(k,?)-sandwich Problems: why not ask for special kinds of bread?

Transcription

1 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 17 #1 Matemátca Contemporânea, Vol. 42, c?2014, Socedade Braslera de Matemátca (k,?)-sandwch Problems: why not ask for specal knds of bread? F. Couto L. Fara S. Klen F. Prott L. T. Noguera Abstract In ths work, we consder the Golumbc, Kaplan, and Shamr decson sandwch problem for a property Π: gven two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ), the queston s: Is there a graph G = (V, E) such that E 1 E E 2 and G satsfes Π? The graph G s called sandwch graph. Note that what matters here s just the fllng of the sandwch. Our proposal s to try dfferent knds of bread for each chosen specal sandwch fllng. In other words, we focus on the complexty of sandwch problems when, beforehand, t s known that G satsfes a property Π, = 1, 2. Let (Π 1, Π, Π 2 )-sp denote the sandwch problem for property Π when G satsfes Π, called sandwch problem wth boundary condtons. When G s not requred to satsfy any specal property, Π s denoted by. A graph G s (k,?) f there s a partton of V (G) nto k ndependent sets and? clques. It s known that (, (k,?), )-sp s NP-complete, for all k +? greater than 2. In order to motvate ths new work proposal, n ths paper we descrbe polynomal-tme algorthms for three sandwch problems wth boundary condtons: (perfect, (k,?), polynomal number of maxmal clques)-sp for all k,? N, (, 2000 AMS Subject Classfcaton: 60K35, 60F05, 60K37. Key Words and Phrases: Graph Sandwch Problems, Boundary Condtons, (k,?)- graphs. *Partally Supported by CNPq, CAPES and FAPERJ.

2 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 18 #2 18 F. Couto, L. Fara, S. Klen, F. Prott and L. T. Noguera (2, 1), trangle-free)-sp, and (, (2, 1), bounded degree)-sp. The frst problem ncludes the case (chordal, (k,?), chordal)-sp. 1 Introducton Golumbc, Kaplan and Shamr ntroduced n [15] the graph sandwch problem for property Π n ts orgnal form, as follows: graph sandwch problem for property Π Instance: Graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ), such that E 1 E 2. Queston: Is there G = (V, E) such that E 1 E E 2 and G satsfes Π? We call E 1 the set of mandatory edges, E 2 \ E 1 the set of optonal edges, and E(G 2 ) the set of forbdden edges. Hence, any sandwch graph G = (V, E) for the par G 1, G 2 must contan all mandatory edges and no forbdden edges. Graph sandwch problems have drawn much attenton because they naturally generalze graph recognton problems and have many applcatons [8, 9, 10, 14, 19]. After studyng many of them, we started questonng why not choose specal knds of bread for a partcular specal stuffed sandwch? After all, we can change ts taste just changng ts bread. So, we propose a generalzed verson of sandwch problems, that we call sandwch problem wth boundary condtons, denoted by (Π 1, Π,Π 2 )-sp n whch the nput graphs G 1 and G 2 satsfy propertes Π 1 and Π 2, respectvely. We can formalze t as follows: graph sandwch problem for property Π wth boundary condtons Instance: Graphs G = (V, E ) satsfyng Π, = 1, 2, such that E 1 E 2. Queston: Is there G = (V, E) such that E 1 E E 2 and G satsfes Π? If G s not requred to satsfy any specal property, we denote Π by. The recognton problem for a class of graphs C s equvalent to a partcular graph sandwch problem where E 1 = E 2. In ths case, the goal s to decde wether G 1 = G 2 = G satsfes Π, where Π s the property of

