Robust algorithms for restricted domains

r

ition,er themise”nput is

inatelution,bustlyalss resulttimeph

posedaph israph

trictede been

ad).

Journal of Algorithms 48 (2003) 160–172

www.elsevier.com/locate/jalgo

Robust algorithms for restricted domains

Vijay Raghavan and Jeremy Spinrad∗

CS Department, Vanderbilt University, Box 1679-B, Nashville, TN 37235, USA

Received 19 September 2001

Abstract

We introduce a new definition of efficient algorithms for restricted domains. Under this definan algorithm is required to be “robust,” i.e., it must produce correct output regardless of whethinput actually belongs to the restricted domain or not. This is to be contrasted with the “proversion of solving problems on restricted domains, in which there is a guarantee that the iin the class, and an algorithm to “solve” the problem need not function correctly or even termif this guarantee is not met. There exist problems that have a polynomial time promise sowhile being NP-hard if required to be robust. We show perhaps the surprising result that rofinding a maximum independent set in awell-coveredgraph (i.e., a graph in which every maximindependent set is of the same size) is NP-hard. An argument can be made that this hardneis more meaningful than the trivial polynomial time promise algorithm. We give a polynomialrobust algorithm for the maximum clique problem in unit disk graphs, i.e., given an input graGin general form, the output is either a maximum clique forG or a certificate thatG is not a unitdisk graph. The existence of this algorithm is to be reconciled with the apparent contradictionby the facts: (1) Recognizing whether an input graph given in general form is a unit disk grNP-hard; in fact, it is not even known to be in NP. (2) Finding a maximum clique in an input ggiven in general form is NP-hard. 2003 Elsevier Inc. All rights reserved.

Keywords:Algorithms; Robustness; Well-covered graphs; Unit disk graphs

1. Introduction

Although many papers have been written that claim to solve a problem on a resinput domain, there are some possible interpretations of this statement which havignored in the literature.

* Corresponding author.E-mail addresses:[email protected] (V. Raghavan), [email protected] (J. Spinr

0196-6774/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0196-6774(03)00048-8

V. Raghavan, J. Spinrad / Journal of Algorithms 48 (2003) 160–172 161

rithmce it isve antwoorithm

longsin itsplanar,

s oftengned;s, it isticular

, thisoducehing;laringto the

for a

(for

itionstep.

obustrithmeen a

ablelutions

r

their

put ise that

e

In all interpretations, if the input instance is in the restricted domain, the algosolves the problem correctly. For some domains, no more needs to be said, sinobvious whether or not the input meets the criteria for the domain. Thus if you haalgorithm that solves the satisfiability problem for input CNF-formulas with at mostliterals per clause, you can scan an input CNF-formula to determine whether the algis applicable.

For other algorithms on restricted domains, it is less obvious whether the input beto the domain of interest. For example, an algorithm may accept an input graphrepresentation as an adjacency list and solve a problem as long as the input graph isor perfect, or any of the numerous classes of graphs that appear in the literature. It inot easy to determine whether the input is of the form for which the algorithm is desithe recognition problem for the class may be open or even NP-hard or worse. Thunot immediately clear whether the special purpose algorithm is applicable to a parinput graph.

In the most common interpretation of solving a problem on a restricted domainissue is deliberately finessed away. If the input is in the domain, the algorithm must prthe correct output, while if the input is not in the domain the algorithm may do anytpossible behavior includes producing correct output, producing incorrect output, decthat the input is not in the domain of interest and running forever. This correspondsstandard definition of “promise” problems [11,17].

We propose a stricter requirement for an algorithm that claims to solve a problemrestricted domain.

A robust algorithm for problemP on domainC must solveP correctly for everyinput inC. For inputs not inC, the algorithm may either produce the correct outputproblemP ) or answer that the input is not inC.

Any “promise” algorithm can be turned into a robust algorithm by running a recognalgorithm for the restricted class for which it was designed as a preprocessingHowever, this affects the running time of the algorithm. We will see that there are ralgorithms that solve a problem on a class much more efficiently than any known algofor the recognition problem on the class, and for such problems the difference betw“promise” algorithm and a robust one can be paramount.

