On Two Segmentation Problems

Transcription

1 Journal of Algorthms 33, Ž Artcle ID jagm , avalable onlne at on On Two Segmentaton Problems Noga Alon* Department of Mathematcs, Raymond and Beerly Sacler Faculty of Exact Scences, Tel A Unersty, Tel A, Israel and Insttute for Adanced Study, Prnceton, New Jersey E-mal: noga@math.tau.ac.l and Benny Sudaov Department of Mathematcs, Raymond and Beerly Sacler Faculty of Exact Scences, Tel A Unersty, Tel A, Israel E-mal: sudaov@math.tau.ac.l Receved January 12, 1999; revsed Aprl 15, 1999 The hypercube segmentaton problem s the followng: Gven a set S of m vertces of the dscrete d-dmensonal cube 0, 14 d, fnd vertces P 1,..., P, P 4 0, 1 d and a parttons of S nto segments S 1,...,S so as to maxmze the sum Ý 1 Ýc S Pc, where s the overlap operator between two vertces of the d-dmensonal cube, defned to be the number of postons they have n common. Ths problem Ž among other ones. was consdered by Klenberg, Papadmtrou, and Raghavan, where the authors desgned a determnstc approxmaton algorthm that runs n polynomal tme for every fxed and produces a soluton whose value s wthn of the optmum, as well as a randomzed algorthm that runs n lnear tme for every fxed and produces a soluton whose expected value s wthn 0.7 of the optmum. Here we desgn an mproved approxmaton algorthm; for every fxed 0 and every fxed our algorthm produces n lnear tme a soluton whose value s wthn Ž 1. of the optmum. Therefore, for every fxed, ths s a polynomal approxmaton scheme for the problem. The algorthm s determnstc, but t s convenent to frst descrbe t as a randomzed algorthm and then to derandomze t usng some propertes of expander graphs. We also consder a segmented verson of the mnmum spannng tree problem, where we show that no approxmaton can be acheved n polynomal tme, unless P NP Academc Press * Research supported n part by a USAIsrael BSF grant, by the Mnerva Center for Geometry at Tel Avv Unversty, by Sloan Foundaton Grant , and by a State of New Jersey grant $30.00 Copyrght 1999 by Academc Press All rghts of reproducton n any form reserved.

2 174 ALON AND SUDAKOV 1. INTRODUCTION Motvated by the study of varous decson-mang procedures arsng n data mnng, Klenberg, Papadmtrou, and Raghavan ntroduced n 9 a new class of optmzaton problems, whch they called segmentaton problems. In these problems, a company has some nformaton about a set of customers C, and ts objectve s to choose Ž and produce. a prescrbed number of polces. The objectve s to optmze, over all possble choces of polces, the sum, over all customers c C, of the value assgned by c to the best polcy among the polces produced, accordng to hs ndvdual utlty functon. Once the polces are chosen, the set of customers s parttoned nto segments, where segment number conssts of all customers that pc polcy number. It turns out that n many cases, even when the optmzaton tas s trval for the nonsegmented case Ž 1., the correspondng optmzaton problem s NP-hard already for 2. In the present paper we study two problems of ths type. The frst one s the followng. THE HYPERCUBE SEGMENTATION PROBLEM. Gven a set S of m customers, each a vertex of the dscrete d-dmensonal cube 0, 14 d, fnd polces P 4 1,...,P, P 0, 1 d and a partton of S nto segments S,...,S so as to maxmze the sum 1 Ý Ý 1 cs P c, where s the overlap operator between two vertces of the d-dmensonal cube, defned to be the number of postons they have n common. Ths problem s consdered n 9, where the authors show that ts precse soluton s NP-hard even for 2. They desgn two approxmaton algorthms for the problem: The frst s a determnstc algorthm that runs n polynomal tme for every fxed and produces a soluton whose value s wthn 2' 2 2 Ž of the optmum. It s based on the nterestng observaton that for every set S 0, 14 d there s some P S so that for every x 0, 14 d, Ý cs Ž '. Ý Pc cs xc. The second s a randomzed algorthm that runs n lnear tme for every fxed and produces a soluton whose expected value s wthn 0.7 of the optmum.

