Gabriel Istrate

The strong equivalence of ET0L grammars

Abstract We define a version of structural equivalence of ET0L grammars , called strong equivalence , that takes into account their matrix structure, and prove its decidability.

Partition into Heapable Sequences, Heap Tableaux and a Multiset Extension of Hammersley’s Process

We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersleys process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is

Counting, structure identification and maximum consistency for binary constraint satisfaction problems

\frac{1+\sqrt{5}}{2}\cdot \ln(n)

Heapability, Interactive Particle Systems, Partial Orders: Results and Open Problems

. Finally we give a heap-based extension of Young tableaux, prove a hook inequality and an extension of the Robinson-Schensted correspondence.

The Language (and Series) of Hammersley-Type Processes

Using a framework inspired by Schaefers generalized satisfiability model [Sch78], Cohen, Cooper and Jeavons [CCJ94] studied the computational complexity of constraint satisfaction problems in the special case when the set of constraints is closed under permutation of labels and domain restriction, and precisely identified the tractable (and intractable) cases. Using the same model we characterize the complexity of three related problems: 1. counting the number of solutions. 2. structure identification (Dechter and Pearl [DP92]). 3. approximating the maximum number of satisfiable constraints. Supported in part by the NSF grant CCR-9701911

Improved approximation algorithms for low-density instances of the Minimum Entropy Set Cover Problem

We outline results and open problems concerning partitioning of integer sequences and partial orders into heapable subsequences (previously defined and established by Byers et al.).

Dimension-Dependent behavior in the satisfability of random k-Horn formulae

We study languages and formal power series associated to (variants of) Hammersleys process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the extension to intervals of the Hammersley process we show that there are two relevant formal languages. One of them leads to the same class of languages as the ordinary Hammersley tree process. The other one yields non-context-free languages. The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the variants of the Hammersleys process, that have the formal languages studies in this paper as their support. We employ these algorithms to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.

The phase transition in random Horn satisfiability and its algorithmic implications

We study the approximability of instances of the minimum entropy set cover problem, parameterized by the average frequency of a random element in the covering sets. We analyze an algorithm combining a greedy approach with another one biased towards large sets. The algorithm is controlled by the percentage of elements to which we apply the biased approach. The optimal parameter choice leads to improved approximation guarantees when average element frequency is less than e.