NP-completeness of generalized Kaboozle

Transcription

1 [DOI: /psjjp ] Regular Paper NP-completeness of generalzed Kaboozle Tetsuo Asano 1 Erk D. Demane 2 Martn L. Demane 2 Ryuhe Uehara 1,a) Receved: August 28, 2011, Accepted: March 2, 2012 Abstract: Kaboozle s a puzzle consstng of several square cards, each annotated wth colored paths and dots drawn on both sdes and holes drlled. The goal s to jon two colored dots wth paths of the same color (and fll all holes) by stackng the cards sutably. The freedoms here are to reflect, rotate, and order the cards arbtrarly, so t s not surprsng that the problem s NP-complete (as we show). More surprsng s that any one of these freedoms reflecton, rotaton, and order s alone enough to make the puzzle NP-complete. Furthermore, we show NP-completeness of a partcularly constraned form of Kaboozle related to 1D paper foldng. Specfcally, we suppose that the cards are glued together nto a strp, where each glued edge has a specfed foldng drecton (mountan or valley). Ths varaton removes the ablty to rotate and reflect cards, and restrcts the order to be a vald folded state of a gven 1D mountan-valley pattern. Keywords: Kaboozle, NP-completeness, orgam, paper foldng puzzle, slhouette puzzle 1. Introducton Kaboozle: The Labyrnth Puzzle s a puzzle created and developed n 2007 by Albatross Games Ltd., London *1. Ths multlayer labyrnth conssts of four square cards; see Fg. 1. (In fact, each card s octagonal, but the pattern on t s a square.) Each card has holes drlled at dfferent locatons, and varous colored paths and dots drawn on both sdes. The goal s to arrange the cards by rotaton, reflecton, and stackng n an arbtrary order to create a contnuous monochromatc path between the corner dots of the same color that s vsble on one sde of the stack. That s, n Kaboozle, we have many combnatons of choces to solve the puzzle. We can rotate and reflect each card, and change the order of layers n any way. Many possble combnatons of these basc operatons make ths smple puzzle wth only four cards more dffcult than t appears at frst glance. On the other hand, snce we have three dfferent basc operatons, t s hard to determne what makes ths puzzle dffcult. The goal of ths paper s to understand what makes ths puzzle NP-complete, when generalzed to n cards nstead of four. Kaboozle s an example of a broader class of puzzles n whch patterned peces wth holes must be arranged to acheve some goal, such as monochromatc sdes. For example, Albatross Games Ltd. places Kaboozle n a seres of puzzles called Transposers *2, whch all have ths style. See Ref. [5] for descrptons, and Ref. [11] for the relevant patent. Our NP-hardness proofs for Kaboozle mmedately mply NP-completeness for ths general famly of puzzles, though there are lkely other specal cases of nterest. An earler form of ths type of puzzle s a slhouette puz- 1 School of Informaton Scence, Japan Advanced Insttute of Scence and Technology (JAIST), Nom, Ishkawa , Japan 2 Computer Scence and Artfcal Intellgence Lab, Massachusetts Insttute of Technology (MIT), Cambrdge, MA 02139, USA a) uehara@jast.ac.jp Fg. 1 The four Kaboozle cards and one of the ten solutons. zle, where peces are regons wth holes (no pattern beyond opaque/transparent) and the goal s to make a target shape. Perhaps the frst slhouette puzzle, and certanly the best known, s the Queston du Lapn or Rabbt Slhouette Puzzle, frst produced n Pars around 1900 (see p.35 n Ref. [7]). Fgure 2 shows the puzzle: gven the fve cards on the left, stack them wth the rght orentatons to obtan one of two dfferent rabbt slhouettes. The puzzle can be played onlne *3. The freedoms n a slhouette puzzle are reflecton and rotaton of the cards; the card stackng order has no effect on the slhouette. (In fact, both rabbts can be obtaned wthout reflectng the cards n Fg. 2, so that puzzle only needs rotaton.) Are these freedoms enough for NP-completeness? We show that ndeed sl- *1 *2 *3 c 2012 Informaton Processng Socety of Japan 713