3 TWO SEGMENTATION PROBLEMS 175 Here we desgn an mproved approxmaton algorthm; for every fxed 0 and every fxed our algorthm produces n lnear tme a soluton whose value s wthn Ž 1. of the optmum. Therefore, for every fxed ths s a polynomal tme approxmaton scheme for the problem. Our algorthm s determnstc, but t s convenent to frst descrbe t as a randomzed one and then to derandomze t usng some propertes of expander graphs. Ths mproves the performance rato as well as the runnng tme of the determnstc algorthm of 9 for all m, d, and. The randomzed algorthm of 9 s slghtly faster than ours for large, but the performance rato of our algorthm s much better. The second segmentaton problem we consder s the followng mnmzaton problem, whch s only mentoned brefly n 9. THE MINIMUM SPANNING TREE SEGMENTATION PROBLEM. Gven a connected graph G Ž V, E. and n nonnegatve functons f : E R, 1 n, fnd spannng trees T 1,...,T of G, so as to mnmze the sum n Ý 1 1j j mn f T, where f Ž T. Ý f Ž e. j e EŽT.. j We show that unless P NP, there s no polynomal tme algorthm that approxmates the optmal soluton of ths problem up to any fnte factor, even f 2. The rest of ths paper s organzed as follows. In Secton 2 we consder the hypercube segmentaton problem, descrbe our randomzed approxmaton algorthm, and present s derandomzaton. In Secton 3 we present the Ž smple. proof that there s no polynomal tme approxmaton algorthm for the mnmum spannng tree segmentaton problem, unless P NP. The fnal secton 4 contans some concludng remars. 2. THE HYPERCUBE SEGMENTATION PROBLEM In ths secton we present a polynomal approxmaton scheme for the hypercube segmentaton problem. Frst we descrbe a randomzed algorthm for ths problem and then we show how t can be derandomzed, usng some propertes of random wals on expanders Random Samplng Let S 0, 14 d be a set of m customers. Denote by fž P, S. Ý 4 c S Pc the total value of polcy P 0, 1 d for the customer set S S, where Pc s the number of coordnates n whch P and c agree.

4 176 ALON AND SUDAKOV A famly of polces P 1,...,P nduces a partton of the entre set S nto segments S 1,...,S by puttng c S nto the set S f Pc Pjc for all j and by breang tes arbtrarly. It s easy to see that ths partton maxmzes the value of the expresson Ý fž P, S. over all possble part- tons of S nto parts S 1,...,S. Therefore the segmentaton problem s equvalent to the problem of fndng a famly of optmal polces. Note that the optmal value of the problem s clearly at least md2. Ths value can be produced wthout segmentaton by pcng the majorty bt n each of the d coordnates. We frst descrbe a smple randomzed approxmaton algorthm for the hypercube segmentaton problem, whch for any fxed 0 produces a soluton whose expected value s wthn Ž 1. from optmal. For any fxed and the runnng tme of ths algorthm s lnear. ALGORITHM A, Ž.. Input: A set S of m customers, each beng a vertex of 0, 14 d. 1. Sample l Ž 2. customers from S wth repettons, randomly and ndependently, accordng to a unform dstrbuton. 2. For all possble parttons of the sample set nto at most segments do: 2.1. For each segment n the partton fnd an optmal polcy: a vector from 0, 14 d whch n each coordnate agrees wth the majorty of the elements of the segment. Ths gves a famly of polces Produce the segmentaton of the entre set S accordng to ths famly of polces. 3. Let S,...,S 1 be the optmal segmentaton obtaned from all possble parttons of the sample set Žnote that some of the sets S may be empty. and let P,1, be the correspondng famly of polces. Output: ŽS, P. for 1. Note that the number of all possble parttons of a set of sze l nto at most parts s O Ž l.. Therefore, the runnng tme of ths algorthm s Ž l1. OŽŽ 2. ln. O md e md. Ths s lnear n the length of the nput md for any fxed and and remans polynomal n ths length for each up to OŽlnŽ md. ln lnž md... To study the performance of the algorthm, let P 1,..., P be the optmal famly of polces and let S 1,...,S be the correspondng segmentaton of S. Denote by X the subset of S obtaned by our random samplng. Let X,..., X be the partton of X defned by X X S and let P,...,P 1 1 be the optmal famly of polces for ths segmentaton of X Žf X s empty, then P can be any vector.. Denote by S 1,...,S the partton of S nduced by the famly P 4. Snce the algorthm A, Ž. checs all possble