An important advantage of robust solutions is that they “compose”; in addition to beto form robust solutions by using robust procedures for a restricted domain, robust soare closed under subsets, intersections, and unions of domains. Clearly, if problemP hasrobust algorithmA for domainC, thenA is also a robust algorithm for any subsetX ofC. Consequently ifA1 andA2 are robust algorithms, respectively, for domainsC1 andC2,then bothA1 andA2 are robust algorithms forC1∩C2 as well. To get a robust algorithm foC1 ∪C2, we run bothA1 andA2. If either one halts with a correct solution forP , then wesimply output the solution; otherwise, both algorithms declare that the input is not indomain of interest and we can output that the input is not inC1 ∪C2. This simple idea canform the basis for solving a hard problem by doing a case analysis of whether the inin any of the known restricted domains for which a robust algorithm is available. Nota similar strategy will not work to find a “promise” solution forC1 ∪ C2 using “promise”algorithmsA1 andA2, respectively, forC1 andC2; if a problem instance belonged to th

162 V. Raghavan, J. Spinrad / Journal of Algorithms 48 (2003) 160–172

n

ework.solved”

thesolvedenerale usedm can

zationcisely,

ceof

g

rk:

red by

ourmple,tion offit the

ion of

ing actionof the

ue inif theas an

aphs,ownen a

symmetric difference ofC1 andC2 andA1 andA2 gave us two different solutions, thethere may be no way to tell which is the correct one!

Composing robust algorithms can also be expressed in a more practical framSuppose that there is a problem we want to solve, and that the problem has been “for a variety of special cases. We would like to design a suite of algorithms, withproperty that if the input falls into the domain of one of the special cases that can beefficiently, we use the solution for that special case; otherwise, we use a slower gsolution method. If the special case solutions require a promise, their results cannot bin this setting, since they may produce incorrect output. By contrast, a robust algorithbe used, even though the special cases may be difficult to recognize.

A decision formulation of robustness can be obtained by considering a generalithat includes ordinary and robust decision problems as special cases. More presuppose that we have two decision problemsP1 andP2, both defined on a common instanspace. The general decision problemD requires an algorithm to output correctly onethree possible responses:

– Yes, P1 is true.– No, P2 is not true.– No, P1 is not true.

Robust decision problems are then obtained by makingP2 the problem of recognizinif the input belongs to the restricted domainC of interest andP1 the problem one wouldlike to solve in the restricted domainC. Normal decision problems also fit this framewoP2 is simply made the same asP1. Note the asymmetry betweenP1 andP2—an algorithmfor solvingP1 would suffice to solveD, while an algorithm forP2 would not necessarilysolveD. This formulation also allows decision problems such as the ones consideValiant and Vazirani in [20] in detecting unique solutions.

The requirement of robustness is implicit in earlier work, but to the best ofknowledge, it has not been recognized as important in its own right. For examany approximation schemes for NP-hard optimization problems use a natural norobustness: either some good approximation is found or the input is declared not torequirement. The well-known approximation schemes [2,21] fornε coloring of 3-colorablegraphs fit this classification.

In this paper, we present two technical results that serve to illustrate the notrobustness.

The first result, in Section 2 of the paper, is concerned with the problem of findmaximum clique in unit disk graphs. Unit disk graphs are defined to be the intersegraphs of unit disks on the plane. Given a model, i.e., the coordinates of the centersdisks, an algorithm due to Clark, Colbourn, and Johnson [9] finds a maximum cliqsuch graphs. An open problem posed by Breu [4, p. 94] and by Prisner [15] asksmaximum clique can be found if one is given only the graph in a standard form, say,adjacency matrix. A complication here is that it is NP-hard to recognize unit disk gras shown by Breu and Kirkpatrick [3]. A solution was proposed in [12], but was shto be incorrect in [4]. In this paper, we give a robust algorithm for the problem: giv


ph or

endentwell-ndeed,ndentwell-utrd. We

oblems

ction(suchates ofinput.ut a

enerallemsf unit

whichnomialat theliquenot a

tegions

m

time,

imum

standard form of a graph, the algorithm either outputs the maximum clique in the gradeclares that the graph is not a unit disk graph.

