A Greedy Algorithm for the Social Golfer and the Oberwolfach Problem
AA Greedy Algorithm for the Social Golfer and theOberwolfach Problem
Daniel Schmand , Marc Schr¨oder , and Laura Vargas Koch Institute of Computer Science, Goethe University, Frankfurt, Germany, [email protected] School of Business and Economics, RWTH Aachen University, Aachen, Germany, [email protected] School of Business and Economics, RWTH Aachen University, Aachen, Germany, [email protected]
July 20, 2020
Abstract
Inspired by the increasing popularity of Swiss-system tournamentsin sports, we study the problem of predetermining the total number ofrounds that can at least be guaranteed in a Swiss-system tournament.For tournaments with n participants, we prove that we can alwaysguarantee n/ k ≥ (cid:98) n/ ( k ( k − (cid:99) rounds and that this bound is tight. This gives rise toa simple polynomial time 1 /k -approximation algorithm for k -somes ingolf tournaments. Up to our knowledge, this is the first analysis of anapproximation algorithm for the social golfer problem.In the Oberwolfach problem, a match corresponds to a group of k players positioned in a cycle and the constraint is that no participantshould meet the same neighbor more than once. We show that thesimple greedy approach guarantees at least (cid:98) ( n + 4) / (cid:99) rounds for theOberwolfach problem. Assuming that El-Zahar’s conjecture is true, weimprove the bound to be essentially tight. Swiss-system tournaments received a highly increasing consideration in thelast years and are implemented in various professional and amateur tourna-1 a r X i v : . [ c s . D M ] J u l ents in, e.g., badminton, bridge, chess, e-sports and card games. A Swiss-system tournament is a non-eliminating tournament format that features apredetermined number of rounds of competition. Assuming an even numberof participants, each player plays exactly one other player in each round andtwo players play each other at most once in a tournament. The number ofrounds is predetermined and publicly announced. The actual planning of around usually depends on the results of the previous rounds to generate asattractive matches as possible and highly depend on the considered sport.Tournament designers usually agree on the fact that one should have atleast log( n ) rounds in a tournament with n participants to ensure that therecannot be multiple players without a loss in the final rankings. Appleton(1995) even mentions playing log( n ) + 2 rounds, so that a player may loseonce and still win the tournament.In this work, we examine a bound on the number of rounds that can be guaranteed by tournament designers. Since the schedule of a round dependson the results of previous rounds, it might happen that at some point inthe tournament, there is no next round that fulfills the constraint that twoplayers play each other at most once in a tournament. This raises thequestion of how many rounds a tournament organizer can announce beforethe tournament starts while being sure that this number of rounds canalways be scheduled. We provide bounds that are independent of the resultsof the matches and the detailed rules for the setup of the rounds.We model the feasible matches of a tournament with n participants as anundirected graph with n vertices. A match that is feasible in the tournamentcorresponds to an edge in the graph. Assuming an even number of partic-ipants, one round of the tournament corresponds to a perfect matching inthe graph. After playing one round we can delete the corresponding perfectmatching from the set of edges to keep track of the matches that are stillfeasible. We can guarantee the existence of a next round in a Swiss-systemtournament if there is a perfect matching in the graph. The largest numberof rounds that a tournament planner can guarantee is equal to the largestnumber of perfect matchings that a greedy algorithm is guaranteed to deletefrom the complete graph. Greedily deleting perfect matchings models thefact that rounds cannot be preplanned or adjusted later in time.Interestingly, our results imply that infeasibility issues can arise in somestate-of-the-art rules for table-tennis tournaments in Germany. There is apredefined amateur tournament series with more than 1000 tournamentsper year that guarantees the 9 to 16 participants 6 rounds in a Swiss-systemtournament (Hessischer Tischtennis-Verband 2020). We can show that atournament with 10 participants might become infeasible after round 5,even when respecting all relevant rules, for more details see Kuntz (2020).Corollary 4 shows that tournament designers could extend the lower boundfrom 5 to 6, by choosing the fifth round carefully.We generalize our results to the famous social golfer problem in which2ot 2, but k ≥ k that contains every vertexof the graph exactly once. In graph theory, a feasible round of the socialgolfer problem is called a clique factor. We address the question of howmany rounds can be guaranteed if clique factors, where each clique has asize of k , are greedily deleted from the complete graph, i.e., without anypreplanning.A closely related problem is the Oberwolfach problem. In the Ober-wolfach problem, we seek to find seating assignments for multiple diners atround tables in such a way that two participants sit next to each other ex-actly once. Half-jokingly, we use the fact that seatings at Oberwolfach semi-nars are assigned greedily, and study the greedy algorithm for this problem.Instead of deleting clique factors, the algorithm now iteratively deletes aset of vertex-disjoint cycles that contains every vertex of the graph exactlyonce. Such a set is called a cycle factor. We restrict attention to the specialcase of the Oberwolfach problem in which all cycles have the same length k .We analyze how many rounds can be guaranteed if cycle factors, in whicheach cycle has length k , are greedily deleted from the complete graph. Our Contribution
