Arie Matsliah

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.

Testing Graph Isomorphism

We deal with the question of how many queries are required to distinguish between the case that two graphs <i>G</i> and <i>H</i> on <i>n</i> vertices are isomorphic, and the case that they are ε-far, that is they differ in more than ε(^<i>n</i><inf>2</inf>) pairs for all possible bijections of their vertices. Querying is defined as probing the adjacency matrix of any one of the two graphs, i.e. asking if a pair of vertices forms an edge of the graph or not.We investigate both one-sided error and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried; and the second setting is where one of the graphs is known to the algorithm in advance.We prove that the query complexity of the one-sided error testing problem is Θ(<i>n</i>^3/2) if both graphs need to be queried, and that it is Θ(<i>n</i>) if one of the graphs is known in advance (where the Θ notation hides polylogarithmic factors in the upper bounds). For the two-sided error testers we prove that the query complexity is Θ(√<i>n</i> when one of the graphs is known in advance, and we show that the query complexity lies between Ω(<i>n</i>) and <i>Õ</i>(<i>n</i>^5/4) if both <i>G</i> and <i>H</i> need to be queried. All of our algorithms are additionally non-adaptive, while all of our lower bounds apply for adaptive testers as well as non-adaptive ones.

Approximate Hypergraph Partitioning and Applications

We show that any partition-problem of hypergraphs has an O(n) time approximate partitioning algorithm and an efficient property tester. This extends the results of Goldreich, Goldwasser and Ron who obtained similar algorithms for the special case of graph partition problems in their seminal paper (1998). The partitioning algorithm is used to obtain the following results: ldr We derive a surprisingly simple O(n) time algorithmic version of Szemeredis regularity lemma. Unlike all the previous approaches for this problem which only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists; ldr For any r ges 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n2r-1) time (deterministic) algorithms. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/isin) queries and constant running time.

Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions f : {0, 1}n → Rcan be tested with O(n/e) queries, for any totally ordered range R. More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(eµ(n))). We construct a family of instances of Ω(2n) pairs in n-dimensional hypercubes such that no more than roughly a 1/√n fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≅ n3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanskis conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not n1/2-δ-realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/e) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.

On the query complexity of testing orientations for being Eulerian

We consider testing directed graphs Eulerianity in the orientation model introduced in Halevy et al. [2005]. Despite the local nature of the Eulerian property, it turns out to be significantly harder to test than other properties studied in the orientation model. We show a nonconstant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sublinear query complexity 2-sided test, for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.

Hardness and Algorithms for Rainbow Connectivity

An edge-colored graph

Relating proof complexity measures and practical hardness of SAT

G

Computing interpolants without proofs

is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connectivity} of a connected graph

Testing st-Connectivity

G

Sound 3-Query PCPPs Are Long

, denoted