The second result, in Section 3 of the paper, is about finding the largest indepset in a well-covered graph. At first glance this seems like a trivial problem since acovered graph is a graph in which every maximal independent set is also maximum. Ione motivation for studying well-covered graphs comes from the fact that the indepeset problem is “easy”; see Plummer’s survey [14] for the extensive work done oncovered graphs. Itis true that finding a “promise” algorithm for this problem is trivial, bwhat about a robust one? We show perhaps the surprising result that this is NP-haargue that results such as this force us to rethink our ideas on what it means for prto be tractable in restricted domains.

2. A robust algorithm for clique on unit disk graphs

As mentioned in the introduction, unit disk graphs are defined to be the intersegraphs of circles of unit radius. A unit disk graph can be given either in standard formas adjacency matrices or adjacency lists), or as a geometric model in which coordincenterpoints for circles corresponding to each vertex are provided as part of theA polynomial time algorithm for the clique problem on unit disk graphs taking as inpgeometric model was first1 given by Clark, Colbourn, and Johnson [9].

Our solution for clique on unit disk graphs can be seen as an example of a gstrategy for providing robust solutions when both recognition and optimization probare hard. Here the clique problem on general graphs is hard and recognition odisk graphs is also hard [3], so we find a larger superclass of unit disk graphssomehow satisfy the requirements of being recognizable as well as having a polytime algorithm for the clique problem. Thus we have no need for even a “promise” thinput is a unit disk graph; if the graph is in the larger class, we solve the maximum cproblem correctly, and if the graph is not in the class we can answer that the input isunit disk graph.

Let G be a unit disk graph, and letM be a unit disk model forG. The algorithm ofClark, Colbourn, and Johnson works as follows. For each edge(x, y) of G, let cx andcybe the centers of circles corresponding tox andy in M. Draw two circles with radiusdistanced(cx, cy), centered atcx andcy , respectively. LetH be the set of vertices thahave centerpoints within the intersection of these two circles. ThenH induces a cobipartitgraphGH , as proved in [9]; the two cliques correspond to circles with centers in the reof the intersection of the two circles that lie on either side of the line fromcx to cy .

Since GH is cobipartite, a maximum clique inGH corresponds to a maximuindependent set in the bipartite graph that is the complement ofGH . The size of themaximum independent set in a bipartite graph is equal to|V | −M, whereM is the sizeof a maximum matching. Since a maximum matching can be found in polynomial

1 Chazelle and Lee [7] solved a different problem and incorrectly claimed to have solved the “maxweighted clique problem for circle intersection graphs.”


clique

t time,l

of

givenraph,

ebors of

wn forf

ve theat will

Fig. 1. Graph given with unit disk representation.

independent set is polynomial for bipartite graphs, and thus we can find a maximumin GH in polynomial time. We find the maximum clique in the subgraphH = H(x,y)

induced by each edge (x, y); the largest such clique is the maximum clique inG, as wewill see below. Assuming that all operations on numbers can be done in constanthis leads to an O(n4.5) algorithm; Breu [4] improves this to O(n3.5 logn) by using speciageometric properties of the graphs in the matching problems.

To see that this algorithm is correct, letC be a maximum cardinality clique inG. Letc1, c2 be two center points of maximum distance inC. Then every other center pointvertices inC is within the two circles of radiusd centered atc1 andc2, respectively, whered is the distance betweenc1 andc2. Therefore, all vertices ofC will be considered whenfinding the maximum clique aroundc1 andc2.

To illustrate the Clark–Colbourn–Johnson algorithm, we may assume that we areinput as in Fig. 1, which shows the centers and circles for a six vertex graph. In this gvertices 2, 3, 4, 5, 6 induce a clique, and vertex 1 is adjacent to 2, 3, and 6.

Figure 2 shows the circles drawn to construct cobipartite graphs for the edges(3,5)and(2,4). The centers included in both circles drawn for(2,4) are 1, 2, 5; note that thsubgraph created does not include 3 and 6, even though these vertices are neighboth 2 and 4. However, centers of 2, 3, 4, 5, and 6 are all included in the circles dra(3,5); they would also be included in the circles drawn for(4,6), which are the vertices othe clique at maximum distance from each other on the plane.

We now show how some ideas from this algorithm can be adapted so as to remodependence on the model as input. We first present an edge ordering property thallow us to find a maximum clique.

