New Formulation for Coloring Circle Graphs and its Application to Capacitated Stowage Stack Minimization
NNew Formulation for Coloring Circle Graphsand its Applicationto Capacitated Stowage Stack Minimization
Masato Tanaka Tomomi MatsuiFebruary 2, 2021
Abstract
A circle graph is a graph in which the adjacency of vertices can be rep-resented as the intersection of chords of a circle. The problem of calculatingthe chromatic number is known to be NP-complete, even on circle graphs.In this paper, we propose a new integer linear programming formulation fora coloring problem on circle graphs. We also show that the linear relaxationproblem of our formulation finds the fractional chromatic number of a givencircle graph. As a byproduct, our formulation gives a polynomial-sized linearprogramming formulation for calculating the fractional chromatic numberof a circle graph.We also extend our result to a formulation for a capacitated stowagestack minimization problem.
This paper addresses problems of coloring circle graphs. A circle graphis a graph in which the adjacency of vertices can be represented as theintersection of chords of a circle. It is well known that circle graphs andoverlap graphs are of the same class (e.g., see [19, 20]).In [14], Even and Itai studied the problem of realizing a given permuta-tion through networks of queues in parallel and through a network of stacksin parallel. The problem was translated into a coloring problem of a cir-cle graph (see also [20]). There are practical applications involving stacksorting including assigning incoming trains [9, 10] or trams [5] to tracks of1 a r X i v : . [ c s . D M ] F e b switching yard or depot; parking buses in parking lots [17]; and stowageplanning for container ships [4, 33]. K¨onig and L¨ubbecke [27] considered analgorithmic view towards stack sorting. Stack and queue layouts of graphsrelated to colorations of circle graphs are discussed in [12].Even in the case of circle graphs, the problems of finding the chromaticnumber [18] and clique covering number [25] are NP-complete. Approxi-mation algorithms for coloring circle graphs are proposed in [7, 31]. In asurvey [13], Dur´an, Grippo, and Safe summarized structural results relatedto circle graphs and presented some open problems. Both the maximumclique problem and maximum independent set problem have polynomialtime algorithms when restricted to circle graphs [2, 8, 19, 30, 32].In this paper, we propose an integer linear programming formulation forcoloring problems on circle graphs. We also show that the linear relaxationproblem of our formulation finds the fractional chromatic number of a givencircle graph. For a general graph, the problem of finding the fractionalchromatic number is NP-complete [21]. Our proposal gives a polynomial-sized formulation for fractional coloring problems on circle graphs.The reminder of this paper is organized as follows. The next sectionpresents some notations and definitions. In Section 3, we propose a new for-mulation for coloring circle graphs. We discuss a relation between the linearrelaxation of our formulation and fractional chromatic number in Section 4.Section 5 reports our computational experiments. In Section 6, we brieflydiscuss an extension of our formulation to a capacitated stowage stack min-imization problem with zero rehandle constraint. Finally, Section 7 makessome closing remarks. Let G = ( V, E ) be an undirected graph with a set of vertices V and set of arcs E . A coloring of a graph is an assignment of a color to each vertex such thatall adjacent vertices are of a different color. The smallest number of colorsneeded to color a graph G is called its chromatic number , denoted by χ ( G ).The coloring problem has long been studied and is known to be NP-completefor general graphs [24]. An independent set is a subset of vertices in a graphsuch that no two are adjacent. The fractional chromatic number χ f ( G ) isthe smallest positive number k ∈ R + for which there exists a probabilitydistribution over the independent sets of G satisfying the following; givenan independent set S drawn from the distribution, Pr[ v ∈ S ] ≥ /k ( ∀ v ∈ V ). Although its computation is NP-complete [21], the fractional chromatic2umber of a general graph can be obtained using linear programming (seeSubsection 4.2).A clique is a subset of vertices such that its induced subgraph is complete.The clique number ω ( G ) of a given graph G is the number of vertices in amaximum clique in G . It is known that ω ( G ) ≤ ϑ ( G ) ≤ χ f ( G ) ≤ χ ( G ),where ϑ ( G ) denotes the Lova´sz number [29, 26] of a given graph G . Apentagon graph (5-cycle) C , which is an example of a circle graph, satisfies( ω ( C ) , ϑ ( C ) , χ f ( C ) , χ ( C )) = (2 , √ , . , circle graph is a graph in which the adjacency of vertices can be rep-resented as the intersection of chords of a circle. The circle and chordscorresponding to a given circle graph G are called a circle diagram of G .Hereinafter, we assume that terminal points of chords in a circle diagramare mutually distinct. Figure 1 (a) and (b) show an example of a circlegraph and its corresponding circle diagram, respectively.Fig. 1: An example of circle graph and related diagram.It is well known that circle graphs and overlap graphs are of the sameclass (e.g., see [19, 20]). A graph is an overlap graph if its vertices areintervals on a line such that two vertices are adjacent if and only if thecorresponding intervals partially overlap (that is, they have non-empty in-tersection), but neither contains the other. It is easy to construct a set of3ntervals representing a