# FPT Algorithms for Diverse Collections of Hitting Sets

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## Abstract

**:**

## 1. Introduction

Vertex Cover | |

Input: | Graph G. |

Solution: | A vertex cover S of G of the smallest size. |

**Air traffic control.**Conflict graphs are used in the design of decision support tools for aiding Air Traffic Controllers (ATCs) in preventing untoward incidents involving aircraft [2,3]. Each node in the graph G in this instance is an aircraft, and there is an edge between two nodes if the corresponding aircraft are at risk of interfering with each other. A vertex cover of G corresponds to a set of aircraft that can be issued resolution commands which ask them to change course, such that afterwards there is no risk of interference.In a situation involving a large number of aircraft, it is unlikely that every choice of ten aircraft to redirect is equally desirable. For instance, in general, it is likely that (i) it is better to ask smaller aircraft to change course in preference to larger craft, and (ii) it is better to ask aircraft which are cruising to change course, in preference to those which are taking off or landing.**Wireless spectrum allocation.**Conflict graphs are a standard tool in figuring out how to distribute wireless frequency spectrum among a large set of wireless devices so that no two devices whose usage could potentially interfere with each other are allotted the same frequencies [4,5]. Each node in G is a user, and there is an edge between two nodes if (i) the users request the same frequency, and (ii) their usage of the same frequency has the potential to cause interference. A vertex cover of G corresponds to a set of users whose requests can be denied, such that afterwards there is no risk of interference.When there is large collection of devices vying for spectrum, it is unlikely that every choice of ten devices to deny the spectrum is equally desirable. For instance, it is likely that denying the spectrum to a remote-controlled toy car on the ground is preferable to denying the spectrum to a drone in flight.**Managing inconsistencies in database integration.**A database constructed by integrating data from different data sources may end up being inconsistent (that is, violating specified integrity constraints) even if the constituent databases are individually consistent. Handling these inconsistencies is a major challenge in database integration, and conflict graphs are central to various approaches for restoring consistency [6,7,8,9]. Each node in G is a database item, and there is an edge between two nodes if the two items together form an inconsistency. A vertex cover of G corresponds to a set of database items in whose absence the database achieves consistency.In a database of large size, it is unlikely that all data are created equal; some database items are likely to be of better relevance or usefulness than others, and so it is unlikely that every choice of ten items to delete is equally desirable.

**The Diverse X Paradigm.**Mike Fellows has proposed the Diverse X Paradigm as a solution for these issues and others [11]. In this paradigm, “X” is a placeholder for an optimization problem, and we study the complexity—specifically, the fixed-parameter tractability—of the problem of finding a few different good quality solutions for X. Contrast this with the traditional approach of looking for just one good quality solution. Let X denote an optimization problem where one looks for a minimum-size subset of some set; Vertex Cover is an example of such a problem. The generic form of X is then:

X | |

Input: | An instance I of X. |

Solution: | A solution S of I of the smallest size. |

Diverse X | |

Input: | An instance I of X, and positive integers $k,r,t$. |

Parameter: | $(k,r)$ |

Solution: | A set $\mathcal{S}$ of r solutions of I, each of size at most k, such that a diversity measure of $\mathcal{S}$ is at least t. |

**Diversity measures.**The concept of diversity appears also in other fields, and there are many different ways to measure the diversity of a collection. For example, in ecology, the diversity of a set of species (“biodiversity”) is a topic that has become increasingly important in recent times—see, for example, Solow and Polasky [12].

#### Our Problems and Results

Feedback Vertex Set | |

Input: | A graph G. |

Solution: | A feedback vertex set of G of the smallest size. |

d-Hitting Set | |

Input: | A finite universe U and a family $\mathcal{F}$ of subsets of U, each of size at most d. |

Solution: | A hitting set S of $\mathcal{F}$ of the smallest size. |

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

- ${2}^{k{r}^{2}}\xb7{\left(kr\right)}^{O\left(1\right)}$ if $\left|U\right|<kr$ and
- ${d}^{kr}\xb7{\left|U\right|}^{O\left(1\right)}$ otherwise.

**Theorem**

**4.**

**Corollary**

**1.**

- ${2}^{k{r}^{2}}\xb7{\left(kr\right)}^{O\left(1\right)}$ if $n<kr$ and
- ${2}^{kr}\xb7{n}^{O\left(1\right)}$ otherwise.