Given an orderingL = e1, e2, . . . , em of the edges of a graphG, let GL[i] be thesubgraph ofG with edge set{ei, ei+1, . . . , em}. For each edgeei = (x, y), we defineNL,ito be the set of vertices that are adjacent to bothx andy in GL[i].


e

edhas a

elique

tydges

.

edyete)

Fig. 2. Intersection of circles for the nonrobust algorithm.

Definition 1. An edge orderingL = e1, e2, . . . , em is a cobipartite neighborhood edgelimination ordering (CNEEO) if for eachei , NL,i induces a cobipartite graph inG.

We will use a greedy algorithm to find a CNEEO inG, and use the ordering to find thmaximum clique. We will prove that ifG admits a CNEEO, the greedy algorithm will finthe ordering and produce a correct maximum clique, and that every unit disk graphCNEEO.

Lemma 1. Given a graphG and a CNEEOL forG, a maximum clique inG can be foundin polynomial time.

Proof. For eachi, NL,i induces a cobipartite graph inG. We use matching to find thmaximum clique in each such cobipartite subgraph; we will show that the largest cfound by this algorithm is a maximum cardinality clique inG.

Let C be a maximum clique inG, and letei be the first edge ofC in L. Since everyvertex of C is in NL,i , we must find a clique with cardinality|C| when we find themaximum clique in the graph induced byNL,i in G. ✷

Our greedy algorithm for finding a CNEEOL works as follows. Starting with an empordering, at each stagei, i � 1, we choose an edge to extend the current ordering of ee1, e2, . . . , ei−1. Each edge ofGL[i] is considered as a possible candidate forei . The firstedgee ofGL[i] such that the neighbors of both endpoints ofe inGL[i] induce a cobipartitegraph inG is then chosen asei . If no such edge exists, we declare thatG has no CNEEOClearly, this algorithm runs in polynomial time.

Lemma 2. If G admits a CNEEO, then the greedy algorithm finds a CNEEO forG.

Proof. LetL be a CNEEO forG and suppose, for the sake of contradiction, that the grealgorithm stops and declares thatG does not have a CNEEO after creating an (incomplorderingL′ = e1, e2, . . . , ei−1. Lete be the earliest edge ofGL′ [i] inL; let j be the positionof e in L. Every neighbor of both endpoints ofe in GL′ [i] is a member ofNL,j , sincee is


urn–a

que

imums but

theig. 1radii.

artite.y ordergraph

n thend

artite;

t

the first edge ofGL′ [i] in L; thus the neighbors of both endpoints ofe in GL′ [i] induce acobipartite graph inG, a contradiction. ✷

Next, we show that every unit disk graph has a CNEEO.

Lemma 3. Every unit disk graph admits a CNEEO.

Proof. Let G be a unit disk graph, and letM be a unit disk model forG. Let d(x, y) bethe distance between the center ofx and the center ofy in M. Order the edges ofG into alist L by nonincreasing distance between the centers of the corresponding circles inM.

For eachei = (u,v), the vertices inNL,i are thosez such thatd(u, z) � d(u, v)

andd(v, z) � d(u, v). These are exactly the vertices considered in the Clark–ColboJohnson algorithm for finding the maximum clique inG givenM, and thus inducecobipartite graph as proved in their paper [9]. Thus,L is a CNEEO. ✷

The following theorem is immediate from the above lemmas.

Theorem 1. There is a polynomial time robust algorithm for finding the maximum clifor unit disk graphs.

Note that we have a polynomial time robust algorithm to recognize and find maxcliques on any graph which admits a CNEEO. This includes not just unit disk graphall intersection graphs of unit spheres in arbitrary dimensions.

We now look at two examples that illustrate the robust algorithm for solvingmaximum clique problem on unit disk graphs. Assume that the input graph from Fis given in adjacency matrix form, rather than being given as a set of centers andThe robust algorithm looks for edges such that the joint neighborhood is cobipSince the entire graph is cobipartite in this case, the edges can be placed in anfor the elimination scheme, and we can solve the clique problem for each cobipartiteconsidered.