given circle graph (overlap graph) from a correspond-ing circle diagram by a simple transformation: cutting the circumference ofthe circle at some point p that is not an endpoint of a chord and unfold-ing it at that point (e.g., see Section 11.3 of [20]). Hereinafter, we assumethat an input of a given circle graph G = ( V, E ) is a corresponding intervalrepresentation I ( G ) = { I ( j ) ⊆ R | j ∈ V } , where I : j (cid:55)→ [ l j , r j ]. We alsoassume that all the terminal points of intervals in I ( G ) are mutually dis-tinct. Here, we note that an interval representation of a given circle graph isnot unique. Figure 1 (c) shows an interval representation of the circle graphin Figure 1 (a).Given an interval representation I ( G ) of a circle graph G = ( V, E ), weintroduce a partial order (cid:22) defined on the vertex set V . For any pair ofvertices i, j ∈ V , we define i (cid:22) j if and only if either i = j or r i ≤ l j holds, where I ( i ) = [ l i , r i ] and I ( j ) = [ l j , r j ]. Obviously, ( V, (cid:22) ) is a partiallyordered set. Although every chain of ( V, (cid:22) ) is an independent set of G , theconverse implication does not hold. Figure 2 shows a partially ordered setcorresponding to the interval representation in Figure 1 (c).Fig. 2: Partially ordered set corresponding to the interval representation inFigure 1 (c). There is an arrow from i to j if and only if i (cid:22) j . Given a circle graph G = ( V, E ) and corresponding interval representation I ( G ), we introduce a directed graph Γ as follows. The vertex-set of Γ isdefined by V ∪ { } , where 0 is an artificial vertex called a root . The arc-setof Γ, denoted by A , is defined by A = { (0 , i ) | i ∈ V } ∪ { ( i, j ) | I ( i ) (cid:41) I ( j ) } . T ⊆ A is called an arborescence if and only if | T | = | V | and each vertex i ∈ V has a unique incoming-arc in T . When a givenarborescence T has an arc ( i, j ), we say that j is a child of i and i is a(unique) parent of j with respect to T . For any arborescence T and a vertex i ∈ V ∪ { } , Ch( T, i ) denotes the set of children of i with respect to T .In the following, we associate each coloring with an arborescence on Γ.Let φ : V → { , , . . . , c } be a c -coloring of G . For each vertex j ∈ V , wedefine a parent of j with respect to φ , denoted by Prt( φ, j ), as follows: if V (cid:48) = { i ∈ V | φ ( i ) = φ ( j ) , I ( i ) (cid:41) I ( j ) } is empty, then we define Prt( φ, j ) =0 (root); else, Prt( φ, j ) denotes a vertex in V corresponding to a unique(inclusion-wise) minimum interval in V (cid:48) . Given a coloring φ of G , T ( φ )denotes an arborescence { (Prt( φ, j ) , j ) ∈ A | j ∈ V } . Figure 3 (a) shows a3-coloring and corresponding arborescence in Γ.Fig. 3: An example of a 3-coloring and its corresponding arborescence. Lemma 3.1.
Let T be an arborescence of Γ . Then, there exists a c -coloring φ of a given circle graph G satisfying T = T ( φ ) if and only if C1: for each i ∈ V , Ch(
T, i ) is a chain of ( V, (cid:22) ) or the empty set and the size of every antichain of ( V, (cid:22) ) contained in Ch( T, is less thanor equal to c . Proof. It is obvious that the size of a minimum chain cover (partition) of aposet is greater than or equal to the size of a maximum antichain. Thus, thedefinition of T ( φ ) implies that if there exists a c -coloring φ of G satisfying T = T ( φ ), then T satisfies C1 and C2.We show the converse implication. In the following, we construct a c -coloring from an arborescence T satisfying C1 and C2. Because T satisfiesC2, Dilworth’s theorem [11] implies that there exists a set of (at most) c chains of ( V, (cid:22) ) partitioning Ch( T, V, (cid:22) ) is anindependent set of a given circle graph, we obtain a c -coloring of a sub-graphof G induced by Ch( T,
0) by assigning a color to each chain. For each vertex i ∈ Ch( T, i to all the descendants of i with respect to T . Denote the map (coloring) obtained above by φ . We only need to showthat if φ ( j ) = φ ( j (cid:48) ) and j (cid:54) = j (cid:48) , then vertices j and j (cid:48) are non-adjacent on agiven circle graph G . When I ( j ) ⊆ I ( j (cid:48) ) or I ( j (cid:48) ) ⊆ I ( j ), the non-adjacencyis obvious. Otherwise, let r (cid:48) be a unique lowest common ancestor of j and j (cid:48) with respect to T . We denote a child of r (cid:48) that is an ancestor of j (or j (cid:48) ) by i (or i (cid:48) ), respectively. If r (cid:48) (cid:54) = 0, then C1 directly implies that I ( i ) ∩ I ( i (cid:48) ) = ∅ .When r (cid:48) = 0, φ ( i ) = φ ( j ) = φ ( j (cid:48) ) = φ ( i (cid:48) ) implies that i and i (cid:48) are containedin a mutual chain in Ch( T,
0) and thus I ( i ) ∩ I ( i (cid:48) ) = ∅ . From the above, I ( j ) ⊆ I ( i ) , I ( j (cid:48) ) ⊆ I ( i (cid:48) ) , and I ( i ) ∩ I ( i (cid:48) ) = ∅ hold. As a consequence, weobtain the non-adjacency of j and j (cid:48) , because I ( j ) ∩ I ( j (cid:48) ) = ∅ . (cid:3) Let P ⊆ R be a set of (positions of) terminal points of intervals in I ( G ).Recall that terminal points of intervals in I ( G ) are mutually distinct andthus | P | = 2 | V | . Let M = ( m pi ) be a 0-1 matrix whose entries are indexedby P × V , satisfying m pi = (cid:26) p ∈ I ( i )) , . (Here, we note that M is a clique matrix (Section 3.4 of [20]) of an inter-val graph corresponding to the set of intervals I ( G ).) Obviously, M is anantichains-versus-vertices incidence matrix of poset ( V, (cid:22) ). In addition, itis easy to see that all the maximal antichains are included. The definitionof M directly implies the following lemma, which characterizes the size of amaximum antichain. Lemma 3.2.