2 houette puzzles are NP-complete, even allowng just rotaton or just vertcal reflecton of the peces. Furthermore, we show that Kaboozle s NP-complete under the same restrcton of just rotaton or just vertcal reflecton. But s reflecton or rotaton necessary for Kaboozle to be NPcomplete? We show that Kaboozle s NP-complete even when the cards can only be stacked n a desred order, wthout rotaton or reflecton. We also show that Kaboozle s NP-complete when restrcted to a class of orderngs that arse from paper foldng, as descrbed below. Our foldng varaton of Kaboozle s nspred by a 1907 patent [4] commercalzed as the (poltcally ncorrect) Pck the Pckannnes puzzle [8]. Ths puzzle conssts of a sngle pece, shown on the left of Fg. 3, wth holes, mages (stars), and crease lnes. The goal s to fold along the crease lnes to make an array of stars, as shown on the rght. Ths type of puzzle severely lmts the vald stackng orders of the parts, whle also effectvely forbddng rotaton and reflecton of the parts. We consder a smple general puzzle along these lnes, by restrctng a generalzed Kaboozle puzzle. Namely, we glue all the cards n the Kaboozle puzzle nto a strp, and specfy the foldng drecton (mountan or valley) on each glued edge (crease). Now the only freedom s to fold the 1D strp of paper down to a unt sze, respectng the foldng drectons. Ths freedom s a weak form of the orderng of the cards; rotaton and reflecton are effectvely forbdden. Ths dea also comes from problems n computatonal orgam. In polynomal tme, we can determne whether a mountan-valley pattern on a 1D strp of paper can be folded flat, when the dstances between creases are not all the same [1]. A recent noton s the foldng complexty, the mnmum number of smple folds requred to construct a unt-spaced mountan-valley pattern (strng) [2]. For example, n pleats alternatng mountan and valley can be folded n a polylogarthmc number of smple folds and unfolds. On the contrary, the number of dfferent ways to fold a Fg. 2 The classc slhouette puzzle Queston du Lapn. unform mountan-valley pattern of length n down to unt length s not well-nvestgated. The number of foldngs of a paper strp of length n to unt length has been computed by enumeraton, and t seems to be exponentally large; the curve fts to Θ(3.3 n )(see the sequence number A n The On-Lne Encyclopeda of Integer Sequences [6]). However, as far as the authors know, the detals are not nvestgated, and t was not known whether ths functon s polynomal or exponental. Recently, the last author showed theoretcal lower and upper bounds of ths functon: t s Ω(3.07 n )ando(4 n ) [9]. These results mply that a gven random mountan-valley pattern of length n has Θ(1.65 n ) foldngs on average, whch s bounded between Ω(1.53 n )ando(2 n ). Related work of foldng of a paper strp of length n can be found n Ref. [10]. Intutvely, the foldng verson of the Kaboozle puzzle seems easy. Perhaps we could apply the standard dynamc programmng technque from one sde of the strp? But ths ntuton s not correct. Essentally, the problem requres foldng a 1D strp of paper, but the strp has labels whch place constrants on the foldng. Despte the stuaton beng qute restrctve, we prove the problem s stll NP-complete. Therefore we conclude that the generalzed Kaboozle problem s NP-complete even f we allow only one of orderng, rotaton, or reflecton of the cards, and n the orderng case, even f the orderng comes from a 1D strp foldng. 2. Prelmnares We generalze the number of the Kaboozle cards to n + 1. Each card s square, wth some fragments of a path drawn on both sdes, and some holes drlled nto t. From the vewpont of the complexty, we assume that each card can be encoded n constant space. For example, the coordnates of holes and fragments of a path are ntegers, and the heght of a card s also a suffcently large nteger. To smplfy, we also assume that the wdth of a card s a unt length. Hence these szes have no effect on the complexty of the problems. We wll use just one color of the path we have to jon. The (potental) endponts of a path are dstngushable from the other fragments. To smplfy, we assume that the cards are numbered 0, 1, 2,...,n. These n + 1 cards should be used n the puzzle. You cannot leave any cards unused. A strp of the cards can be constructed as follows: for each 0 n 1, the rght sde of the card s glued to the left sde of the card + 1, and that sde s called the ( + 1)st crease. Each crease has a label M or V whch means that the strp must be mountan folded or valley folded at the crease. (We defne one Fg. 3 Puzzle commercalzed as Pck the Pckannnes. Fgure from Ref. [4]. c 2012 Informaton Processng Socety of Japan 714

