Orlando Lee

Sorting signed permutations by short operations

BackgroundDuring evolution, global mutations may alter the order and the orientation of the genes in a genome. Such mutations are referred to as rearrangement events, or simply operations. In unichromosomal genomes, the most common operations are reversals, which are responsible for reversing the order and orientation of a sequence of genes, and transpositions, which are responsible for switching the location of two contiguous portions of a genome. The problem of computing the minimum sequence of operations that transforms one genome into another – which is equivalent to the problem of sorting a permutation into the identity permutation – is a well-studied problem that finds application in comparative genomics. There are a number of works concerning this problem in the literature, but they generally do not take into account the length of the operations (i.e. the number of genes affected by the operations). Since it has been observed that short operations are prevalent in the evolution of some species, algorithms that efficiently solve this problem in the special case of short operations are of interest.ResultsIn this paper, we investigate the problem of sorting a signed permutation by short operations. More precisely, we study four flavors of this problem: (i) the problem of sorting a signed permutation by reversals of length at most 2; (ii) the problem of sorting a signed permutation by reversals of length at most 3; (iii) the problem of sorting a signed permutation by reversals and transpositions of length at most 2; and (iv) the problem of sorting a signed permutation by reversals and transpositions of length at most 3. We present polynomial-time solutions for problems (i) and (iii), a 5-approximation for problem (ii), and a 3-approximation for problem (iv). Moreover, we show that the expected approximation ratio of the 5-approximation algorithm is not greater than 3 for random signed permutations with more than 12 elements. Finally, we present experimental results that show that the approximation ratios of the approximation algorithms cannot be smaller than 3. In particular, this means that the approximation ratio of the 3-approximation algorithm is tight.

The Online Connected Facility Location Problem

In this paper we propose the Online Connected Facility Location problem (OCFL), which is an online version of the Connected Facility Location problem (CFL). The CFL is a combination of the Uncapacitated Facility Location problem (FL) and the Steiner Tree problem (ST). We give a randomized O(log2 n)-competitive algorithm for the OCFL via the sample-and-augment framework of Gupta, Kumar, Pal, and Roughgarden and previous algorithms for Online Facility Location (OFL) and Online Steiner Tree (OST). Also, we show that the same algorithm is a deterministic O(logn)-competitive algorithm for the special case of the OCFL with M = 1, where M is a scale factor for the edge costs.

The Eternal Dominating Set problem for proper interval graphs

We solve the Eternal Dominating Set problem for proper interval graphs.We show that, in this case, the optimum equals the largest size of an independent set.For such graphs, there is no advantage in allowing multiple guards at a vertex. In this paper, we solve the Eternal Dominating Set problem for proper interval graphs. We prove that, in this case, the optimal value of the problem equals the largest size of an independent set. As a consequence, we show that the problem can be solved in linear time for such graphs. To obtain the result, we first consider another problem in which a vertex can be occupied by an arbitrary number of guards. We then derive a lower bound on the optimal value of this latter problem, and prove that, for proper interval graphs, it is the same as the optimum of the first problem.

Flow-based formulation for the maximum leaf spanning tree problem

Abstract The maximum leaf spanning tree problem consists in finding a spanning tree of a given graph G with the maximum number of leaves. In this work we propose a flow-based mixed-integer linear programming model for this problem. Computational experiments are executed in two sets of instances from the literature. The results show that the model is competitive with alternative exact approaches.

The Online Prize-Collecting Facility Location Problem

Abstract In this paper we propose the Online Prize-Collecting Facility Location problem (OPCFL), which is an online version of the Prize-Collecting Facility Location problem (PCFL). The PCFL is a generalization of the Uncapacitated Facility Location problem (FL) in which some clients may be left unconnected by paying a penalty. Another way to think about it is that every client has a prize that can only be collected if it is connected. We give a primal-dual O(log n)-competitive algorithm for the OPCFL based on previous algorithms for Online Facility Location (OFL) due to Fotakis [Fotakis, D., A primal-dual algorithm for online non-uniform facility location, Journal of Discrete Algorithms 5 (2007), pp. 141–148] and Nagarajan and Williamson [Nagarajan, C. and D. P. Williamson, Offline and online facility leasing, Discrete Optimization 10 (2013), pp. 361–370].