A more interesting example for the robust algorithm is shown in Fig. 3. Note that ioriginal graph, the joint neighborhood of(1,6) contains the independent set 2, 4, 5, athus is not cobipartite. However, the joint neighborhood of every other edge is cobipone of many possible cobipartite neighborhood edge elimination orderings is(1,4), (4,6),(3,4), (1,6), (1,2), (2,3), (2,6), (1,5), (5,6). Note that when(1,6) is processed, join

Fig. 3. Sample graph for robust algorithm.


s bywhichre areut onat therobustgraph

rs to thelargertioned

bustnsiderave a

put

aphsUD,en aise—

tion.

”robust

neighbors of 1 and 6 using only edges that occur after(1,6) in the ordering are simply 2and 5, and thus induce a cobipartite graph inG.

We note that although the clique problem is solved robustly for unit disk graphsolving the problem for a larger class of graphs (i.e., graphs that admit a CNEEO)can be recognized in polynomial time, not all robust algorithms are of this form. Therobust algorithms that always solve a problem correctly if the input is in a class, bindividual inputs not in the class may either solve the problem correctly or answer thinput is invalid, depending on how the input is presented. For example, there arealgorithms that may give different answers (the correct answer or the answer that theis not in the class) on the same graph, depending on the assignment of vertex numbegraph; thus, robust algorithms are not necessarily simply solving the problem on aclass of graphs. The approximation algorithms for coloring 3-colorable graphs menearlier [2,21] are examples of robust algorithms of this form.

3. Intractability for restricted domains

The change in the notion of “solving a problem in a restricted domain” from nonroto robust necessitates a change in the definition of the term “NP-hard for a class.” Cothe following typical situation that arises when dealing with restricted domains. We hdecision problemΠ over some restricted domainD about which the following is known:

(1) Π is NP-hard if the domain is not restricted.(2) Π is known to be decidable in polynomial time if a promise is given that the in

belongs to domainD.(3) Recognizing if the input belongs to domainD is (NP- or co-NP-)hard.

We are now interested in finding if there is a polynomial timerobustalgorithm forΠ overthe domainD.

This is not unlike the situation that arose in the clique problem on unit disk gr(CPUD) dealt with in the previous section. However, there is a key difference: with CPthe known result was that CPUD was known to be solvable in polynomial time givmodelof the input as the intersection graph of unit disks. A model is more than a promit is a verifiable certificate of membership in the restricted domain.

Suppose that we suspect that it is hard to find a robust solution forΠ overD. Howwould we go about proving this? “Normal” reductions cannot work well in this situaTo see this, consider how we would do a normal reduction from, say, 3-SAT toΠ overD.Given any instanceI of 3-SAT, we would construct, in polynomial time, an instanceI ′of Π whose input is inD such thatI is satisfiable if and only ifI ′ is a “Yes” instanceof Π .

The point is that such a reduction is not achievable unless P= NP, sinceΠ is decidablein polynomial time given a promise that the input is inD. In other words, the “normalreduction cannot distinguish between problems that have a promise solution but nosolution and problems that allow robust solutions.


gh an

t

on isn thermal”ing aust

e

rof

ite aile nottilityered

dentsively

meanyake

le wayrithmte to

ndent

lem ofset

We illustrate how we can prove an NP-hardness result in this situation throuexample. Informally, we use the following “modified” reduction.

Consider a reduction from a known NP-hard problemΠ ′ to a problemΠ over arestricted domainD with the following properties:

(1) If the answer to the instanceI ′ ofΠ ′ is “yes,” the reduction will produce an instanceIof problemΠ in the domainD such thatI is a “yes” instance ofΠ .

(2) If the answer to the instanceI ′ of Π ′ is “no,” the reduction will produce an inpuinstanceI of problemΠ such thatI is a “no” instance.

Note that the difference between a “normal” reduction and a “modified” reductithat if the answer is “no,” the “modified” reduction does not need to produce input irestricted domain. That is, the “modified” reduction relaxes a requirement of a “noreduction. Nevertheless, such a “modified” reduction implies the hardness of findrobust solution forΠ overD unless P= NP. Suppose, to the contrary, that there is a robpolynomial time algorithmA for problemΠ over the domainD and that a “modified”reduction of the above form also exists. Run the algorithmA on the instanceI of Πproduced by the reduction. IfI ′ is a “yes” instance ofΠ ′ thenA must answer “yes,” sincI will be a “yes” instance ofΠ in the domainD, according to the reduction. IfI ′ is a“no” instance ofΠ , thenA may either answer “no” or answerI is not an instance ovethe domainD. Thus,A would answer “yes” if and only if the answer to the instanceΠ ′ is “yes,” and we would have a polynomial time algorithm forΠ ′, a contradiction ifP = NP.