3 Fg. 4 Example of the reducton for F(x 1, x 2, x 3, x 4 ) = (x 1 x 2 x 3 ) ( x 1 x 2 x 4 ) ( x 2 x 3 x 4 ). sde of the strp as the top sde, and creases are mountan or valley folded wth respect to ths sde.) We assume that the label of the frst crease s M wthout loss of generalty, or otherwse specfed. For a strp of the cards, a folded state s a flat foldng of unt length (where the unt s the wdth of a card) such that each crease s consstent wth ts label. (We note that a folded state always exsts for any strng of labels [9].) The man problem n ths paper s the followng. Input: A strp of n + 1 Kaboozle cards, each wth a label of length n. Queston: Determne whether the strp has a folded state that s consstent wth the labels, and exactly one connected path s drawn on a surface of the folded state. We begn wth an observaton for foldng a unt pattern. Observaton 1 A strp of n + 1 cards wth n creases has a unque folded state f and only f the crease pattern s a pleat,.e., MVMV MV or MVMV MVM. Proof. A pleat foldng has no other folded state whch can be shown by a smple nducton on n snce the card n can be stacked unquely. Suppose that a mountan-valley pattern has a unque folded state. Wthout loss of generalty, we assume that the frst crease s a mountan. If the second crease s also a mountan, we have two folded states of the cards 0, 1, and 2: the paper layers stack from bottom as 2, 1, 3or2, 3, 1. Hence the second crease must be a valley. We can repeat the argument for each crease, and obtan the pleat pattern. Usng the pleats, we ntroduce a useful foldng pattern for NP-completeness, namely, the shuffle pattern of length : (MV) 1 MM(VM) 1 *4. By Observaton 1, the left and rght pleats are folded unquely and ndependently. However, these pleats can be combned n any order to fold to unt length. Thus we have ( ) 2 dstnct foldngs of the shuffle pattern of length. We note that the center card of the shuffle pattern of length, the card + 1 n our notaton, always appears on one sde of any folded state. We call ths sde the top of the shuffle pattern, and card + 1 the top card (although t may come to the bottom n a natural foldng). 3. NP-completeness of Generalzed Kaboozle It s easy to see that all the problems n ths paper are n NP. *4 Here we use the standard notaton x k for strng repetton. For example, (MV) 3 MM(VM) 3 = MVMVMVMMVMVMVM. Hence we concentrate on the proofs of NP-hardness. Our reducton uses the followng 1-n-3 SAT problem. Input: A conjunctve normal form (CNF) Boolean formula F(x 1,...,x n ) = c 1 c 2 c m, where each clause c = (l1 l 2 l 3 ) has three lterals l j {x 1,...,x n, x 1,..., x n }. Queston: Determne whether F has a truth assgnment such that each clause contans exactly one true lteral. Ths problem s a well-known NP-complete varant of 3- satsfablty (the problem [LO4] n Ref. [3]). For a gven CNF formula F(x 1,...,x n ) wth n varables and m clauses, we use 4n + 1 Kaboozle cards as follows. Fgure 4 shows an example of the reducton for F(x 1, x 2, x 3 ) = (x 1 x 2 x 3 ) ( x 1 x 2 x 4 ) ( x 2 x 3 x 4 ). Each gray area s a hole n the card, each black lne s a fragment of the unque path, and the black crcles are the endponts of the unque path. Top card: One top cardsplacedatthetopoftheshufflepattern, and t represents m clauses. On the top card, two endponts of the unque path are drawn, and each clause s represented by a hole n the card. Each hole has two dmples correspondng to the borders of the path and that wll be extended to one of three possble drectons by the lteral cards descrbed below. Lteral card: We use 2n lteral cards. Here, the ndex wth 1 n s used to represent the th varable, and the ndex j wth 1 j m s used to represent the jth clause. Each card represents ether x or x. We make m gadgets on the card for the varable x as follows. If nether x nor x appear n clause c j, the card x has a hole at that place. Hence ths card has no nfluence at that place of the clause c j. If x appears n clause c j, the card x has a part of the path at that place. Accordng to the poston (frst, second, or thrd lteral) n the clause, the path s depcted at top, center, or bottom, respectvely, as shown n Fg. 4. If x appears n clause c j, the card x has a cover area of the path at that place. Ths whte area covers the correspondng path drawn on the lteral card x, as shown n Fg. 4. Each lteral card x s symmetrc to the lteral card x, and hence omtted. Blank card: We use 2n blank cards depcted n Fg. 4. They wll be used to jon lteral cards and the top card. They have no nfluence on the appearance of the lteral cards. c 2012 Informaton Processng Socety of Japan 715