A faster algorithm for packing branchings in digraphs

We consider the problem of finding an integral (and fractional) packing of branchings in a capacitated digraph with root-set demands. Schrijver described an algorithm that returns a packing with at most m + n 3 + r branchings that makes at most m ( m + n 3 + r ) calls to an oracle that basically computes a minimum cut, where n is the number of vertices, m is the number of arcs and r is the number of root-sets of the input digraph. Leston-Rey and Wakabayashi described an algorithm that returns a packing with at most m + r - 1 branchings but makes a large number of oracle calls. In this work we provide an algorithm, inspired on ideas of Schrijver and in a paper of Gabow and Manu, that returns a packing with at most m + r - 1 branchings and makes at most ( m + r + 2 ) n oracle calls. Moreover, for the arborescence packing problem our algorithm provides a packing with at most m - n + 2 arborescences-thus improving the bound of m of Leston-Rey and Wakabayashi-and makes at most ( m - n + 5 ) n oracle calls.

Removable cycles in non-bipartite graphs

In this paper we prove the following result. Suppose that s and t are vertices of a 3-connected graph G such that G-s-t is not bipartite and there is no cutset X of size three in G for which some component U of G-X is disjoint from {s,t}. Then either (1) G contains an induced path P from s to t such that G-V(P) is not bipartite or (2) G can be embedded in the plane so that every odd face contains one of s or t. Furthermore, if (1) holds then we can insist that G-V(P) is connected, while if G is 5-connected then (1) must hold and P can be chosen so that G-V(P) is 2-connected.

Packing dicycle covers in planar graphs with no K 5 - e minor

We prove that the minimum weight of a dicycle is equal to the maximum number of disjoint dicycle covers, for every weighted digraph whose underlying graph is planar and does not have K5–e as a minor (K5–e is the complete graph on five vertices, minus one edge). Equality was previously known when forbidding K4 as a minor, while an infinite number of weighted digraphs show that planarity does not guarantee equality. The result also improves upon results known for Woodalls Conjecture and the Edmonds-Giles Conjecture for packing dijoins. Our proof uses Wagners characterization of planar 3-connected graphs that do not have K5–e as a minor.

On Generalizations of the Parking Permit Problem and Network Leasing Problems

Abstract We propose a variant of the parking permit problem, called multi parking permit problem, in which an arbitrary demand is given at each instant and one may buy multiple permits to serve it. We show how to reduce this problem to the parking permit problem, while losing a constant cost factor. We obtain a 4-approximation algorithm and, for the online setting, a deterministic O(K)-competitive algorithm and a randomized O ( lg ⁡ K ) -competitive algorithm, where K is the number of permit types. For a leasing variant of the Steiner network problem, these results imply O(lg n)-approximation and online O ( lg ⁡ K lg ⁡ | V | ) -competitive algorithms, where n is the number of requests and | V | is the size of the input metric. Also, our technique turns into polynomial-time the pseudo-polynomial algorithms by Hu, Ludwig, Richa and Schmid for the 2D parking permit problem. For a leasing variant of the buy-at-bulk network design problem, these results imply: (i) an algorithm which improves the best previous approximation, and (ii) the first competitive online algorithm.

Advances in Aharoni-Hartman-Hoffman's Conjecture for Split digraphs

Abstract Let k be a positive integer and let D be a digraph. A (path) k-pack P k of D is a collection of at most k vertex-disjoint paths in D. The weight of a k-pack P k is the number of vertices covered by it and we say P k is optimal if its weight is maximum. A vertex-coloring C is orthogonal to a k-pack P k if each color class C ∈ C meets min ⁡ { | C | , k } paths of P k . In 1985, Aharoni, Hartman and Hoffman conjectured that for any optimal k-pack of D there exists a coloring orthogonal to it. In this paper we give a partial answer to this question by presenting two special types of k-packs in split digraphs for which we can always find an orthogonal coloring.