Domination - UTKweb.eecs.utk.edu/~cphill25/cs594_spring2015_projects/Domination.pdfLinear Programming (LP) Most problems within graph theory can be thought of as mixed-integer linear

Domination

Abdul Almami, Allan Lu, Tony Rodriguez

April 18, 2015

Abdul Almami, Allan Lu, Tony Rodriguez () Domination April 18, 2015 1 / 45

Overview

1 IntroductionDomination BasicsA Few ApplicationsRelationship to Other ProblemsHistory

2 ComplexityNP-completenessPolynomial-time ApproximationsBest Known Exact Algorithm

3 Linear ProgrammingLinear and Integer ProgrammingDomination as an Integer Program

4 Implementations

5 Results

6 Open Problems

7 Homework


Domination basics

A set S ⊆ V of vertices in a graph G = (V ,E ) iscalled a dominating set if every vertex v ∈ V is eitheran element of S or is adjacent to an element of S .

The domination number of a graph G , denoted γ(G ),is the minimum cardinality among all dominating setsof G .


Examples of Domination

Figure: The black vertices represent minimal dominating sets.


Dominating Queens

Some authors say the origins of dominating sets in graphscan be traced back to this problem.In the 1850s, chess enthusiasts in Europe considered theproblem of determining the minimum number of queensthat can be placed on a chessboard so that all squares areeither attacked by a queen or occupied by a queen.


Covering Codes and Football Pools

In European football (or soccer), there are 3 possibleoutcomes to a game: 1) home wins, 2) draw, 3) awaywins. Suppose there are n matches being played, and youwant to be able to predict the outcomes of these games.You are willing to be wrong on at most one game. Howmany predictions (or bets) do you need to place toguarantee this?

games 1 2 3 4 5 6 7 8 9

bets 1 3 5 9 27 71-73 156-186 402-486 1060-1269


Ice Cream Stands


Ice Cream Stands


Additional Applications

Routing in networks

Camera locations in buildings

Communications tours

Mobile ad hoc network construction

Finding influential nodes in social networks


Vertex Cover and Domination

Every minimum vertex cover is also a dominating set, butnot necessarily a minimum dominating set.


Vertex Cover and Domination

But, not all dominating sets are vertex covers.

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Independent Set and Domination

Every maximal independent set is a dominating set.

But, not every dominating set is a maximal independentset.


History

In 1862, Carl Jaenisch (left)attempted to determine the minimum number of queensrequired to cover an N × N chess board.

In 1892, W. W. Rouse Ball (right) reported three basictypes of problems that chess players studied during thistime.


History

In 1958, Claude Berge(left) wrote a book on graph theory, in which heintroduced the “coefficient of external stability”. Thislater became known as the domination number of agraph. Oystein Ore (right) introduced the terms“dominating set” and “domination number” in his bookon graph theory, which was published in 1962.


History

In 1998, Teresa Haynes (left), Stephen Hedetniemi(center), and Peter Slater (right), published two books ondomination, Fundamentals of Domination in Graphs andDomination in Graphs: Advanced Topics.

In 1975, Steve Hedetniemi and Ernie Cockayne developedthe first algorithm to find a minimum dominating set: alinear-time algorithm for trees.


History

Dr. David S. Johnson

Department of Computer Science, Columbia University

Worked on dominating set in the 1970s


Complexity

The dominating set problem is one of the classicalNP-complete problems given by Garey and Johnson.

Input: a graph G and a positive integer kDecision: Is γ(G ) ≤ k?

This problem is NP-complete. The optimization version ofthe problem (determine γ(G ) for a given graph G ) isNP-hard.


Maybe we can approximate it?

A polynomial time approximation scheme, or PTAS, is analgorithm that runs in polynomial time and approximatesa solution to a problem to within a constant multiple. Forexample, given a maximal matching M on a graph G , weknow that any vertex cover VC satisfies|M | ≤ |VC | ≤ 2|M |. Since finding a maximum matchingis solvable in polynomial time, this is a PTAS for thevertex cover problem.

Since vertex cover isn’t terribly different than domination,maybe we can find a PTAS for domination.


Nope

Theorem. Unless P=NP, no polynomial timeapproximation scheme exists for the dominating setproblem.

In other words, finding a “good” approximation for aminimum dominating set is equivalent to proving P=NP.


Bounds on Domination Number

Theorem. Let ∆(G ) be the maximum degree of anyvertex in G . Then γ(G ) ≥ n/(1 + ∆(G )).