Motivated by applications in sports, the social golfer problem, and the Ober-wolfach problem, we study the greedy algorithm that iteratively deletes aclique, respectively cycle, factor in which all cliques/cycles have a fixed size k , from the complete graph. We prove the following main results for com-plete graphs with n vertices for n divisible by k . • We can always delete (cid:98) n/ ( k ( k − (cid:99) clique factors in which all cliqueshave a fixed size k from the complete graph. In other words, thegreedy procedure guarantees a schedule of (cid:98) n/ ( k ( k − (cid:99) rounds for thesocial golfer problem. In particular, this implies that in a Swiss-systemtournament with two players per match, we can always schedule n/ • The bound of (cid:98) n/ ( k ( k − (cid:99) is tight, in the sense that we can choosethe first (cid:98) n/ ( k ( k − (cid:99) rounds in such a way that no additional feasible3ound exists. If a well-known conjecture by Chen et al. (1994) in graphtheory is true (the conjecture is proven to hold for k ≤ (cid:98) n/ ( k ( k − (cid:99) rounds. Given that we can checkin polynomial time whether our graph satisfies the condition of thisexample after (cid:98) n/ ( k ( k − (cid:99) rounds, we can in such case pick a differentclique factor in the last round such that an additional round can bescheduled for n ≥
4. In particular, by cleverly selecting the last tworounds we can always schedule (cid:98) n/ ( k ( k − (cid:99) + 1 rounds under theassumption that the conjecture is true. • We can always delete (cid:98) ( n + 4) / (cid:99) cycle factors in which all cycles havea fixed size k , where k ≥
3, from the complete graph. This impliesthat our greedy approach guarantees to schedule (cid:98) ( n + 4) / (cid:99) roundsfor the Oberwolfach problem. • If El-Zahar’s conjecture (El-Zahar 1984) is true, we can increase thenumber of cycle factors that can be deleted to (cid:98) ( n + 2) / (cid:99) for k evenand (cid:98) ( n + 2) / − n/ k (cid:99) for k odd. Additionally, we show that thisbound is essentially tight by distinguishing three different cases. Inthe first two cases, the bound is tight, while in the last case, a gap ofone round remains. We follow standard notation in graph theory and for two graphs G and H we define an H -factor of G as a union of vertex-disjoint copies of H thatcontains every vertex of the graph G . For some graph H and n ∈ N ≥ ,a tournament with r rounds is defined as a tuple T = ( H , . . . , H r ) of H -factors of the complete graph K n such that each edge of K n is in at mostone H -factor. The feasibility graph of a tournament T = ( H , . . . , H r ) isa graph G = K n \ (cid:83) i ≤ r H i that contains all edges that are in none of the H -factors.Motivated by Swiss-system tournaments and the importance of greedyalgorithms in real-life optimization problems, we study the greedy algo-rithm that starts with an empty tournament and iteratively extends thecurrent tournament by an arbitrary H -factor in every round until no H -factor remains in the feasibility graph. We refer to Algorithm 1 for a formaldescription. 4 lgorithm 1: Greedy tournament scheduling
Input : number of vertices n and a graph H Output: tournament TG ← K n i ← while there is an H -factor H i in G do delete H i from Gi ← i + 1 endreturn T = ( H , . . . , H i − ) Greedy Social Golfer Problem
In the greedy social golfer problem weconsider tournaments with H = K k , for k ≥
2. The greedy social golferproblem asks for the minimum number of rounds of a tournament computedby Algorithm 1, as a function of n and k . The solution of the greedy socialgolfer problem is a guarantee on the number of K k -factors that can beiteratively deleted from the complete graph without any preplanning. Forsports tournaments this corresponds to n players being assigned to roundswith matches of size k such that each player is in exactly one match perround and each pair of players meets at most once in the tournament. Greedy Oberwolfach Problem
In the greedy Oberwolfach problem weconsider tournaments with H = C k , for k ≥
3. The greedy Oberwolfachproblem asks for the minimum number of rounds calculated by Algorithm 1,given n and k . This corresponds to a guarantee on the number of C k -factorsthat can always be iteratively deleted from the complete graph without anypreplanning.Observe that for k = 3, both problems are equivalent. To avoid trivialcases, we assume throughout the paper that n is divisible by k . This is anecessary condition for the existence of a single round. Usually, in real-lifesports tournaments, additional dummy players are added to the tournamentif n is not divisible by k . The influence of dummy players on the tournamentplanning strongly depends on the sport. There are sports, like e.g., golf orkarting where matches can still be played with less than k players, or otherswhere the match needs to be cancelled if one player is missing, for examplebeach volleyball or tennis doubles. Thus, the definition of a best possibleround if n is not divisible by k depends on the application. We exclude theanalysis of this situation from this work to ensure a broad applicability ofour results and focus on the case n mod k = 0.5 .1 Related Literature For a match size of k ≥ social golfer problem with n ≥ H = K k , i.e., atournament such that feasibility graph is empty, exists. For H = K , sucha complete tournament coincides with a round-robin tournament. Round-robin tournaments are known to exist for every even number of players.Algorithms to calculate such schedules are known for more than a centurydue to Schurig (1886).For H = K k and k ≥
2, complete tournaments are also known as resolv-able balanced incomplete block designs (resolvable-BIBDs). To be precise, a resolvable-BIBD with parameters ( n, k,
1) is a collection of subsets (blocks)of a finite set V with | V | = n elements with the following properties:1. Every pair of distinct elements u, v from V is contained in exactly oneblock.2. Every block contains exactly k elements.3. The blocks can be partitioned into rounds R , R , . . . , R r such thateach element of V is contained in exactly one block of each round.Notice that a round in a resolvable-BIBD corresponds to an H -factor inthe social golfer problem. Similar to the original social golfer problem, aresolvable-BIBD consists of ( n − / ( k −