For any vertex subset (cid:101) V ⊆ V , the size of a maximum antichainof ( V, (cid:22) ) contained in (cid:101) V is equal to the maximum components of vector M (cid:101) x ,where (cid:101) x is the (fixed) characteristic vector of (cid:101) V . i ∈ V ∪ { } , we define a vertex subset V [ i ] = { j ∈ V | ( i, j ) ∈ A } .We define V • = { i ∈ V | V [ i ] (cid:54) = ∅} . Here, we note that V [0] = V and 0 (cid:54)∈ V • hold. For each arc ( i, j ) ∈ A , we introduce a 0-1 variable x ij . The vectorof all 0-1 variables is denoted by x ∈ { , } A . For any vertex i ∈ V • ∪ { } , x [ i ] denotes a subvector of x indexed by arcs emanating from i , and M [ i ] denotes a submatrix of M consisting of column vectors of M indexed by V [ i ] . Then, we have the following. Lemma 3.3.
Given a vector x ∈ { , } A and positive integer c , the con-straints M x [0] ≤ c ,M [ i ] x [ i ] ≤ ( ∀ i ∈ V • ) , (cid:88) i :( i,j ) ∈ A x ij = 1 ( ∀ j ∈ V ) are satisfied if and only if T = { ( i, j ) ∈ A | x ij = 1 } is an arborescencesatisfying conditions C1 and C2. Proof. Let T be an arborescence satisfying conditions C1 and C2. We seta vector x ∈ { , } A as the characteristic vector of T . Then, it is obviousthat x satisfies the above constraints.Now, assume that x ∈ { , } A and a positive integer c satisfy the aboveconstraints. We define T = { ( i, j ) ∈ A | x ij = 1 } . Then, constraints (cid:80) i :( i,j ) ∈ A x ij = 1 ( ∀ j ∈ V ) directly imply that T is an arborescence ofΓ. From Lemma 3.2, the inequality M x [0] ≤ c implies that T satisfiescondition C2. Similarly, Lemma 3.2 and M [ i ] x [ i ] ≤ ( ∀ i ∈ V • ) imply thatfor any i ∈ V • , the size of a maximum antichain in Ch( T, i ) is less thanor equal to 1. From Dilworth’s theorem, Ch(
T, i ) becomes a chain (or theempty set). For any i ∈ V \ V • , Ch( T, i ) = ∅ . Thus, T satisfies conditionC1. (cid:3) The above lemma directly implies the following formulation for a circlegraph coloring problem: 7G : min . c s . t . M x [0] ≤ c ,M [ i ] x [ i ] ≤ ( ∀ i ∈ V • ) , (cid:88) i :( i,j ) ∈ A x ij = 1 ( ∀ j ∈ V ) ,x ij ∈ { , } ( ∀ ( i, j ) ∈ A ) ,c ∈ Z + . Lemma 3.3 directly implies the following.
Theorem 3.4.
A pair ( (cid:98) x , (cid:98) c ) ∈ { , } A × Z + is optimal to CG if and only if (cid:98) c = χ ( G ) and there exists a (cid:98) c -coloring φ satisfying T ( φ ) = { ( i, j ) ∈ A | (cid:98) x ij =1 } . Proof. Lemma 3.1 implies that a pair ( x , c ) ∈ { , } A × Z + is feasible toCG, if and only if, T = { ( i, j ) ∈ A | x ij = 1 } and c satisfies conditions C1and C2. Thus, the optimal value of CG is equal to χ ( G ). We can constructa χ ( G )-coloring φ from an optimal solution of CG by applying a techniquedescribed in the proof of Lemma 3.1. The inverse implication is clear. (cid:3) In this section, we show that the linear relaxation problem of our formulation(CG) finds the fractional chromatic number of a given circle graph.
