the reals or the integers) each containing at most <i>m</i> nonzero elements. We present a new algorithm that multiplies <i>A</i> and <i>B</i> using <i>O</i>(<i>m</i>^0.7<i>n</i>^1.2+<i>n</i>^{2+<i>o</i>(1)}) algebraic operations (i.e., multiplications, additions and subtractions) over <i>R</i>. The naïve matrix multiplication algorithm, on the other hand, may need to perform Ω(<i>mn</i>) operations to accomplish the same task. For <i>m</i>≤<i>n</i>^1.14, the new algorithm performs an almost optimal number of only <i>n</i>^{2+<i>o</i>(1)} operations. For <i>m</i>≤<i>n</i>^1.68, the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses <i>O</i>(<i>n</i>^2.38) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast <i>rectangular</i> matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices. As the known fast rectangular matrix multiplication algorithms are far from being practical, our result, at least for now, is only of theoretical value.

Finding and counting given length cycles

We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results.

Hardness and algorithms for rainbow connection

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In the first result of this paper we prove that computing rc(G) is NP-Hard solving an open problem from Caro et al. (Electron. J. Comb. 15, 2008, Paper R57). In fact, we prove that it is already NP-Complete to decide if rc(G)=2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ε>0, a connected graph with minimum degree at least εn has bounded rainbow connection, where the bound depends only on ε, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

H-Factors in Dense Graphs

The following asymptotic result is proved. For every?>0, and for every positive integerh, there exists ann0=n0(?, h) such that for every graphHwithhvertices and for everynn0, any graphGwithhnvertices and with minimum degreed?((?(H)?1)/?(H)+?)hncontainsnvertex disjoint copies ofH. This result is asymptotically tight and its proof supplies a polynomial time algorithm for the corresponding algorithmic problem.

Connected Domination and Spanning Trees with Many Leaves

Let G=(V,E) be a connected graph. A connected dominating set

Approximation algorithms and hardness results for cycle packing problems

S \subset V

Color-coding: a new method for finding simple paths, cycles and other small subgraphs within large graphs

is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted

AlmostH-factors in dense graphs

\gamma_c(G)