1) rounds. For the existence of aresolvable-BIBD the conditions n mod k = 0 and n − k − k = 3, Ray-Chaudhuri and Wilson (1971) proved thatthese two conditions are also sufficient. Later, Hanani et al. (1972) provedthe same result for k = 4. In general, these two conditions are not sufficient(one of the smallest exceptions being n = 45 and k = 5), but Ray-Chaudhuriand Wilson (1973) showed that they are asymptotically sufficient, i.e., forevery k there exist a constant c ( k ) such that the two conditions are sufficientfor every n larger than c ( k ). These results immediately carry over to theexistence of a complete tournament with n players and H = K k .In greedy tournament scheduling, Algorithm 1 greedily adds one roundafter another to the tournament, and thus extends a given tournament stepby step. The study of the existence of another feasible round in a giventournament with H = K k is related to the existence of an equitable graph-coloring. Given some graph G = ( V, E ), an (cid:96) -coloring is a function f : V → { , ..., (cid:96) } , such that f ( u ) (cid:54) = f ( v ) for all edges ( u, v ) ∈ E . An equitable (cid:96) -coloring is an (cid:96) -coloring, where the number of vertices in any two colorclasses differs by at most one, i.e., |{ v | f ( v ) = i }| ∈ { (cid:4) n(cid:96) (cid:5) , (cid:6) n(cid:96) (cid:7) } for everycolor i ∈ { , ..., (cid:96) } .To relate equitable colorings of graphs to the study of the extendability oftournaments, we consider the complement graph ¯ G of the feasibility graph G = ( V, E ), as defined by ¯ G = K n \ E . Notice that a color class in an6quitable coloring of the vertices of ¯ G is equivalent to a clique in G . Inan equitable coloring of ¯ G with nk colors, each color class has the same size,which is k . Thus, finding an equitable nk -coloring in ¯ G is equivalent to findinga K k -factor in G and thus an extension of the tournament. Questions on theexistence of an equitable coloring dependent on the vertex degrees in a graphhave already been considered by Erd˝os (1964), who posed a conjecture on theexistence of equitable colorings in low degree graphs, that has been provenby Hajnal and Szemer´edi (1970). Their proof was simplified by Kiersteadet al. (2010), who also gave a polynomial time algorithm to find an equitablecoloring. In general graphs, the existence of clique factors with clique sizeequal to 3 (Garey and Johnson 1979, Sec. 3.1.2) and at least 3 (Hell andKirkpatrick 1978, 1983, 1984) is known to be NP-hard.The maximization variant of the social golfer problem for n players and H = K k asks for a schedule which lasts as many rounds as possible. It ismainly studied in the constraint programming community using heuristicapproaches (Dot´u and Van Hentenryck 2005, Triska and Musliu 2012a,b,Liu et al. 2019). Our results give lower bounds for the maximization variantusing a very simple greedy algorithm.For n players and table sizes k , ..., k (cid:96) with n = k + ... + k (cid:96) , the (classical) Oberwolfach problem can be stated as follows. Defining ˜ H = (cid:83) i ≤ (cid:96) C k i theproblem asks for the existence of a tournament of n players, with H = ˜ H which has ( n − / k = k = ... = k (cid:96) , Alspach et al. (1989) showed existence for all odd k and all n odd with n mod k = 0. For k even, Alspach et al. (1989) and Hoffmanand Schellenberg (1991) analyzed a slight modification of the Oberwolfachproblem and showed that there is a tournament, such that the correspondingfeasibility graph G is not empty, but equal to a perfect matching for all even n with n mod k = 0.Liu (2003) studied a variant of the Oberwolfach problem in bipartitegraphs and gave conditions under which the existence of a complete tourna-ment is guaranteed.The question of extendability of a given tournament with H = C k cor-responds to the covering of the feasibility graph with cycles of length k .Covering graphs with cycles is already studied since Petersen (1891). Theproblem of finding a set of cycles of arbitrary lengths covering a graph (if oneexists) is polynomially solvable (Edmonds and Johnson 1970). However, ifcertain cycle lengths are forbidden, the problem is NP-complete (Hell et al.1988). 7igure 1: Consider a tournament with 6 participants and H = K . The leftfigure corresponds to three rounds, where each color denotes the matches ofone round. The right figure depicts the feasibility graph after these threerounds. Consider the example of a tournament with n = 6 and H = K depicted inFigure 1. The coloring of the edges in the graph on the left represents threerounds H , H , H . The first round H is depicted by the set of red edges.Each edge corresponds to a match. In the second round, all blue edges areplayed. The third round H consists of all green edges. After these threerounds, the feasibility graph G of the tournament is depicted on the rightside of the figure. We cannot feasibly schedule a next round as there is noperfect matching in G . Equivalently, we can observe that the tournamentwith 3 rounds cannot be extended, since there is no equitable 3-coloring in¯ G , that is depicted on the left of Figure 1.On the other hand there is a tournament with n = 6 and H = K that consists of 5 rounds. The corresponding graph is depicted in Figure 2.Since this is a complete tournament, the example is a resolvable-BIBD withparameters (6 , , V of the BIBD and the colors in the figure correspond to the rounds in theBIBD. Note that these examples show that there is a complete tournamentwith n = 6 and H = K , where 5 rounds are played while the greedyalgorithm can get stuck after 3 rounds. In the remainder of the paper, weaim for tight bounds on the number of rounds that can be guaranteed byusing the greedy algorithm. The paper is structured as follows. We start with the analysis of Swiss-system tournaments to demonstrate our main ideas. To be more precise,Section 3 considers the setting of greedy tournament scheduling with H = K . Section 4 then generalizes the main results for the greedy social golfer8igure 2: A complete tournament with 6 players and 5 rounds, in whicheach color represents the matches of a round.problem. Lastly, in Section 5, we obtain lower and upper bounds on thenumber of rounds for the greedy Oberwolfach problem. Most sports tournaments consist of matches between two competing players.We therefore first consider the special case of a tournament with H = K .In this setting, the greedy social golfer problem boils down to iterativelydeleting perfect matchings from the complete graph.First, we use Dirac’s theorem to show that we can always greedily deleteat least n perfect matchings from the complete graph. Recall that we assume n to be even to guarantee the existence of a single perfect matching. Proposition 1.