5 TWO SEGMENTATION PROBLEMS 177 parttons of X, we conclude that the value of ts soluton satsfes fž P, S. fž P, S. fž P, S.. Ý Ý Ý It thus suffces to prove that the expected value of Ý fž P, S. s at least Ž 1. of the optmum value Ý fž P, S.. Let S be a segment n the optmal partton of S, and let S Ž r. j, r 0, 1 be the subset of S of all vectors, whose jth coordnate equals to r. Put S Ž. 0 S Ž. 1 S. By step 2.1 of the algorthm A, Ž. j j j the jth coordnate P, n the polcy vector P j s the majorty bt over all the jth coordnates of the elements from X S. Thus we obtan that the event P j Pj can happen only f the number of elements n X whch belong to S Ž. 0 or to S Ž. j j 1 devates from ts expected value by at least Ž 2.ŽS lm.. Note that the value X S Ž r. j j s bnomally dstrbuted wth parameters l and S Ž r. j m. Therefore, usng the standard estmates for bnomal dstrbutons Ž see, e.g., 4, Appendx A. we obtan that Pr P P e j j 2 Ž jžs l m... Each such event contrbutes an addtve factor of S Ž. 0 S Ž. j j 1 S j to the total dfference between the optmal value and the expected result of the algorthm. By the above dscusson, ths mples that the expected value of ths dfference s at most Ý j Ž jž 2 S l m.. Pr P P S Ž 0. S Ž 1. S e. Ý j j j j j j Ž. ct Consder the functon gt te 2. It s easy to chec that gt Ž. attans ts ' Ž. 12 maxmum at t 1 2c and, hence, gt c for any real t. By tang 2 Ž. Ž j Ž S l m.. c to be S lm we obtan that e OŽ m S l. j ' Ž.. Therefore the expected value satsfes Ý Ý Ý j ž / 1 1 j E f P, S f P, S S e ( 2 Ž jžslm.. ž ' / d m Ý fž P, S. O S Ý Ý. l 1 j1 1 ' ' Ý Ž. Ý Ž. Ý( 1 1 j1 Ý Ž. ž ( By Jensen s nequalty Ý1 S Ý S m. Therefore, d m E f P, S f P, S O ' m ž / ž l / ' / f P, S O md. l 1

6 178 ALON AND SUDAKOV Usng the facts that l Ž 2. and that the optmal value of the hypercube segmentaton problem s at least md2, we conclude that wth the rght choce of the constant n the defnton of l, ž / 2 md E fž P, S. fž P, S. Ž 1. fž P, S.. Ý Ý Ý 1 1 Ths completes the proof that the approxmaton rato of the algorthm A, s at least Derandomzaton a random wals Let G Ž V, E. be a connected, nonbpartte, d-regular graph on m vertces. A random wal on G s equvalent to a tme reversble Marov chan. The states of the Marov chan are the vertces of the graph, and for any two vertces u and the transton probablty from u to, p 1d f Ž u,. u s an edge and zero otherwse. Note that by defnton the transton probablty matrx P Ž 1d. A, where A s the adjacency matrx of the graph G, and the unform dstrbuton s the statonary dstrbuton of ths wal Žsee, e.g., 13 for some basc results about random wals on graphs.. The egenvalues of P are reals, and the largest egenvalue Ž n absolute value. s 1. We denote the second largest Žn absolute value. egenvalue by and defne the egenalue gap to be 1. Ths quantty s drectly related to the expanson propertes of the graph G Žsee, e.g., 2, 3, 14, 15.. Roughly speang, s large f and only f G s a good expander. Let U be a subset of vertces of G. Consder a random wal on G startng from a vertex chosen unformly at random. We denote by tl the number of tmes the random wal vsts a vertex of U durng the frst l steps. The followng useful result about the behavor of tl was proved by Gllman Ž 7 followng 1, 5, 8.. THEOREM Let G Ž V, E. be a connected, regular graph on m ertces wth egenalue gap. Consder a random wal on G startng from a ertex chosen unformly at random. Let U be an arbtrary subset of ertces of G, U cm. Then for any l 2 20l Pr t cl 4e. l To use the above result for producng effcent determnstc algorthms, we need an explct constructon of regular graphs wth constant degree and large egenvalue gap. The best nown such constructons were gven by Lubotzy, Phllps, and Sarna 11 and ndependently by Marguls 12. For each d p 1, where p s a prme congruent to 1 modulo 4, they

