Antoni Lozano

The complexity of algorithmic problems on succinct instances

Highly regular combinatorial objects can be represented advantageously by some kind of description shorter than their full standard encoding. For instance, graphs exhibiting enough regularities can be described using encodings substantially shorter than the full adjacency matrix. Anatural scheme for such succinct representations is by means of boolean circuits computing, as a boolean function, the values of individual bits of the binary encoding of the object. The complexity of many algorithmic problems changes drastically when this succinct representation is used to present the input. Two powerful lemmas quantifying exactly this increase of complexity are presented. These are applied to show that previous results in the area can be interpreted assufficient conditions for completeness in the logarithmic time and polynomial time counting hierarchies.

Mining frequent closed rooted trees

Many knowledge representation mechanisms are based on tree-like structures, thus symbolizing the fact that certain pieces of information are related in one sense or another. There exists a well-studied process of closure-based data mining in the itemset framework: we consider the extension of this process into trees. We focus mostly on the case where labels on the nodes are nonexistent or unreliable, and discuss algorithms for closure-based mining that only rely on the root of the tree and the link structure. We provide a notion of intersection that leads to a deeper understanding of the notion of support-based closure, in terms of an actual closure operator. We describe combinatorial characterizations and some properties of ordered trees, discuss their applicability to unordered trees, and rely on them to design efficient algorithms for mining frequent closed subtrees both in the ordered and the unordered settings. Empirical validations and comparisons with alternative algorithms are provided.

Succinct circuit representations and leaf language classes are basically the same concept

This note connects two topics of complexity theory: The topic of succinct circuit representations initiated by Galperin and Wigderson (1983), and the topic of leaf language classes initiated by Bovet et al. (1992). It will be shown for any language that its succinct version is polynomial-time many-one complete for the leaf language class determined by it.

Self-reducible sets of small density

We study the complexity of sets that are at the same time self-reducible and sparse orm-reducible to sparse sets. We show that sets of this kind are low for the complexity classes Δ2p, Θ2p, NP, or P, depending on the type of self-reducibility used and on certain restrictions imposed on the query mechanism of the self-reducibility machines. The proof of some of these results is based on graph-theoretic properties that hold for the graphs induced by the self-reducibility structures.

Seeded tree alignment and planar tanglegram layout

The optimal transformation of one tree into another by means of elementary edit operations is an important algorithmic problem that has several interesting applications to computational biology. We introduce a constrained form of this problem in which a partial mapping of a set of nodes in one tree to a corresponding set of nodes in the other tree is given, and present efficient algorithms for both ordered and unordered trees. Whereas ordered tree matching based on seeded nodes has applications in pattern matching of RNA structures, unordered tree matching based on seeded nodes has applications in co-speciation and phylogeny reconciliation. The latter involves the solution of the planar tanglegram layout problem, for which we give a polynomial-time algorithm.

Mining Frequent Closed Unordered Trees Through Natural Representations

Many knowledge representation mechanisms consist of link-based structures; they may be studied formally by means of unordered trees. Here we consider the case where labels on the nodes are nonexistent or unreliable, and propose data mining processes focusing on just the link structure. We propose a representation of ordered trees, describe a combinatorial characterization and some properties, and use them to propose an efficient algorithm for mining frequent closed subtrees from a set of input trees. Then we focus on unordered trees, and show that intrinsic characterizations of our representation provide for a way of avoiding the repeated exploration of unordered trees, and then we give an efficient algorithm for mining frequent closed unordered trees.

On sparse hard sets for counting classes

Abstract In this paper, we study one-word-decreasing self-reducible sets which are introduced by Lozano and Toran (1991). These are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is lexicographically smaller than the input. We show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact, we can prove that for any class K chosen from {PP, NP, C = P, MOD2P, MOD3 P,…} it holds that (1) if there is a sparse ⩽Pbtt-hard set for K then K ⊆ P, and (2) if there is a sparse ⩽SNbtt-hard set for K then K ⊆ NP ⌢ co-NP. This generalizes the result of Ogiwara and Watanabe (1991) to the mentioned complexity classes.

Subtree Testing and Closed Tree Mining Through Natural Representations

Several classical schemes exist to represent trees as strings over a fixed alphabet; these are useful in many algorithmic and conceptual studies. Our previous work has proposed a representation of unranked trees as strings over a countable alphabet, and has shown how this representation is useful for canonizing unordered trees and for mining closed frequent trees, whether ordered or unordered. Here we propose a similar, simpler alternative and adapt some basic algorithmics to it; then we show empirical evidence of the usefulness of this representation for mining frequent closed unordered trees on real-life data.

The Complexity of Modular Graph Automorphism

Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems, we define and study the modular graph automorphism problems. These are the decision problems ModkGA which consist, for each k > 1, of deciding whether the number of automorphisms of a graph is divisible by k. The ModkGA problems all turn out to be intermediate in difficulty between Graph Automorphism and Graph Isomorphism. We define an appropriate search version of ModkGA and design an algorithm that polynomial-time reduces the ModkGA search problem to the decision problem. Combining this algorithm with an IP protocol, we obtain a randomized polynomial-time checker for ModkGA, for all k > 1.

Tile-Packing Tomography Is

Discrete tomography deals with reconstructing finite spatial objects from their projections. The objects we study in this paper are called tilings or tile-packings, and they consist of a number of disjoint copies of a fixed tile, where a tile is defined as a connected set of grid points. A row projection specifies how many grid points are covered by tiles in a given row; column projections are defined analogously. For a fixed tile, is it possible to reconstruct its tilings from their projections in polynomial time? It is known that the answer to this question is affirmative if the tile is a bar (its width or height is 1), while for some other types of tiles