In this subsection, we discuss a maximum weight independent set problemdefined on a given circle graph G = ( V, E ) with a given vertex weight func-tion w : V → R (incidentally, negative vertex weights are permitted). Foran artificial vertex 0, we define w (0) = 0. We propose a linear program-ming formulation of the problem based on a dynamic programming tech-nique [8, 15, 19]. Our linear programming formulation plays an importantrole in the next subsection.A maximum weight independent set problem finds an independent set S of G that maximizes the weight (cid:80) i ∈ S w ( i ). Throughout this section, weassign a linear ordering on the vertex set by setting V = { , , . . . , n } suchthat if I ( i ) ⊇ I ( j ), then i ≤ j . 8or any vertex i ∈ V • ∪ { } , G [ i ] denotes the subgraph of G induced byvertex subset V [ i ] (note that i (cid:54)∈ V [ i ] ). Every maximum weight independentset S ⊆ V satisfies the following: ∀ i ∈ S ∩ V • , S ∩ V [ i ] is a maximum weightindependent set in G [ i ] . Here, we introduce vertex weights defined by (cid:96) i = w ( i ) + max (cid:40) (cid:88) i ∈ V (cid:48) w ( j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V (cid:48) is an independentset of G [ i ] (cid:41) ( ∀ i ∈ V • ) ,w ( i ) ( ∀ i ∈ V \ V • ) . For any vertex subset S ⊆ V , max S denotes a set of vertices correspondingto (inclusion-wise) maximal intervals in { I ( i ) | i ∈ S } . It is clear that if S is an independent set of G , then max S is a chain of ( V, (cid:22) ). This propertyimplies that the weight of a maximum weight independent set with respectto ( w ( i ) | i ∈ V ) is equal to the weight of a maximum weight chain (of( V, (cid:22) )) with respect to ( (cid:96) i | i ∈ V ). By applying the above idea recursively,it is easy to see that ( (cid:96) i | i ∈ V ∪ { } ) satisfies the following formula (cid:96) i = w ( i ) + max (cid:88) j ∈ V (cid:48) (cid:96) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V (cid:48) ⊆ V [ i ] ,V (cid:48) is a chain of ( V, (cid:22) ) ( ∀ i ∈ V • ∪ { } ) ,w ( i ) ( ∀ i ∈ V \ V • ) , (2)where we define w (0) = 0, and (cid:96) denotes the weight of a maximum weightindependent set of G with respect to ( w ( i ) | i ∈ V ). Because the vertexset V = { , , . . . , n } satisfies “if I ( i ) ⊇ I ( j ), then i ≤ j ,” the above for-mula calculates ( (cid:96) n , (cid:96) n − , . . . , (cid:96) , (cid:96) ) sequentially. Figure 4 shows an examplesolution of recursive formula (2).The following lemma summarizes the above discussion. Lemma 4.1.
The solution of recursive formula (2) satisfies that (cid:96) is equalto the weight of a maximum weight independent set with respect to ( w ( i ) | i ∈ V ) . Next, we describe a linear programming formulation of a subproblemappearing in (2).
Lemma 4.2.
Let (cid:96) = ( (cid:96) j | j ∈ V ) represent given vertex weights. For any i ∈ V • ∪ { } , max (cid:88) j ∈ V (cid:48) (cid:96) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V (cid:48) ⊆ V [ i ] ,V (cid:48) is a chain of ( V, (cid:22) ) = max (cid:88) j ∈ V [ i ] (cid:96) j x ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M [ i ] x [ i ] ≤ , x [ i ] ≥ , (3) where x [ i ] is a vector of continuous variables indexed by V [ i ] . i ( (cid:96) ) denotes the linear programming problemon the right-hand-side of (3). First, we show that LC i ( (cid:96) ) has an optimal0-1 vector solution. We say that a matrix has the consecutive 1’s property(for columns) if and only if 1’s occur consecutively in each column. Clearly,the matrix M [ i ] has the consecutive 1’s property [16] (for columns) and istotally unimodular. The total unimodularity implies that a feasible region { x [ i ] | M [ i ] x [ i ] ≤ , x [ i ] ≥ } is a 0-1 polytope [23, 22]. Therefore, LC i ( (cid:96) ) hasan optimal 0-1 vector solution, denoted by (cid:98) x [ i ] . Lemma 3.2 and M [ i ] (cid:98) x [ i ] ≤ imply that the size of a maximum antichain contained in (cid:98) V = { j ∈ V [ i ] | (cid:98) x ij = 1 } is less than or equal to 1. From Dilworth’s theorem, (cid:98) V becomes achain (or the empty set). (cid:3) For any i ∈ V • ∪ { } , the dual of LC i ( (cid:96) ), denoted by DLC i ( (cid:96) ), isDLC i ( (cid:96) ): min . (cid:88) p ∈ P y ip s . t . (cid:88) p ∈ I ( j ) y ip ≥ (cid:96) j ( ∀ j ∈ V [ i ] ) ,y ip ≥ ∀ p ∈ P ) . We substitute DLC i ( (cid:96) ) for the maximum weight chain problem in (2) to10btain the following recursive formula: (cid:96) i = w ( i ) + min (cid:88) p ∈ P y ip (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) p ∈ I ( j ) y ip ≥ (cid:96) j ( ∀ j ∈ V [ i ] ) ,y ip ≥ ∀ p ∈ P ) ( ∀ i ∈ V • ∪ { } ) ,w ( i ) ( ∀ i ∈ V \ V • ) . (4)Lemmas 4.1 and 4.2 and the strong duality theorem directly imply the fol-lowing. Theorem 4.3.