4 Fg. 5 For F(x 1, x 2, x 3 ) = (x 1 x 2 x 3 ) ( x 1 x 2 x 4 ) ( x 2 x 3 x 4 ), a wrong orderng of the cards that corresponds to a wrong assgnment x 1 = 0, x 2 = 1, x 3 = 0, and x 4 = 1. For ths assgnment, the frst clause c 1 contans one true lteral, the second clause c 2 contans three true lterals, and the thrd clause c 3 contans no true lteral. Fg. 6 The cards joned n a strp. We frst show that generalzed Kaboozle s NP-complete, wthout requrng a strp foldng. Theorem 2 Generalzed Kaboozle s NP-complete, even f we forbd reflecton and rotaton. That s, t s NP-complete f we only consder the orderng of the cards. Proof. We use the top card and 2n lteral cards. Make the cards asymmetrc, e.g., by shftng the gadgets on each card a lttle, to forbd reflectng or rotatng the cards (f that s allowed). Clearly, the reducton can be done n a polynomal tme. Because of the pctures of the endponts of the unque path, the top card must be on top. It s not dffcult to see that card x has no nfluence on cards x j and x j f j. Hence t s suffcent to consder the orderng between each par x and x for = 1, 2,...,n. Suppose F(x 1,...,x n ) has a soluton, that s, each clause c j contans exactly one true lteral l j. Then the correspondng lteral card actvates one of three fragments of the path ncdent to the hole representng c j n the top card. To actvate the fragment, the lteral card l j appears before the lteral card l j n stackng order. In ths case, the lteral card l j actvates the fragments n the clauses c j that contan the lteral l j, and the card also covers the lteral card l j whch deactvates the fragments on the lteral card l j. When a clause c j contans two or three true lterals, the path s runed by two or three fragments. On the other hand, f c j contans no true lteral, the path s dsconnected. For example, consder the (wrong) assgnment x 1 = 0, x 2 = 1, x 3 = 0, and x 4 = 1 for F(x 1, x 2, x 3, x 4 ) from Fg. 4, as shown n Fg. 5. Accordng to the assgnment, we put the card x 1 over the card x 1, the card x 2 over the card x 2, and so on. Then, the card x 1 covers the parts of the path on the card x 1, the card x 2 covers the parts of the path on the card x 2, and so on. Any two cards correspondng to dfferent varables can be stacked n any order. For example, we can arrange top, x 1, x 1, x 2, x 2 ; top, x 1, x 2, x 2, x 1 ; or top, x 1, x 2, x 1, x 2 ; and so on. For ths assgnment, the clause c 1 = (x 1 x 2 x 3 ) satsfes the condton of the 1-n-3 SAT because only x 2 s true. Hence the hole correspondng to c 1 n the top card s flled and the path s joned properly. On the other hand, all lterals are true n the clause c 2, and no lteral s true n the clause c 3. Hence the hole correspondng to c 2 produces loops and the path s dsconnected at the hole correspondng to c 3. Therefore, the two endponts of the path on the top card are joned by one smple path f and only f each c j contans exactly one true lteral. We now turn to the man theorem. Theorem 3 Generalzed Kaboozle s NP-complete even n a strp wth a fxed mountan-valley pattern. That s, t s NP-complete f all cards are joned and hence reflecton and rotaton are nhbted and the orderng s strongly restrcted. Proof. We use the top card, 2n lteral cards, and 2n blank cards. We jon these cards nto a strp as x n -b-x n 1 -b- -b-x 2 -b-x 1 -btop-b-x 1 -b-x 2 -b- -b- x n 1 -b-x n, where b means a blank card. Fgure 6 shows the example from Fg. 4. We glue the blank cards upsde down, whch wll be reflected by foldng to unt length. The mountan-valley pattern s the shuffle pattern of length n; that s, the creases on both sdes of the top card are mountan, and from there, the other creases are defned to form two pleats of length n. Now, the left pleat of the top card makes the sequence of x s, and the rght pleat makes the sequence of x s. For each par of x and x, we can choose the orderng between the correspondng cards wth an approprate shufflng. Ths means that we can assgn true or false to ths varable. Moreover, thanks to the blank cards between the lteral cards, we can arrange the orderng of the cards x and x ndependently for each. Hence, by Theorem 2 and the property of the shuffle pattern, the constructed Kaboozle strp wth a fxed mountan-valley pattern has a soluton f and only f the 1-n-3 SAT has a soluton. Carefully checkng folded states of a strp of cards, we can observe that the orentaton of each card s fxed regardless of the foldng pattern. For example, all lteral cards n the strp n the c 2012 Informaton Processng Socety of Japan 716