For each even n ∈ N and H = K , Algorithm 1 outputs atournament with at least n rounds.Proof. Algorithm 1 starts with an empty tournament and extends it byone round in every iteration. To show that Algorithm 1 runs for at least n iterations we consider the feasibility graph of the corresponding tournament.Recall that the degree of each vertex in a complete graph with n vertices is n −
1. In each round, the algorithm deletes a perfect matching and thus thedegree of a vertex is decreased by 1. After at most n − n . By Dirac’s theorem (Dirac 1952), a Hamiltoniancycle exists. The existence of a Hamiltonian cycle implies the existence ofa perfect matching by taking every second edge of the Hamiltonian cycle.So after at most n − n rounds.9 roposition 2. There are infinitely many n ∈ N for which there exists atournament that cannot be extended after n rounds.Proof. Choose n such that n is odd. We describe the chosen tournament byperfect matchings in the feasibility graph G . Given a complete graph with n vertices, we partition the vertices into a set A with | A | = n and V \ A with | V \ A | = n . We denote the players in A by 1 , . . . , n and the playersin V \ A by n + 1 , . . . , n .In each round r = 1 , . . . , n , player i + n is scheduled in a match withplayer i + r − n ) for all i = 1 , . . . , n . After deleting these n perfectmatchings, the feasibility graph G consists of two disjoint complete graphsof size n , as every player in A has played against every player in V \ A .Given that n is odd, no perfect matching exists and hence the tournamentcannot be extended.A natural follow-up question is to characterize those feasibility graphsthat can be extended after n rounds. Proposition 3 answers this questionand we essentially show that the provided example is the only graph struc-ture that cannot be extended after n rounds. Proposition 3.
Let T be a tournament of n rounds with feasibility graph G and its complement ¯ G . Then T cannot be extended if and only if ¯ G = K n , n and n is odd. Before we prove the proposition we present a result by Chen et al. (1994),which the proof makes use of.
Chen-Lih-Wu theorem (Chen et al. 1994).
Let G be a connectedgraph with maximum degree ∆( G ) ≥ n . If G is different from K m and K m +1 , m +1 for all m ≥
1, then G is equitable ∆( G )-colorable. Proof of Proposition 3.
If the complement of the feasibility graph ¯ G = K n , n and n odd, we are exactly in the situation of the proof of Proposition 2. Toshow equivalence, assume that either ¯ G (cid:54) = K n , n or n even. By using theChen-Lih-Wu Theorem, we show that in this case ¯ G is equitable n -colorable.After n rounds, we have ∆( ¯ G ) = n . We observe that ¯ G = K n if andonly if n = 2 and in this case ¯ G = K , , a contradiction. Thus all conditionsof the Chen-Lih-Wu theorem are fulfilled, and ¯ G is equitable n -colorable.An equitable n -coloring in ¯ G corresponds to a perfect matching in G andhence implies that the tournament is extendable.If a tournament is not extendable after n rounds, we can always choosean alternative perfect matching in round n so that at least n + 1 rounds arefeasible. Corollary 4.
By selecting the perfect matching in round n carefully, thetournament can always be extended by at least one further round. roof. After n − T , the degree of every vertex in G is at least n . By Dirac’s theorem (Dirac 1952), there is a Hamiltoniancycle in G . This implies that two edge-disjoint perfect matchings exist: onethat takes every even edge of the Hamiltonian cycle and one that takes everyodd edge of the Hamiltonian cycle. If we first extend T by taking every evenedge of the Hamiltonian cycle and then extend T by taking every odd edgeof the Hamiltonian cycle, we have a tournament of n + 1 rounds. We generalize the previous results to k ≥
3. This means we analyze tourna-ments with n participants and H = K k . Dependent on n and k , we providetight bounds on the number of rounds that can be scheduled greedily, i.e.,by using Algorithm 1. Remember that we assume that n is divisible by k . Theorem 5.
For each n ∈ N and H = K k , Algorithm 1 outputs a tourna-ment with at least (cid:98) nk ( k − (cid:99) rounds. Before we continue with the proof, we first state a result from graphtheory. In our proof, we will use the Hajnal-Szemeredi theorem and adaptit such that it applies to our setting.
Hajnal-Szemeredi Theorem (Hajnal and Szemer´edi 1970).
Let G be a graph with n ∈ N vertices and maximum vertex degree ∆( G ) ≤ (cid:96) − G is equitable (cid:96) -colorable. Proof of Theorem 5.
Assume for sake of contradiction that there are n ∈ N and k ∈ N such that the greedy algorithm for H = K k terminates with atournament T with r ≤ (cid:98) nk ( k − (cid:99)− G corresponding to T . Recall that the degree of a vertex in a complete graphwith n vertices is n −
1. For each K k -factor ( H , . . . , H r ), every vertex loses k − G has degree n − − r ( k − ≥ n − − ( (cid:98) nk ( k − (cid:99) − k −
1) = n − − nk + k − . We observe that each vertex in the complement graph ¯ G has at most degree nk − k + 1. Using the Hajnal-Szemeredi theorem with (cid:96) = nk , we obtain theexistence of an nk -coloring where all color classes have size k . Since there areno edges between vertices of the same color class in ¯ G , they form a cliquein G . Thus, there exists a K k -factor in G , which is a contradiction to theassumption that Algorithm 1 terminated. This implies that r > (cid:98) nk ( k − (cid:99)− (cid:98) nk ( k − (cid:99) .11 emark 6. Kierstead et al. (2010) showed that finding a clique factor canbe done in polynomial time if the minimum vertex degree is at least n ( k − k .Algorithm 1 can thus be considered as a k -approximation algorithm to themaximum number of rounds. Up to our knowledge, this is the first analysisof an approximation algorithm for the social golfer problem. Our second main result on greedy tournament scheduling with H = K k shows that the bound of Theorem 5 is tight. Theorem 7.