7 TWO SEGMENTATION PROBLEMS 179 constructed an nfnte explct famly of d-regular graphs wth 2' d 1. ŽWe note that these graphs wll not have exactly m vertces for any m, but ths does not cause any real problem as we can tae a graph on n vertces such that m n Ž1 ož 1.. m. In ths case the number of vertces n the subset U, U cm s stll Žc ož 1.. n.. Tang, say, d 6 and 2' 5 we get an egenvalue gap of Ž 6 2' and thus we can use such a 6-regular expander for our purposes. Usng ths constructon together wth the result of Gllman we obtan the followng corollary. COROLLARY 2.2. Gen any set S of sze m and any natural number l we l can construct an explct famly F of sze 6 m of multsets F 4, F S, F l wth the followng property. Let F be a multset, chosen randomly and unformly from F, then for eery subset U S of sze cm l PrŽ F U cl. 4e. Proof. Let G be a 6-regular graph on the vertex set S wth egenvalue gap at least Ž 6 2' Let F F 4 be the famly of all possble wals of length l on G. Clearly the sze of F s 6 l m. A random element F of F corresponds to a random wal of length l on the graph G, whch starts n a unform dstrbuton. Note that by defnton F U t l. Therefore, Theorem 2.1 completes the proof of the corollary. A famly of subsets from Corollary 2.2 s the man ngredent of the followng determnstc algorthm for the hypercube segmentaton problem. ALGORITHM B, Ž.. Input: A set S of m customers, each beng a vertex of 0, 14 d. 1. Construct a famly F F 4 of sze OŽ6 l m. of multsubsets of S, Ž 2 where each F l 2., satsfyng the property n the asser- ton of Corollary For 1 F and for all possble parttons of F nto at most segments do: 2.1. For each segment n the partton fnd an optmal polcy: a vector from 0, 14 d whch n each coordnate agrees wth the majorty of the elements of the segment. Ths gves a famly of polces Produce the segmentaton of the entre set S accordng to ths famly of polces. 3. Let S,...,S 1 be the optmal segmentaton obtaned from all possble parttons of F for 1 F Ž note that some of the sets S may be empty. and let P, 1 be the correspondng famly of the best polces. Output: ŽS, P. for 1.

8 180 ALON AND SUDAKOV Ž l1 l. OŽŽ 2 2. ln. The runnng tme of ths algorthm s O 6 md e md. Therefore t s lnear n md for any fxed and and remans polynomal n md for any up to OŽŽlnŽ md. ln lnž md We clam that the above algorthm produces a soluton of value at least Ž 1. of the optmum. Indeed, consder a multset X Ft chosen randomly and ndpendently from the famly F. Let P 1,...,P be the optmal famly of polces and let S 1,..., S be the correspondng segmentaton of S. Let X 1,..., X be the partton of X defned by X X S and let P,...,P 1 be the optmal famly of polces for ths segmentaton of X Žf X s empty, then P can be any vector.. Denote by S 1,...,S the partton of S nduced by the famly P 4. As explaned n the prevous subsecton, t suffces to prove that the expected value of Ý fž P, S. s at least Ž 1. of the optmum value Ý fž P, S.. Let S be a segment n the optmal partton of S, and let S Ž r. j, r 0, 1 be the subset of S consstng of all vectors whose jth coordnate s r. Put S Ž. 0 S Ž. 1 S. By step 2.1 of the algorthm B, Ž. j j j the jth coordnate P, n the polcy vector P j s the majorty bt over all the jth coordnates of the elements from X S. Thus we obtan that the event P j Pj can happen only f the number of elements n X whch belong to S Ž. 0 or to S Ž. j j 1 devates from ts expected value by at least Ž 2.ŽS lm. j. Note that, by Corollary 2.2, X Ft satsfes the property that l PrŽ X U cl. 4e for any subset U S of sze cm. Therefore, we obtan that Pr P P e j j Ž 2 jž S 2 l m 2.. e ŽlŽ j S m. 2.. Each such event contrbutes an addtve factor of S Ž. 0 S Ž. j j 1 S j to the total dfference between the optmal value and the expected result of the algorthm. By the above dscusson, ths mples that the expected value of ths dfference s at most S j 2 ŽlŽ js m.. ÝPrŽ Pj Pj. SjŽ 0. SjŽ 1. mý e. m j j Ž. ct 2 12 As mentoned n Subsecton 2.1 the functon gt te c for any Ž. Ž. ŽlŽ js m. 2. real t. By tang c to be l we obtan that j Sm e OŽ 1 ' l.. Therefore, the expected value satsfes ž Ý / Ý Ý m S j 2 E f P, S f P, S m e j ŽlŽ S m j / md Ý fž P, S. O. ž ' l 1

