arXiv subjects

Romeo Rizzi

Publications and source records attributed to Romeo Rizzi.

49 records · Page 3Linked to original sources

arXiv2015

Enumerating Cyclic Orientations of a Graph

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the \emph{cyclic} orientations of an undirected connected graph with $n$ vertices and $m$ edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations.We show that the problem is of independent interest from both combinatorial and algorithmic points of view, and that each cyclic orientation can be listed with $\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of $\tilde{O}(m)$ time.

cs.DS↗

arXiv2014

On the complexity of the vector connectivity problem

We study a relaxation of the Vector Domination problem called Vector Connectivity (VecCon). Given a graph $G$ with a requirement $r(v)$ for each vertex $v$, VecCon asks for a minimum cardinality set $S$ of vertices such that every vertex $v\in V\setminus S$ is connected to $S$ via $r(v)$ disjoint paths. In the paper introducing the problem, Boros et al. [Networks, 2014] gave polynomial-time solutions for VecCon in trees, cographs, and split graphs, and showed that the problem can be approximated in polynomial time on $n$-vertex graphs to within a factor of $\log n+2$, leaving open the question of whether the problem is NP-hard on general graphs. We show that VecCon is APX-hard in general graphs, and NP-hard in planar bipartite graphs and in planar line graphs. We also generalize the polynomial result for trees by solving the problem for block graphs.

cs.DM↗

arXiv2014

Efficiently listing bounded length st-paths

The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s. For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971 and 1972], and an $O(K(m+n \log n))$ algorithm for the undirected version by Katoh et al. [Networks, 1982], both using $O(Kn + m)$ space. In this work, we consider a different parameterization for this problem: instead of bounding the number of $st$-paths output, we bound their length. For the bounded length parameterization, we propose new non-trivial algorithms matching the time complexity of the classic algorithms but using only $O(m+n)$ space. Moreover, we provide a unified framework such that the solutions to both parameterizations -- the classic $K$-shortest and the new length-bounded paths -- can be seen as two different traversals of a same tree, a Dijkstra-like and a DFS-like traversal, respectively.

cs.DS↗

arXiv2014

Some results on more flexible versions of Graph Motif

The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Due to the high rate of noise in the biological data, more flexible definitions of the problem have been outlined. We present in this paper two inapproximability results for two different optimization variants of Graph Motif: one where the size of the solution is maximized, the other when the number of substitutions of colors to obtain the motif from the solution is minimized. We also study a decision version of Graph Motif where the connectivity constraint is replaced by the well known notion of graph modularity. While the problem remains NP-complete, it allows algorithms in FPT for biologically relevant parameterizations.

cs.CC↗

arXiv2014

Amortized $\tilde{O}(|V|)$-Delay Algorithm for Listing Chordless Cycles in Undirected Graphs

Chordless cycles are very natural structures in undirected graphs, with an important history and distinguished role in graph theory. Motivated also by previous work on the classical problem of listing cycles, we study how to list chordless cycles. The best known solution to list all the $C$ chordless cycles contained in an undirected graph $G = (V,E)$ takes $O(|E|^2 +|E|\cdot C)$ time. In this paper we provide an algorithm taking $\tilde{O}(|E| + |V |\cdot C)$ time. We also show how to obtain the same complexity for listing all the $P$ chordless $st$-paths in $G$ (where $C$ is replaced by $P$ ).

cs.DS↗

arXiv2013

Combinatorial decomposition approaches for efficient counting and random generation FPTASes

Given a combinatorial decomposition for a counting problem, we resort to the simple scheme of approximating large numbers by floating-point representations in order to obtain efficient Fully Polynomial Time Approximation Schemes (FPTASes) for it. The number of bits employed for the exponent and the mantissa will depend on the error parameter $0 < \varepsilon \leq 1$ and on the characteristics of the problem. Accordingly, we propose the first FPTASes with $1 \pm \varepsilon$ relative error for counting and generating uniformly at random a labeled DAG with a given number of vertices. This is accomplished starting from a classical recurrence for counting DAGs, whose values we approximate by floating-point numbers. After extending these results to other families of DAGs, we show how the same approach works also with problems where we are given a compact representation of a combinatorial ensemble and we are asked to count and sample elements from it. We employ here the floating-point approximation method to transform the classic pseudo-polynomial algorithm for counting 0/1 Knapsack solutions into a very simple FPTAS with $1 - \varepsilon$ relative error. Its complexity improves upon the recent result (Štefankovič et al., SIAM J. Comput., 2012), and, when $\varepsilon^{-1} = Ω(n)$, also upon the best-known randomized algorithm (Dyer, STOC, 2003). To show the versatility of this technique, we also apply it to a recent generalization of the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG, obtaining a faster and simpler FPTAS than the existing one.

cs.DS↗

arXiv2013

A Novel Combinatorial Method for Estimating Transcript Expression with RNA-Seq: Bounding the Number of Paths

