Approximation Algorithms for Union and Intersection Covering Problems

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in a union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, intersection k-MST can formalize the problem of connecting a subset of users to both electricity and water. We present a number of hardness and approximation results for union and intersection versions of several standard optimization problems: MST, Steiner tree, set cover, facility location, TSP, and their partial covering variants

A 1.751.75 LP approximation for the Tree Augmentation Problem

In the Tree Augmentation Problem (TAP) the goal is to augment a tree $T$ by a minimum size edge set $F$ from a given edge set $E$ such that $T \cup F$ is $2$-edge-connected. The best approximation ratio known for TAP is $1.5$. In the more general Weighted TAP problem, $F$ should be of minimum weight. Weighted TAP admits several $2$-approximation algorithms w.r.t. to the standard cut LP-relaxation, but for all of them the performance ratio of $2$ is tight even for TAP. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than $2$ is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than $2$, which is a long time open problem even for TAP. In this paper we introduce such an LP-relaxation and give an algorithm that computes a feasible solution for TAP of size at most $1.75$ times the optimal LP value. This gives some hope to break the ratio $2$ for the weighted case. Our algorithm computes some initial edge set by solving a partial system of constraints that form the integral edge-cover polytope, and then applies local search on $3$-leaf subtrees to exchange some of the edges and to add additional edges. Thus we do not need to solve the LP, and the algorithm runs roughly in time required to find a minimum weight edge-cover in a general graph.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279

Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2