9 TWO SEGMENTATION PROBLEMS 181 Ž 2 Usng the facts that l 2. and that the optmal value of the hypercube segmentaton problem s at least md2, we conclude that wth the rght choce of the constant n the defnton of l, the expected value satsfes ž / 2 md E fž P, S. fž P, S. Ž 1. fž P, S.. Ý Ý Ý 1 1 Thus, there exsts a partcular t and a partton of X Ft whch produces a segmentaton of S whose value s wthn Ž 1. from optmum. But then the algorthm B, Ž. wll fnd t n stage 3. Ths completes the proof of the correctness of the algorthm. 3. THE MINIMUM SPANNING TREE SEGMENTATION PROBLEM A hypergraph H s an ordered par H Ž V, E., where V s a fnte set Ž the ertex set., and E s a famly of dstnct subsets of V Ž the edge set..a hypergraph H Ž V, E. s 3-unform f all edges of H are of sze 3. The chromatc number of H s the mnmum number of colors requred to color all ts vertces so that no edge s monochromatc. Lovasz 10 Žsee also. 6 showed that t s NP-hard to determne whether a 3-unform hypergraph s 2-colorable. In ths secton we present a constructon for reducng the 2-colorablty problem for 3-unform hypergraphs to the segmented verson of the mnmum spannng tree. Usng ths constructon we deduce that unless P NP the mnmum spannng tree segmentaton problem does not have any polynomal tme approxmaton even for an extremely smple grapha path wth three parallel edges between each par of consecutve nodes. Suppose we are gven a 3-unform hypergraph H ŽVŽ H., EH. wth EH m edges. Let G ŽVG, EG. be a path of length m wth three parallel edges between each par of consecutve nodes. Each trple e, e, e of parallel edges n G corresponds to an edge Ž u,, w. u w of the hypergraph H and each edge n the trple s labeled by a vertex of the edge Ž u,, w.. For every vertex u VŽ H. defne a weght functon fu on the edges of G wth f Ž e., e EG u beng one f and only f the edge e s labeled by the vertex u and f Ž e. u 0 otherwse. We clam that the chromatc number of the hypergraph H s equal to two f and only f there exsts a par T, T of spannng trees of G such that 1 2 Ý u 12 uv H mn f Ž T. 0.