The solution of recursive formula (4) satisfies that (cid:96) is equalto the weight of a maximum weight independent set with respect to vertexweights w = ( w ( i ) | i ∈ V ) . Considering all constraints appearing in (4), we construct the following linearprogramming problem:ISD( w ) : min . (cid:96) = (cid:88) p ∈ P y p (5a)s . t . (cid:96) i = w ( i ) + (cid:88) p ∈ P y ip ( ∀ i ∈ V • ) , (5b) (cid:96) i = w ( i ) ( ∀ i ∈ V \ V • ) , (5c) (cid:88) p ∈ I ( j ) y ip ≥ (cid:96) j ( ∀ ( i, j ) ∈ A ) , (5d) y ip ≥ ∀ ( i, p ) ∈ ( V • ∪ { } ) × P ) , (5e)where ( (cid:96) , (cid:96) , . . . , (cid:96) n ) and ( y ip | ( i, p ) ∈ ( V • ∪ { } ) × P ) are vectors of contin-uous variables. Theorem 4.4.
The optimal value of ISD ( w ) is equal to the weight of amaximum weight independent set with respect to vertex weights w = ( w ( i ) | i ∈ V ) . Proof. Let ( (cid:101) (cid:96) , (cid:101) y ) be a solution of (4) and ( (cid:96) ∗ , y ∗ ) be an optimal solution ofISD( w ). Because ( (cid:101) (cid:96) , (cid:101) y ) is feasible to ISD( w ), (cid:101) (cid:96) ≥ (cid:96) ∗ holds.For the reminder of this proof, we show the inequality (cid:101) (cid:96) ≤ (cid:96) ∗ . Here,we note that we assign a linear ordering on the vertex set by setting V = { , , . . . , n } such that if I ( i ) ⊇ I ( j ), then i ≤ j . We show that (cid:101) (cid:96) j ≤ (cid:96) ∗ j foreach j ∈ { n, n − , . . . , } by induction on j . Clearly, vertex n ∈ V \ V • , andthus (cid:96) ∗ n = w ( n ) = (cid:101) (cid:96) n . Assume that (cid:101) (cid:96) j ≤ (cid:96) ∗ j for each j ∈ { n, n − , . . . , i + 1 } .When i ∈ V \ V • , we obviously have (cid:96) ∗ i = w ( i ) = (cid:101) (cid:96) i . We consider the case11hat i ∈ V • ∪ { } . It is obvious that for any i ∈ V • ∪ { } , the subvector( (cid:101) y ip ) p ∈ P of (cid:101) y is optimal to problem DLC i ( (cid:101) (cid:96) ). The subvector ( y ∗ ip ) p ∈ P of y ∗ is feasible to problem DLC i ( (cid:96) ∗ ). The induction hypothesis implies thatthe feasible region of DLC i ( (cid:101) (cid:96) ) includes that of DLC i ( (cid:96) ∗ ). The subvector( y ∗ ip ) p ∈ P of y ∗ is feasible to DLC i ( (cid:101) (cid:96) ), and the corresponding objective valuesatisfies (cid:88) p ∈ P y ∗ ip ≥ (optimal value of DLC i ( (cid:101) (cid:96) )) = (cid:88) p ∈ P (cid:101) y ip . Thus, we obtain (cid:96) ∗ i = w ( i ) + (cid:88) p ∈ P y ∗ ip ≥ w ( i ) + (cid:88) p ∈ P (cid:101) y ip = (cid:101) (cid:96) i , where we let w (0) = 0 for simplicity.From the above discussion, we have shown that (cid:96) ∗ = (cid:101) (cid:96) . (cid:3) In this subsection, we discuss the fractional coloring problem. Given anundirected graph G = ( V, E ), F denotes the incidence matrix of independentsets of G . The rows of F are indexed by V , the columns of F are indexed byall the independent sets of G , and each column vector is the incidence vector(characteristic vector) of a corresponding independent set. The fractionalcoloring problem is defined bymin { (cid:62) q | F q = , q ≥ } , where the variable vector q is indexed by all the independent sets in G , and denotes the all-ones vector. The optimal value of the above problem is calledthe fractional chromatic number and is denoted by χ f ( G ). Generally, theabove linear programming problem has an exponential number of variables.The dual of the above problem ismax { w (cid:62) | w (cid:62) F ≤ (cid:62) } , which finds a vertex weight w : V → R maximizing the total sum w (cid:62) subject to the constraint that the weight of every independent set (withrespect to w ) is less than or equal to 1.Let us discuss the case that a given graph G is a circle graph. Givena vertex weight function w : V → R , the weight of every independent set12with respect to w ) is less than or equal to 1 if and only if the minimizationproblem ISD( w ) has a feasible solution whose objective value is less than orequal to 1. Then, the linear programming problemmax. w (cid:62) s.t. (cid:88) p ∈ P y p ≤ , (6)with the constraints of the ISD problem , gives a formulation for the fractional coloring problem on a given circlegraph, where ( w ( i ) | i ∈ V ), ( (cid:96) i | i ∈ V ), and ( y ip | ( i, p ) ∈ ( V • ∪ { } ) × P )are vectors of continuous variables. Here, we note that ( w ( i ) | i ∈ V ) isa given vector of vertex weights in ISD( w ) and is a variable vector in theabove problem.We eliminate variables ( w ( i ) | i ∈ V ) by applying equalities (5b) and (5c).Then, the objective function becomes w (cid:62) = (cid:88) i ∈ V • w ( i ) + (cid:88) i ∈ V \ V • w ( i )= (cid:88) i ∈ V • ( (cid:96) i − (cid:88) p ∈ P y ip ) + (cid:88) i ∈ V \ V • (cid:96) i = (cid:88) j ∈ V (cid:96) j − (cid:88) i ∈ V • (cid:88) p ∈ P y ip . The obtained problem (6) transforms intoFCP : max . (cid:88) j ∈ V (cid:96) j − (cid:88) i ∈ V • (cid:88) p ∈ P y ip s . t . (cid:88) p ∈ P y p ≤ , (cid:88) p ∈ I ( j ) y ip ≥ (cid:96) j ( ∀ ( i, j ) ∈ A ) ,y ip ≥ ∀ ( i, p ) ∈ ( V ∪ { } ) × P ) , where ( (cid:96) i | i ∈ V ) and ( y ip | ( i, p ) ∈ ( V • ∪ { } ) × P ) are vectors of continuousvariables.It is easy to check that FCP is the dual of the linear relaxation problemof CG obtained by substituting x ij ≥ c ≥ x ij ∈ { , } and c ∈ Z + ,respectively. Summarizing the above discussion, we obtain the followingtheorem. 13 heorem 4.5. Let LR be a linear relaxation problem of CG obtained bysubstituting x ij ≥ and c ≥ for x ij ∈ { , } and c ∈ Z + , respectively.Then, the optimal value of LR is equal to the fractional chromatic numberof a given circle graph. In our experiments, we compared the computational time required to findan optimal solution of our formulation CG, a classical coloring problemformulationCL : min . C (cid:88) c =1 y c , s . t . x ic ≤ y c ( ∀ i ∈ V, ∀ c ∈ { , , . . . , C } ) ,x ic + x i (cid:48) c ≤ ∀ c ∈ { , , . . . , C } , ∀{ i, i (cid:48) } ∈ E ) , C (cid:88) c =1 x ic ≥ ∀ i ∈ V ) ,x ic ∈ { , } ( ∀ i ∈ V, ∀ c ∈ { , , . . . , C } ) ,y c ∈ { , } ( ∀ c ∈ { , , . . . , C } ) , and an asymmetric representative formulation [6]AS : min . (cid:88) i ∈ V x ii s . t . x ij = x ji = 0 ( ∀{ i, j } ∈ E ) ,x ji = 0 ( ∀ i, ∀ j ∈ V, i < j ) ,x ij + x ik ≤ x ii ( ∀{ i, j, k } ⊆ V, { j, k } ∈ E ) , (cid:88) i ∈ V x ij = 1 ( ∀ j ∈ V ) ,x ij ≤ x ii ( ∀ i, ∀ j ∈ V ) ,x ij ∈ { , } ( ∀ ( i, j ) ∈ V ) , where the vertex set V = { , , . . . , n } satisfies deg(1) ≥ deg(2) ≥ · · · ≥ deg( n ) (deg( v ) denotes the degree of vertex v ∈ V ). We set the constant C in the classical formulation to the number of colors required in a coloringobtained by the First Fit heuristic (e.g., see [3]). All the experiments wereconducted on a PC running the Windows 10 Pro operating system with14n Intel(R) Core(TM) i7-7700 @3.60GHz processor and 32 GB RAM. Allinstances were solved using CPLEX 12.8.0.0 implemented in Python 3.6.5and NumPy 1.17.2.We generated instances of circle graphs as follows. First, we randomlyshuffled the numbers { , , . . . , | V |} using the “random.shuffle()” commandof the NumPy Python module. We repeatedly removed the first two num-bers x, y from the shuffled sequence and added a non-empty interval [ x, y ] or[ y, x ] to a set of intervals I . We constructed an overlap graph (circle graph)from the set of intervals I . For example, from a sequence (5 , , , , , I = { [3 , , [1 , , [2 , } . For each n ∈{ , , , . . . , } , we generated 100 circle graphs with n vertices and solvedthe coloring problems using CPLEX.The results are summarized in Table 1. The CL, AS, Ours columnsrepresent the classical formulation, asymmetric representative formulation,and our formulation (CG), respectively. The “ ω = χ ” and “ χ f = χ ”columns list the numbers of instances (out of 100 generated instances)satisfying ω ( G ) = χ ( G ) and χ f ( G ) = χ ( G ), respectively. The “max χ − χ f ”column lists the maximum values of χ ( G ) − χ f ( G ) over the 100 generatedinstances. We omit the computational results of some cases, denoted by“-,” because of time limitation. Table 1 shows that our formulation solvescoloring problems efficiently compared to other formulations.We also generated hard instances G ( m ) proposed by Kostochka (Sec-tion 6 in [28]) for each m ∈ { , , . . . , } . These results are summarized inTable 2. When we employed the classical formulation and/or asymmetricrepresentative