There are infinitely many n ∈ N for which there exists atournament that cannot be extended after (cid:98) nk ( k − (cid:99) rounds.Proof. We construct a tournament with n = j ( k ( k − j to be chosen later. We will define necessary properties of j throughout theproof and argue in the end that there are infinitely many possible integralchoices for j . The tournament we will construct has (cid:98) nk ( k − (cid:99) rounds and wewill show that it cannot be extended. Note that (cid:98) nk ( k − (cid:99) = nk ( k − .The proof is based on a step-by-step modification of the feasibility graph G . We will start with the complete graph K n and describe how to delete nk ( k − K k -factors such that the resulting graph does not contain a K k -factor.This is equivalent to constructing a tournament with (cid:98) nk ( k − (cid:99) rounds thatcannot be extended.Given a complete graph with n vertices, we partition the vertices V in two sets, a set A with (cid:96) = nk + 1 vertices and a set V \ A with n − (cid:96) vertices. We choose all nk ( k − K k -factors in such a way, that no edge { a, b } with a ∈ A and b / ∈ A is deleted, i.e., each K k is either entirely in A orentirely in B . Since a vertex in A has nk neighbours in A and k − K k -factor, all edges within A are deleted after deleting nk ( k − K k -factors.We now first argue that after deleting these nk ( k − K k -factors, no K k -factor exists. Assume that there exists another K k -factor. In this case,each vertex in A forms a clique with k − V \ A . However, since( k − · ( nk + 1) > ( k − nk − | V \ A | there are not enough vertices in V \ A ,a contradiction to the existence of the K k -factor.It remains to show that there are nk ( k − K k -factors that do not containan edge { a, b } with a ∈ A and b / ∈ A . We start by showing that nk ( k − K k -factors can be found within A . Ray-Chaudhuri and Wilson (1973) showedthat given k (cid:48) ≥ c ( k (cid:48) ) such that if n (cid:48) ≥ c ( k (cid:48) ) and n (cid:48) mod k (cid:48) ( k (cid:48) −
1) = k (cid:48) , then a resolvable-BIBD with parameters ( n (cid:48) , k (cid:48) , k (cid:48) = k and n (cid:48) = (cid:96) with j = λ · k + 1 for some λ ∈ N largeenough, we establish (cid:96) ≥ c ( k ), where c ( k ) is defined by Ray-Chaudhuri andWilson (1973), and we get | A | = (cid:96) = nk + 1 = j ( k −
1) + 1 = ( λk + 1)( k −
1) + 1 = k + λk ( k − . (cid:96), k,
1) exists, and there is a com-plete tournament for (cid:96) players with H = K k , i.e., we can find nk ( k − K k -factors in A .It remains to show that we also find nk ( k − K k -factors in V \ A . We definea tournament that we call shifting tournament as follows. We arbitrarilywrite the names of the players in V \ A into a table of size k × ( n − (cid:96) ) /k .Each column of the table corresponds to a K k and the table to a K k -factorin V \ A . By rearranging the players we get a sequence of tables, eachcorresponding to a K k -factor. To construct the next table from a precedingone, for each row i , all players move i − n − (cid:96) ) /k ).We claim that this procedure results in nk ( k − K k -factors that do notshare an edge. First, notice that the step difference between any two playersin two rows i (cid:54) = i (cid:48) is at most k −
1, where we have equality for rows 1 and k . However, we observe that ( n − (cid:96) ) /k is not divisible by ( k −
1) since n/k is divisible by k − (cid:96)/k is not divisible by k − (cid:96)/k ( k −
1) = 1 / ( k −
1) + λ and this expression is not integral. Thus, a playerin row 1 can only meet a player in row k again after at least 2 n − (cid:96)k ( k − rounds.Since 2 n − (cid:96)k ( k − ≥ nk ( k − if n ≥ kk − , the condition is satisfied for n suffi-ciently large.Similarly, we have to check that two players in two rows with a relativedistance of at most k − n − (cid:96)k ( k − ≥ nk ( k − if n ≥ k − k , the condition is also satisfied for n sufficiently large.Observe that there are infinitely many n and (cid:96) such that (cid:96) = nk + 1, n isdivisible by k ( k −
1) and (cid:96) mod k ( k −
1) = k and thus the result follows forsufficiently large n .We turn our attention to the problem of characterizing tournamentsthat are not extendable after (cid:98) nk ( k − (cid:99) rounds. Assuming the Equitable ∆-Coloring Conjecture (E∆CC) is true, we give an exact characterization ofthe feasibility graphs of tournaments that cannot be extended after (cid:98) nk ( k − (cid:99) rounds. The existence of an instance not fulfilling these conditions wouldimmediately disprove the E∆CC. Furthermore, this characterization allowsus to guarantee (cid:98) nk ( k − (cid:99) + 1 rounds in every tournament when the last tworounds are chosen carefully. Equitable ∆ -Coloring Conjecture (Chen et al. 1994). Let G be aconnected graph with maximum degree ∆( G ) ≤ (cid:96) . Then G is not equitable (cid:96) -colorable if and only if one of the following three cases occurs:1. G = K (cid:96) +1 .2. (cid:96) = 2 and G is an odd cycle.3. (cid:96) odd and G = K (cid:96),(cid:96) . 13he conjecture was first stated by Chen et al. (1994) in 1994 and is provenfor | V | = k · (cid:96) and k = 2 , ,
4. See the Chen-Lih-Wu theorem for k = 2 andKierstead and Kostochka (2015) for k = 3 ,
4. Both results make use ofBrooks’ theorem (Brooks 1941). For k >
4, the conjecture is still open.
Proposition 8.
If E ∆ CC is true, a tournament with (cid:98) nk ( k − (cid:99) rounds cannotbe extended if and only if K nk +1 is not a subgraph of the complement graph ¯ G . Before we start the proof, we state the following claim, which we willneed in the proof.
Claim 9.