5 Fg. 7 Gadgets for rotaton and reflecton. Fg. 8 A slhouette puzzle for a rectangle. proof of the man theorem are facng up lke the top card n any folded state. Ths orentaton s n fact determned by the party of the poston of the card n the strp. That s, the mountan-valley pattern does not matter for the orentaton. Usng ths fact, we can also let the mountan-valley pattern be free. That s, n the strp form, the foldng drecton s not essental to the dffculty of the puzzle. Corollary 4 Generalzed Kaboozle s NP-complete n the strp form f we do not specfy the mountan-valley pattern. Proof. We use the same strp n the proof of Theorem 3. Even f the mountan-valley pattern s not specfed, the top card should be on top; otherwse, the endponts of the path dsappear. Hence both creases borderng the top card are mountans. If the 1-n-3 SAT nstance has a soluton, the constructed Kaboozle puzzle has a soluton by the foldng n the proof of Theorem 3. On the other hand, f the Kaboozle puzzle has a soluton, we can extract the orderng between x and x for each wth 1 n from the folded state. From these orderngs, we can construct the soluton to the 1-n-3 SAT nstance. By combnng gadgets, we can show that generalzed Kaboozle s also NP-complete f we allow only ether rotaton or reflecton. Note that we can rotate a card 180 by the combnaton of a horzontal reflecton and a vertcal reflecton. To forbd ths knd of cheatng wth cards, we restrct the reflecton to be vertcal. Theorem 5 Generalzed Kaboozle s NP-complete even f the card orderng s fxed (or free), and (1) only 180 rotaton of the cards s allowed, or (2) only vertcal reflecton of the cards s allowed. Proof. As n the proof of Theorem 2, we prepare the top card and 2n lteral cards. Now, the top card s enlarged to twce the sze of the orgnal cards; see Fg. 7 (1). Rotaton: For each varable x, two lteral cards x and x are glued so that a 180 rotaton exchanges them; see Fg. 7 (2). Vertcal reflecton: For each varable x, two lteral cards x and x are glued so that a vertcal reflecton exchanges them; see Fg. 7 (3). It s easy to see that the orderng of the cards has no nfluence, except the top card whch should be the top, and the resultant Kaboozle has a soluton f and only f the 1-n-3 SAT nstance has a satsfyng truth assgnment. In combnaton wth Theorem 2, we can conclude that one of three basc operatons (reflecton, rotaton, and orderng) s enough to make the generalzed Kaboozle NP-complete. As mentoned n the Introducton, the orderng of the cards has no effect n a slhouette puzzle. Along smlar lnes, we can show that slhouette puzzles are NP-complete even f we allow to use one of two operatons. Theorem 6 Slhouette puzzles are NP-complete even f (1) only 180 rotatons of the cards are allowed, or (2) only vertcal reflectons of the cards are allowed. Proof. We reduce the regular (not 1-n-3) SAT, mmckng the gadgets n Fg. 7. The top card has one hole per clause, all n the top half of the card (Fg. 8 (1)). As shown n Fg. 8 (2) and Fg. 8 (3), each lteral card reverses the top and bottom halves for the true and false lterals. The top and bottom sdes are rotatons or vertcal reflectons of each other accordng to the varaton. If the lteral s true n a clause, the correspondng sold patch totally covers the hole of the clause n the top card. Therefore, the slhouette of the stacked cards s a rectangle (wthout holes) f and only f the formula s satsfable. References [1] Arkn, E.M., Bender, M.A., Demane, E.D., Demane, M.L., Mtchell, J.S.B., Setha, S. and Skena, S.S.: When can you fold a map?, Comput. Geom. Theory Appl., Vol.29, No.1, pp (2004). [2] Cardnal, J., Demane, E.D., Demane, M.L., Imahor, S., Ito, T., Kyom, M., Langerman, S., Uehara, R. and Uno, T.: Algorthmc Foldng Complexty, Graphs and Combnatorcs, Vol.27, pp , DOI: /s (2011). [3] Garey, M. and Johnson, D.: Computers and Intractablty A Gude to the Theory of NP-Completeness, Freeman (1979). [4] Lehman, F.H.: Puzzle. U.S. Patent 856, 196 (June 1907). [5] Scherphus, J.: Jaap spuzzlepage: Transposer/Trxxy/Staned (2009), avalable from c 2012 Informaton Processng Socety of Japan 717