**Related Work.**The parameterized complexity of finding a diverse collection of good-quality solutions to algorithmic problems seems to be largely unexplored. To the best of our knowledge, the only existing work in this area consists of: (i) a privately circulated manuscript by Fellows [11] which introduces the Diverse X Paradigm and makes a forceful case for its relevance, and (ii) a manuscript by Baste et al. [21] which applies the Diverse X Paradigm to vertex-problems with the treewidth of the input graph as an extra parameter. In this context, a vertex-problem is any problem in which the input contains a graph G and the solution is some subset of the vertex set of G that satisfies some problem-specific properties. Both Vertex Cover and Feedback Vertex Set are vertex-problems in this sense, as are many other graph problems. The treewidth of a graph is, informally put, a measure of how tree-like the graph is. See, e.g., ([22], Chapter 7) for an introduction of the use of the treewidth of a graph as a parameter in designing $\mathsf{FPT}$ algorithms. The work by Baste et al. [21] shows how to convert essentially any treewidth-based dynamic programming algorithm, for solving a vertex-problem, into an algorithm for computing a diverse set of r solutions for the problem, with the diversity measure being the sum ${\mathrm{div}}_{\mathrm{total}}$ of Hamming distances of the solutions. This latter algorithm is $\mathsf{FPT}$ in the combined parameter $(r,w)$, where w is the treewidth of the input graph. As a special case, they obtain a running time of $\mathcal{O}\left({\left({2}^{k+2}(k+1)\right)}^{r}k{r}^{2}n\right)$ for Diverse Vertex Cover. Furthermore, they show that the r-Diverse versions (i.e., where the diversity measure is ${\mathrm{div}}_{\mathrm{total}}$) of a handful of problems have polynomial kernels. In particular, they show that Diverse Vertex Cover has a kernel with $\mathcal{O}\left(k\right(k+r\left)\right)$ vertices, and that Diverse d-Hitting Set has a kernel with a universe size of $\mathcal{O}({k}^{d}+kr)$.

**Organization of the rest of the paper.**In Section 2, we list some definitions which we use in the rest of the paper. In Section 3, we describe a generic framework which can be used for computing solution families of maximum diversity for a variety of problems whose solutions form subsets of some finite set. We prove Theorem 1 in Section 3.3 and Theorem 2 in Section 4. In Section 5, we discuss some potential pitfalls in using ${\mathrm{div}}_{\mathrm{total}}$ as a measure of diversity. In Section 6, we prove Theorems 3 and 4. We conclude in Section 7.

## 2. Preliminaries

**Auxiliary problems.**We define two auxiliary problems that we will use in some of the algorithms presented in Section 3. In the Maximum Cost Flow problem, we are given a directed graph G, a target $d\in {\mathbb{R}}^{+}$, a source vertex $s\in V\left(G\right)$, a sink vertex $t\in V\left(G\right)$, and for each edge $(u,v)\in E\left(G\right)$, a capacity $c(u,v)>0$, and a cost $a(u,v)$. A $(s,t)$-flow, or simply flow in G is a function $f:E\left(G\right)\to \mathbb{R}$, such that for each $(u,v)\in E\left(G\right)$, $f(u,v)\le c(u,v)$, and for each vertex $v\in V\left(G\right)\backslash \{s,t\}$, ${\sum}_{(u,v)\in E\left(G\right)}f(u,v)={\sum}_{(v,u)\in E\left(G\right)}f(v,u)$. The value of the flow f is ${\sum}_{(s,u)\in E\left(G\right)}f(s,u)$ and the cost of f is ${\sum}_{(u,v)\in E\left(G\right)}f(u,v)\xb7a(u,v)$. The objective of the Maximum Cost Flow problem is to find the maximum cost $(s,t)$-flow of value d.

## 3. A Framework for Maximally Diverse Solutions

#### 3.1. Optimal Augmentation

**Theorem**

**5.**

**Proof.**

**Proposition**

**1**

**.**Suppose a maximum-cost flow among all flows of value v from s to t is given. Let P be a maximum-cost augmenting path from s to t. If we augment the flow along this path, this results in a new flow, of some value ${v}^{\prime}$. Then, the new flow is a maximum-cost flow among all flows of value ${v}^{\prime}$ from s to t.