Given an equitable m -coloring for every connected component G i of some graph G , there is an equitable m -coloring for G .Proof. In every connected component G i there are l i = | G i | mod m largecolor classes , i.e., color classes with more than | G i | m vertices. Since l i = | G i | mod m , it follows that (cid:88) l i mod m = (cid:88) | G i | mod m = 0 . This means we can rename the color classes such that exactly (cid:80) l i m large colorclasses are part of each color class of G , which is an equitable m -coloring. Proof of Proposition 8.
After (cid:98) nk ( k − (cid:99) rounds, the degree of all vertices v inthe complement of the feasibility graph is ∆( v ) = ( k − (cid:98) nk ( k − (cid:99) ≤ nk . Weapply E∆CC with (cid:96) = nk to each connected component ¯ G i of ¯ G . If there is anequitable nk -coloring for every connected component, then by Claim 9 thereis an equitable nk -coloring of ¯ G and thus a K k -factor in G . To complete theproof, we will show that every connected component ¯ G i is either equitable nk -colorable or ¯ G i = K nk +1 , which by the proof of Theorem 7 implies thatthe tournament is not extendable.Assume that there is a connected component ¯ G i that is not equitable nk -colorable. By E∆CC , ¯ G i = K nk +1 , or nk = 2 and ¯ G i is an odd cycle, or nk is odd and ¯ G i = K nk , nk . If ¯ G i = K nk +1 , we are done.So first, assume that nk = 2 and ¯ G i is an odd cycle. An odd cycle canonly be formed from a union of complete graphs K k if there is only oneround with k = 3. Thus, we have that n = 6. In this case, nk = 2 and ¯ G i being an odd cycle is equivalent to ¯ G i = K = K nk +1 .Second, assume that nk is odd and ¯ G i = K nk . Given that k >
2, we willderive a contradiction. Let V and V denote the vertices of the partition of K nk , nk , i.e., K nk , nk only contains edges ( v , v ) with v ∈ V and v ∈ V . Let( v , v ) be an edge in ¯ G . Then v , v and k − k . Let v be such a vertex. Since v is either in V or in V , we have a contradiction with the fact that there are only edges of theform ( v , v ) with v ∈ V and v ∈ V .14 emark 10. The condition in Proposition 8 can be checked in polynomialtime. In order to check whether a connected component corresponds to thecomplete graph with nk + 1 vertices, we consider every vertex and check if allits neighbors are connected. If this is not the case, this vertex is not part ofthe complete graph. This procedure clearly runs in polynomial time. Note that any tournament with H = K k and (cid:98) nk ( k − (cid:99) rounds which doesnot satisfy the condition in Proposition 8 would disprove the E∆CC . Proposition 11.
If E ∆ CC is true, then by choosing round (cid:98) nk ( k − (cid:99) care-fully, there always exists a tournament with (cid:98) nk ( k − (cid:99) + 1 rounds,Proof. A tournament with (cid:98) nk ( k − (cid:99) rounds is either extendable or by Propo-sition 8, a connected component of the complement of the feasibility graphto equal to K nk +1 . In the former case, we are done. So assume the lat-ter case. Denote the vertices in the connected component by A . We firstshorten the tournament by eliminating the last round and then extend it bytwo other rounds.Clearly, the last round of the original tournament corresponds to a K k -factor in the feasibility graph of the shortened tournament. By the assumedstructure of the feasibility graph, all cliques K k are either completely within A or completely within V \ A . All edges between A and V \ A still exist.Select one clique K k in A and one clique K k in V \ A , and exchange onepair of vertices ( v , v ) with v ∈ A and v ∈ V \ A . Extend the shortenedtournament with this particular K k -factor. After extending the tournament,no connected component in the complement of the feasibility graph corre-sponds to K nk +1 . By Proposition 8, the tournament can be extended to have (cid:98) nk ( k − (cid:99) + 1 rounds. In this section we consider tournaments with H = C k for k ≥
3. Dependenton the number of participants n and k we derive bounds on the number ofrounds that can be scheduled greedily in such a tournament.Before we continue with the theorem, we first state a classical result fromAigner and Brandt and a conjecture by El-Zahar. Aigner-Brandt Theorem (Aigner and Brandt 1993).
Let G be agraph with minimum degree δ ( G ) ≥ n − . Then G contains any graph H with at most n vertices and maximum degree ∆( H ) = 2 as a subgraph. El-Zahar’s Conjecture (El-Zahar 1984).
