Linear time Constructions of some dd-Restriction Problems

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions

Improved Constructions of Frameproof Codes

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest cardinality of a $q$-ary $c$-frameproof code of length $l$ and $R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$, $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. In this paper, we give a recursive construction for $c$-frameproof codes of length $l$ with respect to the alphabet size $q$. As applications of this construction, we establish the existence results for $q$-ary $c$-frameproof codes of length $c+2$ and size $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $q\equiv 4\pmod{6}$ when $c=3$. Furthermore, we show that $R_{c,c+2}=(c+2)/c$ meeting the upper bound given by Blackburn, for all integers $c$ such that $c+1$ is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o

Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most $p$ into a given universe generalizes the Maximum Matching problem, which is recovered by taking $p=2$. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problem --- $r$-Simple $k$-Path --- that can be solved by a randomized algorithm with running time $O^*(2^{O(k \frac{\log r}{r})})$. That is, the complexity of the problem decreases with $r$. In this paper we pursue further the direction sketched by Abasi et al. Our main contribution is a derandomization tool that provides a deterministic counterpart of the main technical result of Abasi et al.: the $O^*(2^{O(k \frac{\log r}{r})})$ algorithm for $(r,k)$-Monomial Detection, which is the problem of finding a monomial of total degree $k$ and individual degrees at most $r$ in a polynomial given as an arithmetic circuit. Our technique works for a large class of circuits, and in particular it can be used to derandomize the result of Abasi et al. for $r$-Simple $k$-Path. On our way to this result we introduce the notion of representative sets for multisets, which may be of independent interest. Finally, we give two more examples of problems that were already studied in the literature, where the same relaxation phenomenon happens. The first one is a natural relaxation of the Set Packing problem, where we allow the packed sets to overlap at each element at most $r$ times. The second one is Degree Bounded Spanning Tree, where we seek for a spanning tree of the graph with a small maximum degree