6 [6] Sloane, N.J.A.: The On-Lne Encyclopeda of Integer Sequences (2011), avalable from [7] Slocum, J. and Botermans, J.: Puzzles Old and New: How to Make and Solve Them, Unversty of Washngton Press (1988). [8] Stegmann, R.: Rob s Puzzle Page: Foldng Paper/Card (2010), avalable from stegmann/ allother.htm#fold-paper. [9] Uehara, R.: Stamp foldngs wth a gven mountan-valley assgnment, ORIGAMI 5, CRC Press (2011). [10] Umesato, T., Satoh, T., Uehara, R. and Ito, H.: Complexty of the stamp foldng problem, 5th Annual Internatonal Conference on Combnatoral Optmzaton and Applcatons (COCOA 2011), Lecture Notes n Computer Scence, Vol.6831, pp , Sprnger-Verlag (2011). [11] Wenreb, C.R.: Puzzle. Internatonal Patent WO 99/15248 (and GB , EP , US ) (Apr. 1999), avalable from Tetsuo Asano receved hs B.E., M.E., and Ph.D. degrees from Osaka Unversty, Japan, n 1972, 1974, and 1977, respectvely. In 1977 he joned Osaka Electro- Communcaton Unversty as a lecturer and moved to JAIST (Japan Advanced Insttute of Scence and technology) n He s now a professor n School of Informaton Scence. Hs research nterest ncludes algorthms and data structures, especally n computatonal geometry, combnatoral optmzaton, computer graphcs, computer vson usng geometrc nformaton, and VLSI layout desgn. He s fellow of ACM (2001), IPSJ (2004), and IEICE of Japan (2010). Currently he s the drector of Center for Graduate Educaton Intatve at JAIST. Erk D. Demane receved a B.Sc. degree from Dalhouse Unversty n 1995, and M.Math. and Ph.D. degrees from Unversty of Waterloo n 1996 and 2001, respectvely. Snce 2001, he has been a professor n computer scence at the Massachusetts Insttute of Technology. Hs research nterests range throughout algorthms, from data structures for mprovng web searches to the geometry of understandng how protens fold to the computatonal dffculty of playng games. In 2003, he receved a MacArthur Fellowshp as a computatonal geometer tacklng and solvng dffcult problems related to foldng and bendng movng readly between the theoretcal and the playful, wth a keen eye to revealng the former n the latter. He cowrote a book about the theory of foldng, together wth Joseph O Rourke (Geometrc Foldng Algorthms, 2007), and a book about the computatonal complexty of games, together wth Robert Hearn (Games, Puzzles, and Computaton, 2009). Martn L. Demane s an artst and mathematcan. He started the frst prvate hot glass studo n Canada and has been called the father of Canadan glass. Snce 2005, he has been the Angelka and Barton Weller Artst-n-Resdence at the Massachusetts Insttute of Technology. Both Martn and Erk work together n paper, glass, and other materal. They use ther exploraton n sculpture to help vsualze and understand unsolved problems n mathematcs, and ther scentfc abltes to nspre new art forms. Ther artstc work ncludes curved orgam sculptures n the permanent collectons of the Museum of Modern Art (MoMA) n New York, and the Renwck Gallery n the Smthsonan. Ther scentfc work ncludes over 60 publshed jont papers, ncludng several about combnng mathematcs and art. Ryuhe Uehara receved hs B.E., M.E., and Ph.D. degrees from the Unversty of Electro-Communcatons, Japan, n 1989, 1991, and 1998, respectvely. He was a researcher n CANON Inc. durng In 1993, he joned Tokyo Woman s Chrstan Unversty as an assstant professor. He was a lecturer durng , and an assocate professor durng at Komazawa Unversty. He moved to Japan Advanced Insttute of Scence and Technology (JAIST) n 2004, and he s now a professor n School of Informaton Scence. Hs research nterests nclude computatonal complexty, algorthms and data structures, and graph algorthms. Especally, he s engrossed n computatonal orgam, games and puzzles from the vewponts of theoretcal computer scence. He s a member of EATCS, ACM, and IEEE. c 2012 Informaton Processng Socety of Japan 718