On Structural Parameterizations of Hitting - PowerPoint Presentation

Slide1

On Structural Parameterizations of Hitting Set:Hitting Paths in Graphs Using 2-SAT

Bart M. P. Jansen

June 19th, WG 2015, Munich, GermanySlide2

The Hitting Set Problem

Input: A family

of subsets of a finite universe

, and an integer Question

:

of size that intersects all sets in A solution set is called a hitting setHard problem:NP-completeNo constant-factor approximation unless -complete parameterized by Probably no algorithm with runtime

2

 Slide3

Tree hypergraphsA

set family over universe

is a tree hypergraph if:there is a tree

on vertex set such thatfor each the elements of induce a subtree of

A tree representation can be computed from

in poly time

 3

 Slide4

Hitting Set is easy on tree hypergraphs

Root the tree and select a deepest node

such that its subtree

contains at least one set entirelyThere is a minimum hitting set that contains Recurse on

4Slide5

Structural parameterizations of Hitting Set

Since -Hitting Set

is -complete, we search for parameterizations that are

fixed-parameter tractableRuntime where

only depends on

parameter

As Hitting Set is easy on tree hypergraphs, we parameterize by how close the instance is to a tree hypergraphHitting Set on a tree-like set system Hitting a list of connected subgraphs in a tree-like graph 5Can Hitting Set be solved efficiently if there is a tree-like graph on , such that all sets induce connected subgraphs?

 Slide6

Measures for treelike-nessComplexity depends on measure of treelike-ness

We consider two measures:Size of a minimum feedback vertex set

Number of vertex deletions needed

to break all cyclesSize of a minimum feedback edge set Number of edge deletions needed

to

break all cyclesAlso known as cyclomatic numberIn connected graphs, cyclomatic number is  6Slide7

Negative results

Without further restrictions, not much can be doneParameterization by feedback vertex number not FPT(Unless

)

Parameterization by cyclomatic number not FPT(Unless )Not even when all subgraphs are 3-leaf trees

7

Theorem. Hitting a list of connected subgraphs in a graph of feedback vertex number 2 is -complete Theorem. Hitting a list of connected subgraphs is -hard parameterized by the cyclomatic number

 Slide8

Positive results

Hitting Paths in a Graph

Input

: An undirected simple graph , a list of simple paths

in

, and an integer

Question: of size that hits all paths in In a tree e, a minimum vertex set that hits a prescribed set of paths can be found efficiently, even if the graph is largeModels the hitting set problem where the sets correspond to paths in a tree plus edgesParameterization by feedback vertex number is para-NP-c. 

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Theorem.

Hitting Paths in a Graph parameterized by the cyclomatic number is FPT and can be solved in time  Slide9

Hitting paths in graphs

9Slide10

Removing degree-1 vertices from the graphIf

and

is singleton target path:Remove and all target paths through , decrease

by

If

and is not a singleton target path:Remove from and from all target pathsCyclomatic number is not affected 10Slide11

Connecting the graphIf the graph is not connected:Add an edge between two connected components

Cyclomatic number and hitting sets are not affected

11

Observation.

The preprocessing steps allow us to work with connected graphs of minimum degree at least

 Slide12

Properties of the cyclomatic number (I)

Consider a connected graph

of cyclomatic number

and minimum degree at least two 12

Lemma 1.

There are at most vertices of degree  Slide13

Properties of the cyclomatic number (II)

Consider a connected graph

of

cyclomatic number and minimum degree at least twoLet

be the vertices of degree

13Lemma 2. has at most

components

 Slide14

Hitting sets on subpaths

Consider a component of

in a preprocessed graph

For each such component we define opt: Min. hitting set size for target paths contained fully in

Can be found efficiently by a greedy approach

14Slide15

Budgets on degree- paths

If a hitting set

contains at least opt vertices from

:

Replace solution within

by opt, add the neighbors of the endpoints of to the solution since is a component of degree- verticesThe replacement hits all paths using a vertex of  15

Lemma 3. There is a minimum hitting set for the target paths that contains either

opt

or opt vertices from each component of  Slide16

Algorithm outlineWe solve

Hitting Paths in a Graph in time

The parameter

is the cyclomatic number

Algorithm manipulates the set

of degree

verticesFor each , find a hitting set with Delete the vertices of and the paths they hitIn that branch, the vertices of

are

undeletableRemainder of the solution comes from linear forest

For each component in , branch on using opt or opt vertices from the component in the solutionWe have

subproblems

16Slide17

Simplifying a subproblem

Subproblem gives

and

for

Find minimum hitting set

with

, such thatfor each component of :

opt

17Slide18

Simplifying a subproblem

Remove all paths hit by

and remove the vertices of

The remaining vertices of degree are undeletableMerge

them

into a single undeletable vertex 18Slide19

Simplifying a subproblem19

Target

paths

may become cyclic due to

merging

If

target path fully contains a component with positive budget: remove since it will be hit on If fully contains of budget zero: remove from  Slide20

Hitting paths in a flower with budgets

Input

: A flower graph

with a core , a list of simple paths

in

,

and for each petal of a budget Question: that hits all target paths and contains exactly vertices from each petal

20Slide21

Canonical solutions

within petals

A priori, it is not clear

how to spend the budget on a petalA vertex further left will hit more paths that enter the

petal

from the left, and vice versaIf we fix the leftmost vertex from that goes in the hitting set, there is a greedy optimal extension for the given budgetRepeatedly find the earliest ending path that is not hit yet, pick its rightmost vertexWe call this the canonical solution for that leftmost index

21

opt

 Slide22

Requirements imposed on canonical solutions

Using

canonical solutions, the problem

turns into:For each petal , choose an

index

such that the corresponding canonical solutions hit all target pathsWe can encode the requirement that a path must be hit by constraining the indices that may be chosenLeftmost vertex on is far enough left to hit on

or leftmost

vertex on

is far enough right that canonical solution hits on  22Petal  Petal

 Slide23

Reduction to multi-valued 2-SAT

We obtain

constraints on the indices of canonical solutions

Each constraint is a disjunction of two literalsA literal is “at least”:

or “at most”:

We are

asking if there is a satisfying assignment to a 2-CNF-SAT formula in multi-valued logic such as:

Chosen

canonical solutions hit all paths formula is satisfiedThe problem can be solved in polynomial time on flowers! 23Theorem (Manyà 2001). 2-SAT with truth value set

and literals

and

can be solved in polynomial time Slide24

Algorithm summary

The algorithm branches into

directions

Each branch is simplified into a budgeted flower problemThe subproblem Hitting paths in a

flower

with budgets is in PReduces to multi-valued 2-SAT, which reduces to 2-SAT 24Theorem. Hitting Paths in a Graph parameterized by the cyclomatic number is FPT and can be solved in time

 Slide25

ConclusionWe

analyzed hitting set problems in graphs

, motivated by structural parameterizations of

hitting setHow to compute, given a set system , a tree-like graph in which

there

is a simple path representing every set? 25THANK YOU!ParameterHitting 3-leaf trees

Hitting

paths

Cyclomatic number-hard algorithmNo kernel ()

No

under ETH

Feedback vertex number

Para-NP-completePara-NP-completeParameterHitting 3-leaf treesHitting pathsCyclomatic numberFeedback vertex numberPara-NP-completePara-NP-complete