When there is a promise solution for a problem, the “modified” reduction has qudelicate existence since it seeks to prove that there can be no robust solution whruling out the possibility of a promise solution. The example we use to show the uof this kind of reduction is that of finding a maximum independent set on well-covgraphs.

A graphG is well coveredif every maximal independent set is a maximum indepenset. Well-covered graphs were first introduced in [13], and have been studied extensince then [14].

Clearly, the definition of well-covered graphs implies a trivial polynomial ti“promise” algorithm for the maximum independent set problem, by simply takingmaximal independent set. If the input is not well covered, this algorithm can mmistakes, but it is not clear whether this algorithm can be made robust. One possibto convert the promise algorithm to a robust one is to first run a recognition algoto verify that an input graph is indeed well covered; however, it is co-NP-complerecognize well-covered graphs [6,8,18].

We now show that determining whether a well-covered graph contains an indepeset of sizek cannot be solved in polynomial time by a robust algorithm unless P= NP.We use a reduction from the maximum independent set problem (IS), i.e., the probdeciding, given a graphG= (V ,E) and an integerk, whether there is an independentof sizek.


,delong

e the

n,

s,

t

cent

ther alumn-a

r theuctionet hasbe twot set ofce

nt

4. The reduction

Let 〈G= (V ,E), k〉 be an instance of IS. LetV = {v1, v2, . . . , vn}.Construct a graphG′ from 〈G,k〉 as follows. For each vertexvi , 1� i � n of G, create

k copiesvi,1, vi,2, . . . , vi,k of vi as vertices ofG′. Also, createk(k − 1) additional verticesui,j ,1 � i = j � k. Thus the total number of vertices inG′ is k(n+ k − 1). In the sequelwe shall refer to vertices of the former kind asv-typevertices and vertices of the latter kinasu-typevertices. Furthermore, vertices with the same second index shall be said to bto the samerow and vertices with the same first index shall be said to the samecolumn.

The edges ofG′ are completely specified by the following rules:

(1) Row edgesexist between all pairs of vertices in the same row, i.e., row edges aredges in the set

{(x, y): x = y and∃i 1 � i � k (x = vj,i or x = uj,i) and

(y = vj,i or y = uj,i )}.

(2) Column edgesexist only betweenv-type vertices if they belong to the same columi.e., column edges are the ones in the set

{(vi,j , vi,j ′ ): j = j ′, 1 � i � n, 1� j, j ′ � k

}.

(3) Diagonal edgesexist only betweenu-type vertices if they belong to different columni.e., diagonal edges are the ones in the set

{(ui,j , ui′,j ′): i = i ′, i = j, i ′ = j ′, 1 � i, i ′, j, j ′ � k

}.

(4) Cross edgesexist betweenu- and v-type vertices if the first index of theu vertexcorresponds to the second index ofv vertex, i.e., cross edges are the ones in the se

{(ui,j , vj ′,i): i = j, 1 � i, j � k, 1 � j ′ � n

}.

(5) G-edgesexist only betweenv-type vertices if the corresponding vertices are adjain G, i.e.,G-edges are the ones in the set

{(vi,j , vi′,j ′): (vi , vi′) ∈E, 1 � i, i ′ � n, 1 � j, j ′ � k

}.

Figure 4 shows part of the graph constructed in the reduction for determining whepath 1-2-3-4 contains an independent set of size 3. In the figure, we omit row- and coedges in order to make the picture less crowded. Also,v-type vertices that are copies ofvertexi are labeled simplyi, i ′, andi ′′.