#### 3.2. Faster Augmentation

**Proposition**

**2**

**.**A maximum-weight b-matching in a bipartite graph with ${N}_{1}+{N}_{2}$ nodes on the two sides of the bipartition and M edges that have integer weights between 0 and W can be found in time $O({N}_{1}Mlog(2+\frac{{{N}_{1}\phantom{\rule{-0.166667em}{0ex}}}^{2}}{M}log\left({N}_{1}W\right)))$.

#### 3.3. Diverse Hitting Set

**Lemma**

**1.**

**Theorem**

**1.**

**Proof.**

## 4. Diverse Feedback Vertex Set

**Theorem**

**2.**

**Lemma**

**2.**

**Proof.**

- If there is a $v\in C$ such that ${\delta}_{{G}^{\prime}[B\cup C]}\left(v\right)\le 1$, we delete v from ${G}^{\prime}$.
- If there is an edge $\{u,v\}\in E\left({G}^{\prime}\left[C\right]\right)$ such that ${\delta}_{{G}^{\prime}[B\cup C]}\left(u\right)={\delta}_{{G}^{\prime}[B\cup C]}\left(v\right)=2$, we contract u in ${G}^{\prime}$ and set $\ell \left(v\right):=\ell \left(v\right)\cup \ell \left(u\right)$.

**Case 1:**The vertex v has at least two neighbors in B (in the graph ${G}^{\prime}$).If there is a path in B between two neighbors of v, then we have to put v in A, as otherwise this path together with v will induce a cycle. If there is no such path, we branch on both possibilities, inserting v either into A or into B.**Case 2:**The vertex v has at most one neighbor in B.Since v is a leaf in ${G}^{\prime}\left[C\right]$, it has at most one neighbor also in C. On the other hand, we know that v has degree at least 2 in ${G}^{\prime}[B\cup C]$. Thus, v has exactly one neighbor in B and one neighbor in C, for a degree of 2 in ${G}^{\prime}[B\cup C]$. Let p be the neighbor in C. Again, as we have reduced $({G}^{\prime},A,B,\ell )$, the degree of p in ${G}^{\prime}[B\cup C]$ is at least 3. Thus, either it has a neighbor in B, or, as v is a deepest leaf, it has another child, say w that is also a leaf in ${G}^{\prime}\left[C\right]$, and w has therefore a neighbor in B. We branch on the at most ${2}^{3}=8$ possibilities to allocate v, p, and w if considered, between A and B, taking care not to produce a cycle in B.

**Proof of Theorem**

**2.**

## 5. Modeling Aspects: Discussion of the Objective Function

## 6. Maximizing the Smallest Hamming Distance

**Theorem**

**3.**

- ${2}^{k{r}^{2}}\xb7{\left(kr\right)}^{O\left(1\right)}$ if $\left|U\right|<kr$ and
- ${d}^{kr}\xb7{\left|U\right|}^{O\left(1\right)}$ otherwise.

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Conclusions and Open Problems

**Question**

**1**

**.**Is there a problem Π with solution size k, such that Π is $\mathsf{FPT}$ parameterized by k, while Diverse $\mathrm{\Pi}$, asking for r solutions, is $\mathsf{W}\left[\mathsf{1}\right]$-hard parameterized by $k+r$?

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The network. The middle layer between the vertices ${T}_{i}$ and ${V}_{j}$ is a complete bipartite graph, but only a few selected arcs are shown. A potential augmenting path is highlighted.

**Figure 2.**Part of the modified network for a solution T which is specified by $b=3$ sets ${L}_{1}=\{1,2\},{L}_{2}=\left\{3\right\}$, and ${L}_{3}=\{4,5,6\}$.

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**MDPI and ACS Style**

Baste, J.; Jaffke, L.; Masařík, T.; Philip, G.; Rote, G. FPT Algorithms for Diverse Collections of Hitting Sets. *Algorithms* **2019**, *12*, 254.
https://doi.org/10.3390/a12120254

**AMA Style**

Baste J, Jaffke L, Masařík T, Philip G, Rote G. FPT Algorithms for Diverse Collections of Hitting Sets. *Algorithms*. 2019; 12(12):254.
https://doi.org/10.3390/a12120254

**Chicago/Turabian Style**

Baste, Julien, Lars Jaffke, Tomáš Masařík, Geevarghese Philip, and Günter Rote. 2019. "FPT Algorithms for Diverse Collections of Hitting Sets" *Algorithms* 12, no. 12: 254.
https://doi.org/10.3390/a12120254