3 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 19 #3 (k,?)-sandwch Problems: why not ask for specal knds of bread? 19 belongng to C. Golumbc et al. [15] remark that (, Π, )-sp s nterestng when property Π s polynomally recognzable. In contrast, we observe that (Π 1, Π, Π 2 )-sp s nterestng both when (, Π, )-sp s polynomally solvable and when (, Π, )-sp s NP-complete: If (, Π, )-sp s polynomally solvable then there may exst algorthms wth better complexty to solve (Π 1, Π, Π 2 )-sp, usng propertes Π 1 or Π 2 ; If (, Π, )-sp s NP-complete, then there stll remans the possblty that (Π 1, Π, Π 2 )-sp admts a polynomal-tme algorthm, snce (Π 1, Π, Π 2 )-sp s not more dffcult than (, Π, )-sp. In ths paper we consder the followng stuaton: (, Π, )-sp s NPcomplete and (Π 1, Π, Π 2 )-sp s polynomally solvable. We focus on a partcular property Π related to the (k,?)-partton problem [11, 12]. A graph G s (k,?) f V (G) can be parttoned nto k stable sets and? clques. In [1, 2, 4, 12], the problem of recognzng (k,?)-graphs was shown to be NP-complete f k 3 or? 3 and polynomally solvable otherwse, whle the problem (, (k,?), )-sp was shown to be NP-complete f k +? 3 [7] and polynomally solvable otherwse. A perfect graph G s a graph n whch the chromatc number of every nduced subgraph H s equal to the sze of the largest clque of H. Bpartte, chordal, strongly chordal, comparablty graphs and cographs are mportant classes of perfect graphs [16]. perfect graph recognton s n P [5], and colorng a perfect graph G wth χ(g) colors (the chromatc number of G) s a polynomal problem [3]. In Secton 2, we study three dfferent sandwch problems wth boundary condtons: (perfect, (k,?), polynomal number of maxmal clques)-sp, (, (2, 1), trangle-free)-sp, and (, (2, 1), bounded degree)-sp. We wll prove that these problems can be solved n polynomal tme. Several sandwch problems wth boundary condtons ft on the framework (perfect, (k,?), polynomal number of maxmal clques)-sp

4 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 20 #4 20 F. Couto, L. Fara, S. Klen, F. Prott and L. T. Noguera, snce there are many subclasses of perfect graphs and other mportant ones havng a polynomal number of maxmal clques, such as chordal, trangle free, and bounded degree graphs. Therefore, as a byproduct, the problems (chordal, (k,?), chordal)-sp, (comparablty, (k,?), bounded degree)-sp, (cograph, (k,?), strongly chordal)-sp are examples of polynomally solvable problems. 2 Three Sandwch Problems wth Boundary Condtons Let k,? 0 be fxed ntegers. Defnton 2.1. polynomal number of maxmal clques, or smply pnmc, stands for any nfnte famly of graphs for whch there exsts a polynomal q(n) such that the number of maxmal clques of any graph G n the famly s bounded by O(q(n)), where n = V (G). As an example, pnmc may stand for chordal graphs, by takng q(n) = n 1. Theorem 2.1. (perfect, (k,?), pnmc)-sp s n P. The proof of Theorem 2.1 s based on Algorthm 1. Algorthm 1: Algorthm for solvng (perfect, (k,?), pnmc)-sp Let C be the collecton of maxmal clques of G 2 ; for each subcollecton {C 1, C 2,, C? } of C do let C? = V (C 1 ) V (C 2 ) V (C? ) ; f G 1 \C? s k-colorable then return G = (V, E 1 E(C 1 ) E(C? )) return there s no (k,?)-graph G such that E 1 E(G) E 2 Now we consder the problem (, (2,1), trangle-free)-sp: Theorem 2.2. (, (2,1), trangle-free)-sp s n P.

5 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 21 #5 (k,?)-sandwch Problems: why not ask for specal knds of bread? 21 Algorthm 2: Algorthm for solvng (, (2, 1), trangle-free). f G 1?= G 2 then for each (u, v) E 2 \E 1 do V? := V \{u, v} ; G? = G 1 [V? ] ; f G? s bpartte then return G = (V, E 1 {(u, v)}) return there s no (2, 1)-graph G such that E 1 E(G) E 2 f G 1 s (2, 1) then return G = G 1 = G 2 return there s no (2, 1)-graph G such that E 1 E(G) E 2 Algorthm 2 solves (, (2,1), trangle-free) and runs n O((n+m)m) tme. Fnally, we state the followng result: Theorem 2.3. (, (2,1), bounded degree)-sp s n P. Algorthm 3 solves (, (2,1), bounded degree)-sp by lstng all maxmal clques of G 2 and testng, for each maxmal clque, f the deleton of ts vertces n G 1 yelds a bpartte graph. It runs n O(mn k+1 ) tme, where k = (G 2 ). Algorthm 3: Algorthm for solvng (,(2,1), bounded degree)- sp let C = {C 1,, C l } be the collecton of maxmal clques of G 2 ; for each C C do f G 1 \V (C ) s bpartte then return G = (V, E 1 E(C )) return there s no (2, 1)-sandwch graph G = (V, E) such that E 1 E E 2 We can generalze ths result to graphs G 2 havng a polynomal num-