10 182 ALON AND SUDAKOV 4 To prove ths clam assume, frst, that c : V H 1, 2 s a 2-colorng of the vertces of H wth no monochromatc edges. Denote by G, 1, 2, the subgraph of G spanned by all edges, correspondng to the vertces of H wth color. Snce no edge of the hypergraph H s monochromatc we obtan that from every trple of parallel edges n G at least one belongs to G, 1, 2. Therefore each of the subgraphs G1 and G2 contans a spannng tree T of G. Thus, t s enough to prove that for any u VŽ H. at least one of the values f Ž G. u, 1, 2, s equal to zero. To do so, consder G for 3 cu. By defnton G contans no edges corre- spondng to u and thus f Ž G. 0. Ths mples that Ý u 0 mn f Ž T. mn f Ž G. 0. Ý u u uv H uv H Now assume that there exsts a par T1 and T2 of spannng trees of G such that mn f Ž T. 0 for every vertex u VŽ H.. Let V, 1, 2, 1 2 u be the subset of vertces of H whch correspond to the labels of the edges n the tree T. Consder a vertex colorng of the hypergraph H by two colors such that all vertces n V, 1, 2, are colored by the color and all the remanng vertces are colored arbtrarly. We need to prove that no edge of H s monochromatc and no vertex gets two colors smultaneously. Ths wll mply that the hypergraph H s 2-colorable. Note frst, that the subsets V1 and V2 are dsjont. Indeed, assume ths s false and let u be a vertex of H whch belongs to V V. Then the spannng trees T and T both contan an edge of G whch s labeled by the vertex u. By the defnton of the functon f t follows that f Ž T., f Ž T. u u 1 u 2 0, contradc- ton. Hence each vertex of H gets only one color. Snce T s a spannng tree of G t has at least one edge from every trple of parallel edges n G. Therefore V, 1, 2, ntersects every edge of the hypergraph H n at least one vertex. Ths mples that there are no monochromatc edges. Applyng now the result of Lovasz, we get that f P NP, then there s no polynomal algorthm to decde whether the optmal value n the mnmum spannng tree segmentaton problem s strctly postve, even for the case of two trees. Ths mples the followng. THEOREM 3.1. Gen a connected graph G Ž V, E., V n, and a famly f,1m, of nonnegate weght functons on EŽ G., t s mpossble to approxmate n polynomal tme n m and n the optmal alue of the mnmum spannng tree segmentaton problem for G Ž een for 2. wthn any fnte factor, unless P NP. 4. CONCLUDING REMARKS The class of segmentaton problems contans several nterestng algorthmc questons, and our present paper deals wth two of them.

11 TWO SEGMENTATION PROBLEMS 183 It s not dffcult to extend the problem and results of Secton 2 to hypercubes over a larger fxed alphabet. We omt the smple detals. The hardness result for the mnmum spannng tree segmentaton problem holds, as mentoned n Secton 3, even f the nput graph s the path wth three parallel edges between every two consecutve vertces. Smlarly, t holds for many other graphs, ncludng every n vertex graph that contans a spannng subgraph whch s a subdvson of a graph obtaned from any path by replacng each of Žn. edges by three parallel ones. Ths ncludes the graphs of the d-cubes, as well as many other ones. Some of our technques here are useful n the study of other segmentaton problems. One of these s the catalog segmentaton problem Žn the dense case. consdered n 9. Suppose that we have a set of n customers and a set U of m tems. For each customer we are gven a subset of tems ths customer les. We wsh to create a catalog wth r tems to be sent to the customers. Our objectve s to maxmze the sum, over all tems, of the number of customers that le ths tem. The smple optmal soluton s, of course, to select the r most popular tems. However, f nstead of one catalog we can create dfferent ones, each wth r tems, sendng one of them to each customer, we can sometmes ensure a much bgger value than that gven by a sngle catalog. Ths leads to the catalog segmentaton problem, whose precse formulaton s the followng. THE CATALOG SEGMENTATION PROBLEM. Gven a set U of sze m and a famly S 1,...,Sn of n subsets of U, fnd subsets X 1,..., X of U, each of sze r, so as to maxmze the sum n Ý 1 1j j max S X. Ths problem s consdered n 9, where the authors prove that t s NP-hard even for 2. For fxed and for the specal Ž dense. case n whch each customer les at least a fracton of all tems Žthat s, S U for all., they desgn a randomzed polynomal tme approxma- ton scheme. Our technque here can be used to provde a determnstc polynomal tme approxmaton scheme for ths specal case. Wthout the densty assumpton, the problem appears to be much more dffcult, and as mentoned n 9, even the problem of mprovng the trval 12 approxmaton for 2 n polynomal tme seems dffcult. REFERENCES 1. M. Ajta, J. Komlos, and E. Szemered, Determnstc smulaton n logspace, n Proc. of the 19th ACM STOC, pp , Assoc. Comput. Mach., New Yor, N. Alon, Egenvalues and expanders, Combnatorca 6 Ž 1986., 8396.

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