Theorem. Let S be a maximal independent set of aconnected graph G . Then both S and V \ S aredominating sets of G .

Theorem. For a connected graph G with n ≥ 2,γ(G ) ≤ n/2.

So, we have n/(1 + ∆(G )) ≤ γ(G ) ≤ n/2.


Best Known Exact Algorithm

The first exact algorithm to solve the minimumdominating set problem with running time O(cn) forc < 2 appeared in 2004. The best known exact algorithmfor an arbitrary graph has a running time of O(1.5134n)with polynomial memory, or O(1.5063n) with exponentialmemory.In a nutshell, it transforms the Dominating Set probleminto the Set Cover problem, and solves the Set Coverproblem with reductions and branching.


Linear Programming (LP)

Most problems within graph theory can be thought of asmixed-integer linear programming problems. A linearprogram (LP) is a type of constrained optimization inwhich the objective function and the constraints are bothlinear functions of the variables.


LP Example

Maximize:2x + 3y

Subject to:8x + 3y ≤ 20x , y ≥ 0

The solution to this is x = 0, y = 6.66667 for an objectivevalue of 20.


(Mixed) Integer Linear Programming

A linear program where some of the variables are requiredto be integer is called a mixed integer linear program, orMIP.A linear program where all of the variables are required tobe integer is called an integer linear program, or IP.


Domination as an IP

We’ll construct an IP to model the minimum dominatingset problem for the following graph:

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Domination as an IP

Create a decision variable xi for each vertex i . Setxi = 1 if vertex i is in the dominating set, and 0otherwise.

We want to minimize the number of vertices in thedominating set. So the objective is to minimize

∑xi .

Each vertex needs to be in the set, or have a neighborin the set. So we get a constraint for each vertex ofthe form

∑i∈N[v ] xi ≥ 1 for all v ∈ V .


Domination as an IP

Minimizex1 + x2 + x3 + x4 + x5

Subject to(x1) x1 + x2 + x3 ≥ 1(x2) x1 + x2 + x4 ≥ 1(x3) x1 + x3 + x4 + x5 ≥ 1(x4) x2 + x3 + x4 + x5 ≥ 1(x5) x3 + x4 + x5 ≥ 1xi ∈ {0, 1}


LPs vs. MIPs

In general, solving continuous LPs is fairly simple. Thesimplex method, developed by George Dantzig, is acommon approach used. This method can potentially runin exponential time, but rarely does in practice.

MIPs, however, are a different monster altogether. Ingeneral, MIPs are hard.


Gurobi

Robert Bixby Zonghao Gu Edward Rothberg

From Gurobi.com:”Gurobi Optimization is in the business of helpingcompanies solve their hardest problems by providing thebest optimization solver possible.”


Tabu Search

1 Initialize S to a random set of k vertices. Initialize f to thenumber of vertices not dominated by S .

2 Two solutions S and S ′ are neighbors if S ′ can be formed from Sby removing a single vertex from S and adding a single vertexfrom V \ S . Determine which neighbor S ′ of S has the lowestobjective value f ′. Set S = S ′ and f = f ′. Store which verticeswere removed and added. Removed vertices cannot be added fora certain number of iterations, and added vertices cannot beremoved for a certain number of iterations.

3 If f = 0, a dominating set of size k has been found. Go to step 1with a new, smaller k . If the iteration count reaches a preset limitwithout finding a dominating set, stop. Otherwise, go to step 2.


Greedy Algorithm

Greedy Algorithm

Get the advantage of both vertex cover and maximumindependent set

It uses maximum degree pairing to construct thedominating set


Greedy Algorithm

Sort vertices by degree, in descending orderSpecial case:

For all vertex with max degree==1 & not covered & its neighbor, u, isnot in FDS (Final Dominating Set)

Add the neighbor, u, to FDSMark all the neighbors of u as covered

Do greedy dominating set search (see next slide)Store the proposed DS in PDS

Do Max Degree pairingStore the proposed DS in PDS

Choose the minimum dominating set in PDS


Greedy Algorithm

Greedy dominating set search

While (all vertices are not covered)For the max degree in G

If it is not in FDS

Add it to FDSSet current max degree to 0Mark all neighbors as covered

End while

Store results in PDS


Greedy Algorithm

Max Degree pairing

Count the number of vertices with the same degreeIf there is more than one

start with the max degreeFind the max number fo vertices covered by two pairs

If equal number, add another member to the pair (three vertices)

Add the pair of vertices to PDS

Continue to the next max degree

Store results


Greedy Example


Exact Algorithm


Results

We allowed each program to run up to 30 minutes. If the program wasn’tfinished at that time, it was terminated, it’s best solution was recorded,and the elapsed time it took to produce that solution is reported inparentheses. Times are reported in seconds.

