Antimatroids

Gregory Gutin

Department of Computer Science

Royal HollowayUniversity of London

Egham Surrey TW EX UK

gutincsrhulacuk

Anders Yeo

Department of Computer Science

Royal HollowayUniversity of London

Egham Surrey TW EX UK

anderscsrhulacuk

Abstract

Weintro duce an antimatroid as a family F of of a ground

set E for which there exists an assignmentofweights to the elements

of E such that the to compute a maximal set with

resp ect to inclusion in F of minimum weight nds instead the unique

maximal set of maximum weight Weintro duce a sp ecial class of anti

matroids I antimatroids and show that the Asymmetric and Sym

metric TSP as well as the Assignment Problem are I antimatroids

Keywords Greedy algorithm combinatorial optimization TSP

Assignment Problem matroids

Intro duction

Manycombinatorial optimization problems can b e formulated as follows

where E is a nite set and F is a family of We are given a pair E F

subsets of E andaweight function c that assigns a real weight ceto

every elementofE The weight cS of S F is dened as the sum of the

weights of the elements of S It is required to nd a maximal with resp ect

to inclusion set B F of minimum weight The greedy algorithm starts

from the elementofE of minimum weight that b elongs to a set in F In

Corresp onding author

every iteration the greedy algorithm adds a minimum weight unconsidered

element e to the current set X provided X feg is a of a set in F

It is well known that the greedy algorithm pro duces an optimal solution

to the problem ab ove when E F is a matroid While the greedy algorithm

do es not necessarily nd optima for nonmatroidal pairs E F one might

think that the greedy algorithm always pro duces a solution that is b etter

than many others It was shown by Gutin Yeo and Zverovich that

for every n there is an instance of the Asymmetric TSP ATSP on

n vertices for which the greedy algorithm nds the unique worst p ossible

tour The same result holds for the Symmetric TSP STSP These results

contradict somewhat our intuition on the greedy algorithm

In the present pap er weintro duce the notion of an antimatroid An

antimatroid is a pair E F such that there is an assignmentofweights to

the elements of E for which the greedy algorithm for nding a maximal set

B in F of minimum weight constructs the unique maximal set of maximum

weight The ab ove mentioned results on the TSP indicate that b oth STSP

and ATSP are antimatroids

Similarly to matroids weintro duce I antimatroids and prove that every

nontrivial I antimatroid is an antimatroid I antimatroids are of inter

est not only since they are somewhat close to matroids but also b ecause

they include the STSPATSP and the Assignment Problem AP Thus in

particular we obtain an easy and uniform pro of that the ab ovementioned

problems are antimatroids The fact that the AP is an I antimatroid is

of particular interest since the AP is p olynomial time solvable unlike the

ATSP and STSP provided PNP

I Antimatroids

An I independence family is a pair consisting of a nite set E and a family

F of subsets called independent setsofE such that II are satised

I the is in F

I If X F and Y is a subset of X then Y F

I All maximal sets of F called bases are of the same cardinality k

If S F then let I S fx S fxg FgS This means that

I S contains all elements dierent from S which can b e added to S in

order to have an indep endent set An I indep endence family E F isan

I antimatroid if

 

