SUPERVIVENCIA ACUMULADA A LOS 5 AÑOS

In a similar vein to Dehofet al.we also implement a fixed parameter tractable (FPT) based ap- proach.55_{Given a molecular graphG= (V,E)which is a tree, the electron distribution problem}

can be easily solved using dynamic programming, i.e. recursively splitting the problem into smaller sub problems and solving the sub problems. Of course, not all molecular graphs are trees, but their generally sparse nature means that they are ‘tree-like’.

Definition 9.4. LetGbe a graph,T a tree, and letV= (Vt)t∈T be a family of vertex setsVt⊆

V(G)indexed by the nodes t of T. The pair (T,V)is called a tree-decomposition ofG if it satisfies the following three conditions:

T1) V(G) =t∈TVt;

T2) for every edgee∈Gthere exists at∈Tsuch that both ends ofelie inVt;

T3) Vt1∩Vt3⊆Vt2 whenevert1,t2,t3∈Tsatisfyt2∈t1T t3.

Conditions T1 and T2 together say thatGis the union of the subgraphsG[Vt]; we call these sub-

graphs and the setsVt themselves thepartsof(T,V)and say that(T,V)is a tree-decomposition

ofGinto these parts. Condition T3 implies that the parts of(T,V)are organised roughly like a tree. Figure 9.1b shows a tree decomposition of the graph in figure 9.1a. Thewidthof(T,V)is the number

max|Vt| −1 :t∈T

,

and thetree-widthtw(G)ofGis the least width of any tree-decomposition ofG.

Following from this definition, the verticestof a tree-decomposition will be referred to as the nodesof the tree-decomposition. Any mention of vertices will refer to the underlying graphG. To simplify the algorithm we utilise nice tree-decompositions.

9. Bond Order and Formal Charge Assignment

Definition 9.5. A tree decomposition(T,V)ofGis calledniceif it satisfies the following con- ditions

N1) T is rooted at a leaf noderandVr=/0;

N2) For every leafl∈T,Vl=/0;

N3) Every nodet∈T has at most two children;

N4) Ift∈T has two children,pandqthenVt=Vp=Vqand is known as a join node;

N5) Ift∈T has one child,pthen one of the following conditions is true:

a) Vt ⊂Vp and |Vt|=|Vp| −1 and is known as a forget node with forgotten vertex

vtf :=Vp\Vt.

b) Vt⊃Vpand|Vt|=|Vp|+1 and is known as an introduce node with introduced vertex

vti:=Vt\Vp.

Forget and introduce nodes are defined in relation to the pathtl...tr. Figure 9.1c shows a nice

tree decomposition as produced from the tree decomposition in figure 9.1b.

The process for obtaining a nice tree decomposition of a molecular graph is as follows. A min- imum width tree decomposition of a molecular graph is obtained using the QuickBB method69 as implemented in the LibTW library.70_{In accordance with condition N2 for nice tree decompo-}

sitions, additional nodesiwithVi=/0 are added to each leaf of the tree decomposition, and a root

node is carefully chosen from amongst them to minimise the total number of partial solutions (condition N1). The tree decomposition is then converted from an undirected tree to a directed tree with edges pointing away from the root node.

Condition N3 is met as follows. As long asT has a nodet with more than two children, let t1,t2...tpbe the children oft. A new vertextis added to(T,V)withVt=Vt in the following

manner. The parent oftis set totand for each childtioft withi≥2tis made the parent ofti.

This shifts the over-degree node problem fromttot, but with degree reduced by one, thus the process is repeated until condition N3 is met.

Moving on to condition N4, as long asThas a nodetwith childrent1andt2withVt=Vt1=Vt2, additional nodest₁ andt₂ are added to(T,V)withVt=Vt_i fori∈ {1,2}. The parent oft1 and

t₂ is set totand the parent oft1andt2changed tot1 andt2 respectively. The two children are

treated independently such that ifVt=Vt1butVt=Vt2, only(t1,Vt1)is added to(T,V).

The final step is to makeT satisfy conditions N5a and N5b. For every nodet with childt1

ifVt Vt1 andVtVt1, an additional nodet1 withVt₁ =Vt∩Vt1 as added to(T,V). The parent oft₁ is set tot and the parent oft1is set tot1. This means that forget nodes can be added as

9.4. Optimisation Methods A C B E D G F H (a) D G A B C A E B C D B H C D C F (b) H A B E B C G B C C D C F D G B H B C D A B C A E C D C D B C D B C D D C A B C (c)

Figure 9.1: (a) A graph G = (V,E) with V = {A,B,C,D,E,F,G,H} and E = {(A,B),(A,C),(A,E),(B,D),(B,H),(C,D),(C,F),(D,G)}; (b) a tree-decomposition (T,V) of G as per definition 9.4; (c)a nice tree-decomposition of that in b as per definition 9.5. The nodes of the nice tree-decomposition are coloured red for the root node, green for leaf nodes, white for introduce nodes, blue for forget nodes and orange for join nodes.