RNA-Seq technology offers new high-throughput ways for transcript identification and quantification based on short reads, and has recently attracted great interest. The problem is usually modeled by a weighted splicing graph whose nodes stand for exons and whose edges stand for split alignments to the exons. The task consists of finding a number of paths, together with their expression levels, which optimally explain the coverages of the graph under various fitness functions, such least sum of squares. In (Tomescu et al. RECOMB-seq 2013) we showed that under general fitness functions, if we allow a polynomially bounded number of paths in an optimal solution, this problem can be solved in polynomial time by a reduction to a min-cost flow program. In this paper we further refine this problem by asking for a bounded number k of paths that optimally explain the splicing graph. This problem becomes NP-hard in the strong sense, but we give a fast combinatorial algorithm based on dynamic programming for it. In order to obtain a practical tool, we implement three optimizations and heuristics, which achieve better performance on real data, and similar or better performance on simulated data, than state-of-the-art tools Cufflinks, IsoLasso and SLIDE. Our tool, called Traph, is available at http://www.cs.helsinki.fi/gsa/traph/

q-bio.QM↗

arXiv2013

Indexes for Jumbled Pattern Matching in Strings, Trees and Graphs

We consider how to index strings, trees and graphs for jumbled pattern matching when we are asked to return a match if one exists. For example, we show how, given a tree containing two colours, we can build a quadratic-space index with which we can find a match in time proportional to the size of the match. We also show how we need only linear space if we are content with approximate matches.

cs.DS↗

arXiv2012

Set graphs. II. Complexity of set graph recognition and similar problems

A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we continue the study of set graphs and related topics, focusing on computational complexity aspects. We prove that set graph recognition is NP-complete, even when the input is restricted to bipartite graphs with exactly two leaves. The problem remains NP-complete if, in addition, we require that the extensional acyclic orientation be also `slim', that is, that the digraph obtained by removing any arc from it is not extensional. We also show that the counting variants of the above problems are #P-complete, and prove similar complexity results for problems related to a generalization of extensional acyclic digraphs, the so-called `hyper-extensional digraphs', which were proposed by Aczel to describe hypersets. Our proofs are based on reductions from variants of the Hamiltonian Path problem. We also consider a variant of the well-known notion of a separating code in a digraph, the so-called `open-out-separating code', and show that it is NP-complete to determine whether an input extensional acyclic digraph contains an open-out-separating code of given size.

cs.DM↗

arXiv2012

Optimal Listing of Cycles and st-Paths in Undirected Graphs

We present the first optimal algorithm for the classical problem of listing all the cycles in an undirected graph. We exploit their properties so that the total cost is the time taken to read the input graph plus the time to list the output, namely, the edges in each of the cycles. The algorithm uses a reduction to the problem of listing all the paths from a vertex s to a vertex t which we also solve optimally.

cs.DS↗

arXiv2010

Pure Parsimony Xor Haplotyping

The haplotype resolution from xor-genotype data has been recently formulated as a new model for genetic studies. The xor-genotype data is a cheaply obtainable type of data distinguishing heterozygous from homozygous sites without identifying the homozygous alleles. In this paper we propose a formulation based on a well-known model used in haplotype inference: pure parsimony. We exhibit exact solutions of the problem by providing polynomial time algorithms for some restricted cases and a fixed-parameter algorithm for the general case. These results are based on some interesting combinatorial properties of a graph representation of the solutions. Furthermore, we show that the problem has a polynomial time k-approximation, where k is the maximum number of xor-genotypes containing a given SNP. Finally, we propose a heuristic and produce an experimental analysis showing that it scales to real-world large instances taken from the HapMap project.

cs.CE↗

arXiv2005

Online Permutation Routing in Partitioned Optical Passive Star Networks

This paper establishes the state of the art in both deterministic and randomized online permutation routing in the POPS network. Indeed, we show that any permutation can be routed online on a POPS network either with $O(\frac{d}{g}\log g)$ deterministic slots, or, with high probability, with $5c\lceil d/g\rceil+o(d/g)+O(\log\log g)$ randomized slots, where constant $c=\exp (1+e^{-1})\approx 3.927$. When $d=Θ(g)$, that we claim to be the "interesting" case, the randomized algorithm is exponentially faster than any other algorithm in the literature, both deterministic and randomized ones. This is true in practice as well. Indeed, experiments show that it outperforms its rivals even starting from as small a network as a POPS(2,2), and the gap grows exponentially with the size of the network. We can also show that, under proper hypothesis, no deterministic algorithm can asymptotically match its performance.

cs.DC↗

arXiv2001

Routing Permutations in Partitioned Optical Passive Star Networks

It is shown that a POPS network with g groups and d processors per group can efficiently route any permutation among the n=dg processors. The number of slots used is optimal in the worst case, and is at most the double of the optimum for all permutations p such that p(i)<>i for all i.

cs.DC↗