Let G be a graph with n = k + ... + k (cid:96) . If δ ( G ) ≥ (cid:100) k (cid:101) + ... + (cid:100) k (cid:96) (cid:101) , then G contains (cid:96) vertex disjointcycles of lengths k , ..., k (cid:96) . 15 heorem 12. For each n ∈ N and H = C k , Algorithm 1 outputs a tour-nament with at least (cid:98) n +46 (cid:99) rounds. If El-Zahar’s conjecture is true, thisimproves to (cid:98) n +24 (cid:99) for k even and (cid:98) n +24 − n k (cid:99) for k odd.Proof. Recall that Algorithm 1 starts with the empty tournament and thecorresponding feasibility graph is the complete graph, where the degree ofevery vertex is n −
1. In each iteration of the algorithm, a C k -factor is deletedfrom the feasibility graph and thus every vertex loses 2 edges. We observethat as long as the constructed tournament has at most (cid:98) n − (cid:99) rounds, thedegree of every vertex in the feasibility graph is at least n − −(cid:98) n − (cid:99) ≥ n − .Since a C k -factor with n vertices has degree 2, by the Aigner-Brandt theorem G contains a C k -factor. It follows that the algorithms runs for anotheriteration. In total, the number of rounds of the tournament is at least1 + (cid:98) n − (cid:99) = (cid:98) n +46 (cid:99) .Assuming El-Zahar’s conjecture, we derive an improved bound. First,we assume that k is even. Since we look for C k -factors, we choose k = k = · · · = k (cid:96) = k in El-Zahar’s conjecture and derive the following statement. For k even, there is a C k -factor of G if δ ( G ) ≥ n and n = k · (cid:96) for k, (cid:96) ∈ N ≥ . As long as Algorithm 1 runs for at most (cid:98) n − (cid:99) iterations, the degreeof every vertex in the feasibility graph is at least n − − · (cid:98) n − (cid:99) ≥ n − − n − = n . Hence by El-Zahar’s conjecture, a C k -factor exists and thusanother iteration is possible. This implies that Algorithm 1 is guaranteedto construct a tournament with (cid:98) n − (cid:99) + 1 rounds.Second, we assume that k is odd. Using that in C k -factors all cycleshave the same length, El-Zahar’s conjecture for k odd implies the following: For k odd, there is a C k -factor of G if δ ( G ) ≥ n + n k and n = k · (cid:96) for k, (cid:96) ∈ N ≥ . As long as Algorithm 1 runs for at most (cid:98) n − − n k (cid:99) iterations, the degreeof every vertex in the feasibility graph is at least n − − n − + n k = n + n k . Hence by El-Zahar’s conjecture, a C k -factor exists and the constructedtournament can be extended by one more round. This implies that thealgorithm outputs a tournament with at least (cid:98) n − − n k (cid:99) + 1 rounds.In the rest of the section, we show that the bound presented in Theorem12 is essentially tight. Through a case distinction, we provide matchingexamples that show the tightness of the bounds provided by El-Zahar’sconjecture for two of three cases. For k even but not divisible by 4, anadditive gap of one round remains. All other cases are tight. Note thatthis implies that any improvement of the lower bound via an example byjust one round (or by two for k even but not divisible by 4) would disproveEl-Zahar’s conjecture. Theorem 13.
There are infinitely many n ∈ N for which there exists atournament with H = C k that is not extendable after1. (cid:98) n +24 − n k (cid:99) rounds if k is odd , . (cid:98) n +24 (cid:99) rounds if k mod 4 = 0 ,3. (cid:98) n +24 (cid:99) + 1 rounds if k mod 4 = 2 .Proof of 1. Assume that k is odd. Let n = 2 k (cid:80) ij =0 k j for some integer i ∈ N . We construct a tournament with n participants and H = C k . Todo so, we start with the empty tournament and partition the set of verticesof the feasibility graph into two disjoint sets A and B . The sets are chosensuch that A ∪ B = V , and | A | = n − n k + 1 = ( k − (cid:80) ij =0 k j + 1 = k i +1 , | B | = n + n k − | A | ≤ | B | , since n k ≥
1. Weconstruct a tournament such that in the feasibility graph all edges betweenvertices in A are deleted. To do so, we use a result of Alspach et al. (1989),who showed that there is a solution for the Oberwolfach problem for all odd k with n mod k = 0 and n odd.Observe that | A | mod k = 0, thus | B | mod k = 0. Furthermore, | A | − n is even this also applies to | B | −
1. By using theequivalence of the Oberwolfach problem to complete tournaments, thereexists a complete tournament within A and within B . We combine thesecomplete tournaments to a tournament for the whole graph with min {| A | − , | B | − } / | A |− = n − n k rounds. Since | A | is odd, the number ofrounds is integral.Considering the feasibility graph of this tournament, there are no edgesbetween vertices in A . Thus, every cycle of length k can cover at most k − vertices of A . We conclude that there is no C k -factor for the feasibilitygraph, since nk · k − = n − n k , so we cannot cover all vertices of A . Thus,we constructed a tournament with n − n k = (cid:98) n +24 − n k (cid:99) rounds that cannotbe extended. Proof of 2.
Assume that k is divisible by 4. Let n = i · k for some oddinteger i ∈ N . We construct a tournament with n participants by dividingthe vertices of the feasibility graph into two disjoint sets A and B such that | A | = | B | = n = i · k . Liu (2003) showed that there exist n/ C k -factors in a complete bipartite graph with n/ n/ C k -factor. Since n/ n/ (cid:98) n +24 (cid:99) rounds such that in the feasibility graphthere are only edges within A and within B left. Since i is odd, | A | = i · k is not divisible by k . Thus, it is not possible to schedule another round bychoosing only cycles within sets A and B . Proof of 3.