For this example, there is only one possible answer of a robust algorithm foindependent set problem on well-covered graphs. The graph produced from the redis well covered, so any robust algorithm must find that the maximum independent ssize two. On the other hand, if a vertex 5 was added adjacent only to 4, there wouldvalid answers for the robust algorithm. It could answer that there is an independensize three (for example, 1, 5′, 3′′), or it could answer that the input is not well covered, sinthere are maximal independent sets such as 1, u23, and 1,4′ of size 2 as well as independesets of size 3.


graphs

in

ize

m

n

Fig. 4. Edges used in the reduction.

A maximum independent set of size 2 will be found in the graph if and only ifG had noindependent set of size 3, so a robust algorithm for independent set on well-coveredwould allow us to determine whether the input had an independent set of size 3.

Our result depends on the following claim.

Claim. (a) If G has no independent set of sizek, then every maximal independent setG′ has sizek − 1.

(b) If G has an independent set of sizek thenG′ has maximal independent sets of sk andk − 1.

In other words,G′ is well covered and has maximal independent sets of sizek − 1 ifand only ifG has no independent set of sizek.

Proof of (a). Suppose thatG has no independent set of sizek. Let S be a maximalindependent set inG′. If S contains au-type vertexui,j , thenS cannotcontain:

– Any other vertex from rowi, because of the row edges fromui,j ;– Any v-type vertex from rowj , because of the cross edges fromui,j ; and– Any u-type vertex from any column other thani, because of the diagonal edges froui,j .

Consequently,|S| � k − 1. Also, if S does not contain anyu-type vertex, then the columedges andG-edges ensure thatS corresponds to an independent set inG; under theassumption that there is no independent set inG of size k, |S| � k − 1 in this case aswell.

We now show thatS must be of size exactlyk − 1 if it is maximal.First, supposeS contains au-type vertexui,j . From the above, all the remainingu-type

vertices inS come from the same columni and all thev-type vertices inS come from rowsother thani. Moreover, if a rowh = i is not represented by av-type vertex inS, then thevertexui,h can be safely added to the setS without violating independence. Hence, ifS ismaximal, all rows other thani are represented exactly once giving a size ofk − 1.


um

t

erost a

lems.ption,

n theons in

tionsnownfullof theereasn onlyblems

er theypart of

reticalinguish

Next, supposeS contains nou-type vertex. Then ifS is of size less thank − 1, then atleast two distinct rowsi andj are not represented inS. But thenui,j may be added toSwithout violating independence, contradicting the maximality ofS.

Proof of (b). Suppose thatG has an independent set of sizek with vertices indexedi1, i2, . . . , ik. Thenvi1,1, vi2,2, . . . , vik,k is an independent set of sizek in G′. Also notethat any maximal independent set that contains au-type vertex must be of sizek − 1 fromthe argument in part (a).✷

This leads to the main result of this section.

Theorem 2. There is no polynomial time robust algorithm for finding the maximindependent set for well-covered graphs unless P= NP.

Proof. Suppose that there is a robust polynomial time algorithmA for the independent seproblem on well-covered graphs. We show how to useA to solve IS in polynomial time toachieve the desired contradiction.

For any instance〈G,k〉 of IS, construct the graphG′ as in the reduction above. RunAon the graphG′ to decide if there is an independent set inG′ of sizek. From the claimabove,G′ is well covered and has maximal independent sets of sizek − 1 if and only ifGhas no independent set of sizek; consequently,A will answer “yes” or say thatG′ is notwell covered if and only ifG has an independent set of sizek. In both these cases, answ“yes” to the instance of IS, otherwise answer “no.” Clearly, the time involved is at mpolynomial in the size of〈G,k〉. ✷

5. Conclusions

This paper introduces the notion of a robust algorithm for restricted domain probWe believe that this notion is more practical than the more standard “promise” assumin which an algorithm is allowed to behave in any manner when the input is not idomain. We also introduce a new form of NP-hardness reduction to deal with situatiwhich a problem admits a promise solution but no robust solution unless P= NP.

Many specific problems conform to the scenario of having polynomial time solugiven a model but being open with respect to promise or robust solutions. One well ksuch problem is finding a maximum clique in visibility graphs [1,10]. Cryptology isof such problems, for example, the RSA scheme [16] is compromised if a modelcompositeness of the modulus is given in the form of its prime factorization, whthe standard cryptographic assumption is that the scheme is secure if one is givea promise of the compositeness. It is not clear for these problems whether the probecome easy because the problem is tractable for the class of inputs, or whethbecome solvable only because we are given the answer to a difficult problem asthe input.