9. Bond Order and Formal Charge Assignment

descendants oft₁ and introduce nodes as ancestors. For every vertextwith childt1, ifVt ⊃Vt1 and|Vt|=|Vt1|+kfor somek>1, a new introduce nodet1 is introduced to(T,V)as the child

oftand parent oft1. Letu∈Vt\Vt1be a vertex ofG, thenVt1 =Vt1∪ {u}. As a simple means to increase efficiency of the optimisation algorithm,uis chosen such that it minimises the number of partial solutions introduced when going fromt1tot1. This process is repeated until all nodes

along the patht1...t satisfy condition N5b. Similarly, for every nodet with childt1, ifVt⊂Vt1 and|Vt1|=|Vt|+kfor somek>1, a new forget nodet1is introduced to(T,V)as the child oftand

parent oft1withVt₁ =Vt1\ {u}whereu∈Vt1\Vt is a vertex ofG, repeating until condition N5a is met. In the same manner as the introduce node case,uis chosen to maximise the number of partial solutions removed when going fromt1tot1. At this point, all conditions for a nice tree

decomposition are met.

The electron distribution optimisation process requires determining energies for both atoms and bonds. Explicitly adding edges into the tree decomposition bags enables this, and simplifies the algorithm at the expense of increasing the tree width. With explicit edges, the notion thatV is the family of vertex setsVt⊆V(G)no longer holds. RatherVt⊆V(G)∪E(G). To determine

the formal charge of an atom, electrons on the atom and all incident bond orders need to be provided. To determine the bond order of a bond electrons in the bond need to be provided. However, if charged bonds are in use, the energy of a bond requires the electrons in the bond and the formal charges of the atoms of the bond in order to be calculated. This requires that before a bond can be scored, all other bonds incident to either of the atoms of the bond must also be present. This is achieved by adding an introduce vertex for every not yet introduced edge immediately proceeding the introduce vertex of the first atom component of the edge. A forget vertex for each edge is added immediately proceeding the forget vertex of the second atom component of the edge. Of course, the edge is additionally added to all bags along the path from introduce vertex to forget vertex. Edges are introduced to the tree decomposition immediately after conversion to a directed tree.

9.4.4.1. Algorithm

Lett ∈T be node of the nice tree-decomposition of a graph. ThenXt is the set of forgotten

verticesvsf ∈Vs associated with the forget nodes of the subtreeTt induced onT below (and

including) the nodet. The total number of electrons to placeeT and the multisetPof positions

at which the electrons can be placed are determined as per section 9.2. Each nodetis assigned a score tableStindexed by the ordered pair(l,k)∈Lt×Ktwhere

Lt={nmin,...,nmax} (9.9)

9.4. Optimisation Methods

nmin=max{0,eT− |P|+|P∩Xt|} (9.10)

nmax=min{eT,|P∩Xt|} (9.11)

Kt=X1× ··· ×Xj (9.12)

Xj={(j,k): j∈Vt,0≤k≤mult(P,j)}. (9.13)

withSt[l,k]being the minimum energy of forgotten verticesXt withl∈Lt forgotten electrons

and the additional constraint of further partial electron distributionk∈Kt. Beginning from the

leaves of the nice tree decomposition, and scoring only when all children of a node have been scored, the algorithm distinguishes the kind of each node and determines the score matrix as follows:

Leaf Nodes Leaf nodes are empty sets, so the score table of a leaf node is also empty. Introduce Nodes Lett∈T be the introduce node with childc, andxt=Xt\Xc. Then

St[l,k] = ⎧ ⎨ ⎩ Sp[l,k\xt] ifk\xt=/0, ∞ otherwise. (9.14)

Forget Nodes Lett∈Tbe the forget node with childc, andxt=Xc\Xt. Then

St[l,k] = min n∈{0,...,mult(P,xt)} p∈Lc:p+n=l E(xt,k∪Xt,n) +Sc[p,k∪xt] (9.15)

whereE(xt,k∪Xt,n) is the energy ofxt with nelectrons positioned and the partial electron

distributionk∪Xt.

Join Nodes Lett∈Tbe the parent ofc1andc2withVt=Vifori∈1,2. Then

St[l,k] = min (p,q)∈Lc₁×Lc₂:p+q=l Sc1[p,k] +Sc2[q,k] (9.16)

Root node Each nice tree decomposition has only one root noder∈T which is formally a forget node. However the score table of the root node is unpopulated asKr=/0. Rather than fill

a score table, the minimised electron distribution energy is determined. Letcbe the child ofr withxr=Xc. Then the minimum energy is given by

Emin= min n∈{0,...,mult(P,xr)} p∈Lc:p+n=eT E(xr,Xr,n) +Sr[p,xr] (9.17)

9. Bond Order and Formal Charge Assignment