Assume that k is even, but not divisible by 4. Let n = i · k forsome odd integer i ∈ N ≥ . We construct a tournament with n participantsthat is not extendable after n +24 +1 rounds in two phases. First, we partitionthe vertices into two disjoint sets A and B , each of size n , and we constructa base tournament with n − rounds such that in the feasibility graph only17 a a a a b b b b b a a a a a b b b b b Figure 3: Construction of the base tournament. We transform two cycles oflength 5 into one cycle of length 10.edges between sets A and B are deleted. Second, we extend the tournamentby two additional carefully chosen rounds. After the base tournament, thefeasibility graph consists of two complete graphs A and B connected by aperfect matching between all vertices from A and all vertices from B . Weuse the additional two rounds to delete all of the matching-edges except forone. Using this, we show that the tournament cannot be extended.In order to construct the base tournament, we first use a result of Alspachet al. (1989). It states that there always exists a solution for the Oberwolfachproblem with n (cid:48) participants and cycle length k (cid:48) if n (cid:48) and k (cid:48) are odd and n (cid:48) mod k (cid:48) = 0.We choose n (cid:48) = n/ k (cid:48) = k/
2, apply the result by Alspach et al.(1989) and use an idea by Archdeacon et al. (2004). Fix the solution forthe Oberwolfach problem with n/ k , andapply this solution to A and B separately. Consider one round of thetournament and denote the C k -factor in A by ( a j , a j , . . . , a k + j ) for j = 0 , k , k, . . . , n − k . By symmetry, the C k -factor in B can be denoted by( b j , b j , . . . , b k + j ) for j = 0 , k , k, . . . , n − k . We design a C k -factor in thefeasibility graph of the original tournament. For each j ∈ { , k , k, . . . , n − k } , we design a cycle ( a j , b j , a j , . . . , a k + j , b j , a j , b j , . . . , b k + j )of length k in G . These edges are not used in any other round due to theconstruction and we used the fact that k is odd. We refer to Figure 3 foran example of one cycle for k = 10. Since each vertex is in one cycle in eachround, the construction yields a feasible round of a tournament. Applyingthis procedure to all rounds yields the base tournament with n − rounds.For each edge e = { a ¯ j , a j } with j (cid:54) = ¯ j which is deleted in the feasibilitygraph of the tournament within A , we delete the edges { a ¯ j , b j } and { a j , b ¯ j } in the feasibility graph. After the base tournament, all edges between A and B except for the edges ( a , b ) , ( a , b ) , . . . , ( a n , b n ) are deleted in thefeasibility graph.In the rest of the proof, we extend the base tournament by two addi-tional rounds. These two rounds are designed in such a way that after the18 a a a a b b b b b Figure 4: An example of one cycle in the construction that is used for theextension of the base tournament.rounds there is exactly one edge connecting a vertex from A with one from B . To extend the base tournament by one round construct the cycles ofthe C k -factor in the following way. For j = 0 , k , k, . . . , n − k , we constructcycles ( a j , b j , b j , a j , ..., b k − j b k − j , b k + j , a k + j , a k − j ), see Fig-ure 4. Since all edges within A and B are part of the feasibility graph aswell as all edges ( a j (cid:48) , b j (cid:48) ) for j (cid:48) ∈ { , ..., n } this is a feasible construction ofa C k -factor and thus an extension of the base tournament.After the extension of the base tournament by one round the feasibilitygraph has the following structure. The degree of all vertices equals n − A and B are (cid:26) ( a k − j , b k − j ) | j ∈ (cid:26) , k , k, . . . , n − k (cid:27)(cid:27) . We will construct one more round such that after this round, there is onlyone of the matching edges remaining in the feasibility graph.In order to do so, we will construct the C k -factor with cycles ( C , . . . , C nk )by a greedy procedure as follows. Cycles C , . . . , C n k − will all contain twomatching edges and the other cycles none. In order to simplify notation weset A M = (cid:26) a k − j | j ∈ (cid:26) , k , k, . . . , n − k (cid:27)(cid:27) , and A − M = A \ A M . We have | A − M | = n − nk . We define B M and B − M analogously. For some cycle C z , z ≤ n k − , we greedily pick two of thematching edges. Let ( a l , b l ) and ( a r , b r ) be these two matching edges. Tocomplete the cycle, we show that we can always construct a path from a l to a r by picking vertices from A − M and from b (cid:96) to b r by vertices from B − M .Assuming that we have already constructed cycles C , . . . , C z − , there arestill n − nk − ( z − k − A − M . Even after choosing some vertices for cycle z the number of unused vertices in A − M is at least n − nk − z ( k − ≥ n − nk − z k ≥ n − nk − n k k n − nk ≥ n . Let N ( v ) denote the neighborhood of vertex v . The greedy procedure theconstructs a path from a (cid:96) to a r works as follows. We set vertex a l active.For each active vertex v , we pick one of the vertices a ∈ N ( v ) ∩ A − M , delete a from A − M and set a active. We repeat this until we have chosen k − N ( v ) ∩ A − M ∩ N ( a r ) in order to ensurethat the path ends at a r . Since | A − M | ≥ n , we observe | N ( v ) ∩ A − M ∩ N ( a r ) | ≥ n − − , so there is always a suitable vertex as n ≥ k ≥
54. The construction forthe path from b (cid:96) to b r is analogous.For cycles C n k + , . . . , C nk , there are still n + k leftover vertices within A and within B . The degree of all of these vertices within the set of remainingvertices is at least n + k −
3. This is large enough to apply the Aigner-Brandttheorem as i ≥ k ≥
6. In this way, we construct a C k -factor in thefeasibility graph. This means we can extend the tournament by one moreround. In total we constructed a tournament of n +24 + 1 rounds, which isobviously equal to (cid:98) n +24 (cid:99) + 1.To see that this tournament cannot be extended further, consider thefeasibility graph. Most of the edges within A and B are still present, whilebetween A and B there is only one edge left. This means a C k -factor canonly consist of cycles that are entirely in A or in B . Since | A | = | B | and thenumber of cycles nk = i is odd, there is no C k -factor in the feasibility graphand thus the constructed tournament is not extendable. In this work, we studied the social golfer problem and the Oberwolfachproblem from an optimization perspective. We presented bounds on thenumber of rounds that can be guaranteed by a greedy algorithm. For thesocial golfer problem the provided bounds are tight. Assuming El-Zahar’sconjecture (El-Zahar 1984) holds, a gap of one remains for the Oberwolfachproblem. Up to our knowledge, this gives the first performance guaranteefor the optimization variant of both problems. Since a clique-factor canbe found in polynomial time, the greedy algorithm is a 1 /k -approximationalgorithm for the social golfer problem.While the greedy algorithm for the social golfer problem runs in polyno-mial time, this is still open for the Oberwolfach problem. In each round, our20pproach needs to calculate a C k -factor in the feasibility graph. Althoughit is proven that a C k -factor exists in a graph with high degree, finding onein polynomial time remains open.Furthermore, given some tournament it would be interesting to analyzethe complexity of deciding whether the tournament can be extended by anadditional round. Proving NP -hardness seems particularly complicated sinceone cannot use any regular graph for the reduction proof, but only graphsthat are feasibility graphs of a tournament.Finally, our considerations lead to the question whether El-Zahar’s con-jecture can be proven for the case where all cycles have the same length. Acknowledgement
This research started after supervising the master’s thesis of David Kuntz.We thank David for valuable discussions.
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