In these situations, proving hardness results using standard complexity-theoassumptions is not very easy. That is, such assumptions do not seem sufficient to dist


a modelfor aP.

lopber of

senserems.erifiedhms is

128.(1)

ritish

Ser.

ath. 55

c-key

Institut,hnical

ag/gd/

-key

–381.7–262.ociety,

(1986)

(1984)

between problems that are solvable given a mere promise and problems that needto be solvable. This is because the existence of a polynomial time model solutiondecision problemΠ in restricted domainX places the problem in both NP and co-NTherefore one would need an assumption which is strong enough to separate NP∩ co-NP from P in order to claim thatΠ is hard. More work needs to be done to devetechniques for proving hardness results in this situation, which applies to a numnatural problems.

It has been pointed out to us that the concept of a robust algorithm was in someanticipated by Cameron and Edmonds’ [5] idea of EP (existentially polytime) theoAn EP theorem has the property that every condition of the theorem can be vin polynomial time. The correspondence between EP theorems and robust algoritdiscussed at greater length in [19].

References

[1] J. Abello, H. Lin, S. Pisupati, On visibility graphs of simple polygons, Congr. Numer. 90 (1992) 119–[2] A. Blum, D. Karger, AnO(n3/14)-coloring algorithm for 3-colorable graphs, Inform. Process. Lett. 61

(1997) 49–53.[3] H. Breu, D. Kirkpatrick, Unit disk graph recognition is NP-hard, Comput. Geom. 9 (1998) 3–24.[4] H. Breu, Algorithmic aspects of constrained unit disk graphs, PhD dissertation, University of B

Columbia, 1996.[5] K. Cameron, J. Edmonds, Existentially polytime theorems, polyhedral combinatorics, DIMACS

Discrete Math. Theoret. Comput. Sci. 1 (1990) 83–100.[6] Y. Caro, A. Sebö, M. Tarsi, Recognizing greedy structures, J. Algorithms 20 (1996) 137–156.[7] B.M. Chazelle, D.T. Lee, On a circle placement problem, Computing 36 (1986) 1–18.[8] V. Chvátal, P.J. Slater, A note on well-covered graphs, Quo vadis, graph theory?, Ann. Discrete M

(1993) 179–181.[9] B.N. Clark, C.J. Colbourn, D.S. Johnson, Unit disk graphs, Discrete Math. 86 (1990) 165–177.

[10] D. Dobkin, H. Edelsbrunner, Searching for empty convex polygons, Algorithmica 5 (1990) 561–571.[11] S. Even, A. Selman, Y. Yacobi, The complexity of promise problems with applications to publi

cryptography, Inform. Control 61 (2) (1984) 159–173.[12] A. Gräf, Coloring and recognizing special graph classes, PhD thesis, Musikwissenschaftliches

Abteilung Musikinformatik, Johannes Gutenberg-Universität Mainz, 1995; Also available as TecReport Bericht 20.

[13] M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98.[14] M.D. Plummer, Well-covered graphs: A survey, Quaestiones Math. 16 (1993) 253–287.[15] E. Prisner, Journey through intersection graph country, http://www.math.uni-hamburg.de/sp

mitarbeiter/prisner/Pris/Rahmen.html3/24/99.[16] R.L. Rivest, A. Shamir, L.M. Adleman, A method for obtaining digital signatures and public

cryptosystems, Commun. ACM 21 (2) (1978) 120–126.[17] A.L. Selman, A taxonomy of complexity classes of functions, J. Comput. Syst. Sci. 48 (2) (1994) 357[18] R. Sankaranarayana, L. Stewart, Complexity results for well covered graphs, Networks 22 (1992) 24[19] J. Spinrad, Efficient Graph Representations, Fields Institute Monographs, American Mathematical S

2003.[20] L.G. Valiant, V.V. Vazirani, NP is as easy as detecting unique solutions, Theoret. Comput. Sci. 47

85–93.[21] A. Wigderson, Improving the performance guarantee for approximate graph coloring, J. ACM 30

729–735.

Robust algorithms for restricted domains

Documents

Transcript of Robust algorithms for restricted domains