Gurobi Greedy Exact Tabu

K 3 3 3 3 3 2 – 0.00 3 – 0.008 2 – 0.00001 2 – 0.0001

random 1000.9 2 – 51.42 3 – 19.272 2 – 0.5505 2 – 3.00

random 1000.1 26 – (161) 28 – 0.242 N/A – (1800) 27 – (1693)

random 500.2 12 – (104) 26 – 0.068 N/A – (1800) 12 – (197)

parkinson 26.3 8 – 0.00 8 – 0.009 8 – 0.00142 8 – 0.0001

parkinson 67.3 21 – 0.00 20 – 0.008 N/A – (1800) 21 – 0.009

S 5 K 4 5 – 0.00 6 – 0.01 5 – 0.000044 5 – 0.001242

C 14 5 – 0.00 5 – 0.01 5 – 0.000028 5 – 0.0003

graph6 6 – 0.00 6 – 0.01 6 – 0.000027 6 – 0.00008


Results

In general, it looks like Gurobi lived up to its promise ofproviding the best optimization solver possible. For sparsegraphs, Gurobi was able to solve to optimality almostinstantly. However, on denser graphs, Gurobi becamemuch slower.


Open Problems

Prove or disprove Vizing’s Conjecture: For any twographs G and H , γ(G )γ(H) ≤ γ(G�H), where G�Hdenotes the Cartesian product of G and H .

Let i(G ) represent the cardinality of the smallestmaximal independent set of G . This is also called theindependent domination number of G . Classify thegraphs for which i(G ) = γ(G ).

A set S is a total dominating set if every vertex isadjacent to some element of S . The total dominationnumber of a graph G , γT (G ) is defined in the obviousway. Classify the graphs for which γT (G ) = γ(G ).


Homework, 1

Email homework solutions to [email protected]

1) The Dominating Queens problem has a few solutions ofsize 5. In other words, you only need 5 queens tocompletely cover the chess board with queens. Find onesuch solution.


Homework, 2

2)


Homework, 3

3) Determine γ(Cn) and γ(Pn), where Cn is the cycle on nvertices and Pn is the path on n vertices.

Figure: C4 and P5


References I

T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination inGraphs. Marcel Dekker, Inc., New York (1998).

T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs: AdvancedTopics. Marcel Dekker, Inc., New York (1998).

D. B. West, Introduction to Graph Theory, 2nd ed. Prentice-Hall, Englewood Cliffs,NJ (2000).

N. Bray, Dominating Set. MathWorld–A Wolfram Web Resource, created by EricW. Weisstein. http://mathworld.wolfram.com/DominatingSet.html

http://en.wikipedia.org/wiki/Dominating set

J. M. M. Van Rooij and H. L. Bodlaender, Design by Measure and Conquer, AFaster Exact Algorithm for Dominating Set. arXivhttp://arxiv.org/pdf/0802.2827.pdf (20 Feb 2008).

W. Myrvold, A Simple Algorithm for Dominating Set.


References II

The Ice Cream Stands Problemhttp://www.c3.lanl.gov/mega-math/workbk/dom/doice.html

A. Sasireka et al., Applications of Dominating Set of Graph in Computer Networks.International J. of Engr. Sci. & Research Technology (2014).

http://www.academia.edu/6099967/Applicationsof Dominating Set of Graph in Computer Networks

J. M. Tarr, Domination in Graphs. Graduate Theses and Dissertationshttp://scholarcommons.usf.edu/etd/1786 (2010).

http://www.cs.iastate.edu/∼chaudhur/cs611/Sp07/notes/lec22.pdf

F. Kuhn and R. Wattenhoffer, Constant-Time Distributed Dominating SetApproximation. (2003)

F. Kuhn, T. Moscibroda, and R. Wattenhoffer, What Cannot Be ComputedLocally. (2004)

http://www.gurobi.com/


The End


Domination - UTKweb.eecs.utk.edu/~cphill25/cs594_spring2015_projects/Domination.pdfLinear Programming (LP) Most problems within graph theory can be thought of as mixed-integer linear

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