6 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 22 #6 22 F. Couto, L. Fara, S. Klen, F. Prott and L. T. Noguera ber of maxmal clques, that can be lsted n polynomal tme (see for nstance [20]). Theorem 2.4. (, (2, 1), pnmc)-sp s n P. 3 Conclusons We observe that (k,?) chordal s a polynomally recognzable property [13, 17, 18]. Recently the versons (, chordal (2, 1), )-sp and (, strongly chordal-(2, 1), )-sp have been proved to be NP-complete [6]. In an ongong work, we are tryng to choose propertes Π 1 and Π 2 to analyze the complexty of (Π 1, chordal-(2, 1), Π 2 )-sp and (Π 1, strongly chordal- (2, 1), Π 2 )-sp. In Tables 1 and 2 we summarze the results of ths paper and the correspondng results n the lterature. (Π 1, (2, 1), Π 2 ) Π 1 \Π 2 chordal trangle-free = k chordal O(mn) O(m 2 ) O(mn k+1 )? trangle-free O(mn) O(m 2 ) O(mn k+1 )? = k O(mn) O(m 2 ) O(mn k+1 )? O(mn) O(m 2 ) O(mn k+1 ) NP-c [7] Table 1: Complexty results and open problems for (Π 1, (2, 1), Π 2 ), where propertes Π 1, Π 2 are n {chordal, trangle-free, bounded degree k, }.

7 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 23 #7 (k,?)-sandwch Problems: why not ask for specal knds of bread? 23 (Π 1, (k,?), Π 2 ) Π 1 \Π 2 pnmc perfect P?? NP-complete f k +? 3 and P otherwse[7] Table 2: Complexty results and open problems for (Π 1, (k,?), Π 2 ), where propertes Π 1, Π 2 are n {perfect, pnmc, }. References [1] A. Brandstadt, Parttons of graphs nto one or two ndependent sets and clques, Dscrete Mathematcs 152(1-3) (1996), [2], Corrgendum, Dscrete Mathematcs 186 (2005), 295. [3] A. Brandstädt, V. B. Le, and J. P. Spnrad, Graph classes: a survey, Socety for Industral and Appled Mathematcs, Phladelpha, PA, USA, [4] A. Brandstadt, V. B. Le, and T. Szymczak, The complexty of some problems related to graph 3-colorablty, Dscrete Appled Mathematcs 89(1-3) (1998), [5] M. Chudnovsky, G. Cornuéjols, X. Lu, P. Seymour, and K. Vuskovc, Recognzng Berge Graphs, Combnatorca 25 (2005), [6] F. Couto, Problemas Sanduíche para Grafos-(2,1) com Condções de Contorno (n Portuguese), M.Sc. thess, Unversdade Federal do Ro de Janero, Brazl, [7] S. Dantas, C.M. de Fgueredo, and L. Fara, On decson and optmzaton (k,l)-graph sandwch problems, Dscrete Appled Mathematcs 143 (2004),

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9 Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 25 #9 (k,?)-sandwch Problems: why not ask for specal knds of bread? 25 [17] P. Hell, S. Klen, L. T. Noguera, and F. Prott, Parttonng chordal graphs nto ndependent sets and clques, Dscrete Appled Mathematcs 141 (2004), [18], Packng r-clques n weghted chordal graphs, Annals of OR 138 (2005), [19] R. Srtharan, Chordal bpartte completon of colored graphs, Dscrete Mathematcs 308 (2008), [20] S. Tsukyama, M. Ide, H. Arujosh, and H. Ozak, A new algorthm for generatng all the maxmal ndependent sets, SIAM J. Comput. 6 (1977), COPPE, Unversdade Federal do Ro de Janero Brazl {nandavdc, sula}@cos.ufrj.br Unversdade do Estado do Ro de Janero Brazl luerbo@cos.ufrj.br Unversdade Federal Flumnense Brazl {fabo, loana}@c.uff.br