formulation, the execution time exceeded 1800s even in thecase of G (4).When we employed our formulation (CG), CPLEX found optimal solu-tions for all instances reported in Tables 1 and 2 at the root node (withoutany branching process). Tables 1 and 2 show that all the generated instancessatisfy χ ( G ) − < χ f ( G ) ≤ χ ( G ).Ageev [1] constructed a triangle-free graph G A = ( V, E ) with chro-matic number equal to 5, where | V | = 220 and | E | = 1395. We cal-culated the fractional chromatic number of the graph and obtained that( ω ( G A ) , χ f ( G A ) , χ ( G A )) = (2 , . · · · , | V | | E | computation time [s] ω = χ χ f = χ χ − χ f In this section, we briefly discuss an extension of our formulation to a ca-pacitated stowage stack minimization problem. Let G = ( V, E ) and I ( G ) bea given circle graph and the corresponding interval representation, respec-tively. Throughout this section, H denotes a positive integer representing“capacity.” For any independent set S of G , the height of S is equal tothe size of a maximum antichain of ( V, (cid:22) ) contained in S . In this section,Ξ H denotes the set of independent sets (of G ) whose heights are less thanor equal to H . We introduce a 0-1 matrix F H indexed by V × Ξ H whosecolumns are the incidence vectors of corresponding independent sets in Ξ H .We consider the following 0-1 integer programming problem:P H : min { (cid:62) q | F H q = , q ∈ { , } Ξ H } , where the variable vector q is indexed by Ξ H . The above problem is essen-tially equivalent to a capacitated stowage stack minimization problem withzero rehandle constraint [33].We say that a c -coloring of G is H -admissible if each color class is anindependent set in Ξ H . It is obvious that each feasible solution of P H corre-sponds to an H -admissible coloring. In a similar manner to that in Section 3,16able. 2: Computational results for hard instances G ( m ). m | V | computation time [s] ω χ f χ Ours2 24 0.031 5 6.500 73 62 0.031 8 10.833 114 122 0.141 10 15.750 165 205 0.359 13 21.000 216 316 1.468 15 26.833 277 453 4.282 18 32.857 338 617 13.531 20 39.062 409 812 59.656 22 45.611 4610 1039 155.328 25 52.450 5311 1294 767.766 28 59.318 6012 1584 3878.625 31 66.500 6713 1904 9235.875 33 73.731 7414 2258 33087.641 35 81.143 8215 2647 119019.062 38 88.733 89we introduce a directed graph and associate an H -admissible coloring of G with a directed tree.First, we define a directed graph Γ H = ( V H , A H ) as follows. For eachvertex i ∈ V , we construct a set of H copies of i denoted by V H ( i ) = { ( i · , ( i · , . . . , ( i · H ) } . We introduce an artificial vertex (0 ·
0) and definea vertex set V H = (cid:83) i ∈ V V H ( i ) ∪ { (0 · } . The set of arcs A H is defined by A H = { ((0 · , ( i · | i ∈ V }∪{ (( i · h ) , ( j · h +1)) | I ( i ) (cid:41) I ( j ) , h ∈ { , , . . . , H − }} . From this definition, it is clear that Γ H = ( V H , A H ) is a directed acyclicgraph. Figure 5 shows Γ corresponding to the interval representation inFigure 1 (c).Given an H -admissible coloring φ (cid:48) , we define a subset of directed edgesof A H as follows. Let T ( φ (cid:48) ) be an arborescence of Γ, defined in Section 3.For each vertex i ∈ V , Hgt( φ (cid:48) , i ) denotes the length (number of edges) in aunique path in T ( φ (cid:48) ) from the root 0 to i . Obviously, we have Hgt( φ (cid:48) , i ) ≤ H ( ∀ i ∈ V ). We define a set of arcs T H ( φ (cid:48) ) of Γ H by T H ( φ (cid:48) ) = { ( i · h − , ( j · h ) | ( i, j ) ∈ T ( φ (cid:48) ) } where h = Hgt( φ (cid:48) , j ). Figure 5 shows an arc set T ( φ ) in Γ correspondingto the 3-admissible 3-coloring φ in Figure 3 (b).17ig. 5: Arc subset T ( φ ) in Γ corresponding to a 3-admissible 3-coloring φ in Figure 3 (b).For any vertex ( i · h ) ∈ V H , δ I ( i · h ) and δ O ( i · h ) denote a set of arcsin-coming to ( i · h ) in A H and a set of arcs emanating from ( i · h ) in A H ,respectively. Given an arc subset T (cid:48) ⊆ A H and vertex ( i · h ) ∈ V H , we definea set of vertices Ch( T (cid:48) , ( i · h )) = { j ∈ V | (( i · h ) , ( j · h + 1)) ∈ T (cid:48) } . Then,we have the following property. Lemma 6.1.