I There exists a base B F B fx x x g such that the

 k



following holds for every base B F B B

k 

X

jI x x x B j kk

 j

j 

An I antimatroid is nontrivial if k

Note that if we replace I in II by the following condition we

obtain one of the denitions of a matroid

then there exists x U V such I If U and V are in F and jU j jV j

that V fxg F

In fact I I and I dene a matroid with I b eing an implication

of the three conditions

Theorem For every nontrivial I antimatroid E F there exists a

weight function c from E to the set of positive such that the greedy

algorithm nds the unique worst solution for the problem of nding a mini

mum weight base



Pro of Let B fx x g b e a base that satises I Let Mk

k



and let cx iM and cx jM if x B x I x x x

i  j 



but x I x x x Clearly the greedy algorithm constructs B and

 j



cB Mkk

Let B fy y y g By the choice of c made ab ove wehave that

 k

cy faM aM g for some p ositiveinteger a Then clearly

i

y I x x x

i  a

but y I x x x so y lies in I x x x B provided j

i  a i  j

y is counted a times in the sum in I Hence a Thus

i

k k 

X X

cB cy k M jI x x x B j

i  j

i j 



k M k k k M cB



which is less than the weightofB as Mk Since the greedy algorithm



nds B and B is arbitrarywe see that the greedy algorithm nds the

unique heaviest base

Theorem implies that no nontrivial I antimatroid is a matroid Let

us consider one of the dierences b etween matroids and I antimatroids



For a matroid E F andtwo distinct bases B and B fx x x g

 k

by I wehave that jI x x x B jk j for j k Thus

 j

k 

X

jI x x x B jk k

 j

j 

This inequality b ecomes equality for the matroid E F of the whose

E consists of columns columns are of I and I where I is the identitymatrix

of I jI and F of sets of linearly indep endent columns This shows that

I is sharp in a sense in the denition of I antimatroid

Corollary Nontrivial ATSP STSP and AP are al l antimatroids

Pro of By Theorem it is enough to show that the three problems

are nontrivial I antimatroids For the nontrivial AP E consists of edges

of a complete G with partite sets of cardinality k

and F of matchings in G including the empty one Clearly II



hold Let B fx z x z g B fu v u v g b e a pair of distinct

k k k k

p erfect matchings in G Observe that I x z x z B consists of edges

j j



of B b elonging to the subgraph G of G induced by the vertices fx z

p p



j p k g The cardinality of the p erfect in G is k j

and hence jI x z x z B jk j for j k Moreover

j j



jI x z x z B j k j for some j since B B This inequalities

j j

imply I and thus the nontrivial AP is an I antimatroid

For the STSP ATSP E consists of edges of the complete undirected

ertices z z and every set in F is a subset directed graph K on k v

k

of edges of a Hamilton in K Clearly II hold for b oth ATSP and

STSP

Toverify I for the ATSP transform K into a bipartite graph G with

  

p q k g z and edges fz z partite sets z z and z

p k

q

k



An arc z z of K corresp onds to the edge z z in G Observe that every

p q p

q

Hamilton cycle in K corresp onds to a p erfect matching in G but not vice

versa in general It follows that I is satised for the ATSP since I

holds for the nontrivial APThus the ATSP is an I antimatroid

A similar pro of can b e provided for the STSP but nowevery edge z z

p q

 

of K corresp onds to the pair z z z z of edges in G Hence the STSP is

p q

q p

an I antimatroid

It would b e interesting to have general examples of antimatroids that

are not I antimatroids and are related to wellstudied combinatorial opti

mization problems

It is worth noting that the results of this pap er can b e placed within the

topic of domination analysis of combinatorial optimization algorithms For

the sake of simplicity and claritywe dene the domination numb er only for a

heuristic for the ATSP The reader can easily extend this denition to other

combinatorial optimization problems The domination number of a heuristic

A for the ATSP is the maximum dn such that for every instance

I of the ATSP on n vertices A pro duces a tour T whichisnotworse than

at least dntoursin I including T itself Observe that an exact algorithm

for the ATSP has domination number n For a survey on domination

numb er of the ATSP and STSPsee The main result of our pap er is

that the domination numb er of the greedy algorithm for nontrivial I anti

matroids is It is worth noting that there are p olynomial time heuristics

for the ATSP and STSP of much larger domination numb er for heuristics

of domination numberatleastn see eg

Acknowledgements

We thank for a question on the Assignment Problem that ini

tiated this pap er and the organizers and participants of the workshop

on and its applications to problems of so cietyatDIMACS

Rutgers University where part of the pap er was written We are grateful to

the referee for careful reading and some useful comments

GGs research has b een partially supp orted by an EPSRC grant When

the pap er was written AYworked in BRICS Department of Computer

Science UniversityofAarhus Aarhus Denmark AYwould like to thank

the grant Research Activities in Discrete from the Danish

Natural Science Research Council for nancial supp ort

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