Let T (cid:48) be an arc subset of Γ H . Then, there exists an H -admissible c -coloring φ (cid:48) of a given circle graph G satisfying T (cid:48) = T H ( φ (cid:48) ) ifand only if D0: for any vertex i ∈ V , T (cid:48) includes a unique arc in (cid:83) Hh =1 δ I ( i · h ) , D1: for each ( i · h ) ∈ V H \ { (0 · } , Ch( T (cid:48) , ( i · h )) is a chain of ( V, (cid:22) ) or theempty set; if Ch( T (cid:48) , ( i · h )) is a (non-empty) chain, then T (cid:48) containsa unique in-coming arc to ( i · h ) , and the size of every antichain of ( V, (cid:22) ) contained in Ch( T (cid:48) , (0 · is lessthan or equal to c . Proof (outline). Given a set of arcs T (cid:48) ⊆ A H satisfying conditions D0, D1,and D2, we define a set of arcs T (cid:48)(cid:48) of Γ by T (cid:48)(cid:48) = { ( i, j ) ∈ A | ∃ h ∈ { , , , . . . , H − } , (( i · h ) , ( j · h + 1)) ∈ T (cid:48) } . Condition D0 implies that T (cid:48)(cid:48) is an arborescence of Γ. From conditionsD1 and D2, T (cid:48)(cid:48) satisfies conditions C1 and C2 in Lemma 6.1, and thusthere exists a c -coloring, denoted by φ (cid:48)(cid:48) , of G . The definition of Γ H andcondition D1 imply that Hgt( φ (cid:48)(cid:48) , i ) ≤ H ( ∀ i ∈ V ), which implies that φ (cid:48)(cid:48) is H -admissible.The converse implication is obvious. (cid:3) Now, we give an integer linear programming formulation. For each arc(( i · h ) , ( j · h + 1)) ∈ A H , we introduce a 0-1 variable x i · hj . The vector of all0-1 variables is denoted by x ∈ { , } A H . For any vertex ( i · h ) ∈ V H , x [ i · h ] denotes a subvector of x indexed by set of arcs δ O ( i · h ), or δ O ( i · h ) = ∅ .Lemma 6.1 implies a new formulation of P H as follows:CG H : min . c s . t . M x [0 · ≤ c ,M [ i ] x [ i · h ] ≤ (cid:88) ( j · h − ∈ δ I( i · h ) x j · h − i ( ∀ ( i · h ) ∈ V • × { , . . . , H − } ) , H (cid:88) h =1 (cid:88) ( i · h − ∈ δ I( j,h ) x i · h − j = 1 ( ∀ j ∈ V ) ,x i · hj ∈ { , } ( ∀ (( i · h ) , ( j · h + 1)) ∈ A H ) ,c ∈ Z + . It is not difficult to show the following.
Theorem 6.2.
The optimal value of the linear relaxation problem of CG H is equal to the optimal value of the linear relaxation problem of P H . Proof is omitted. 19
Conclusion
In this paper, we proposed a new formulation for coloring circle graphs. Ourformulation is based on an interval representation of a given circle graphand uses a hierarchical structure of a set of intervals corresponding to eachindependent set. By employing Dilworth’s theorem, we obtain a simplesystem of inequality constraints represented by a clique matrix of an intervalgraph defined by a given interval representation.An advantage of our formulation is that the corresponding linear relax-ation problem finds the fractional chromatic number of a given circle graph.Thus, our formulation also gives a polynomial-sized formulation for a frac-tional coloring problem on a circle graph.We confirmed by computational experiments that a commercial IP solvercan find a coloration quickly under our formulation. When we employed ourformulation, CPLEX found optimal solutions for all the instances randomlygenerated in our computational experiments at the root node (without anybranching process). The results of our computational experiments indicatethat the chromatic number χ ( G ) of a circle graph G is very close to its frac-tional chromatic number χ f ( G ). We conjecture that there exists a constant C satisfying χ ( G ) − C ≤ χ f ( G ) for any circle graph G .We extended our result to a formulation for a capacitated stowage stackminimization problem. Future work is required to evaluate the computa-tional performance of the proposed formulation. References [1] A. A. Ageev, “A triangle-free circle graph with chromatic number 5,”
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