Updating the dynamic programming table

For allP ∈ P containing m >1 edges, and for all P₁, P₂ ∈ P withE(P₁)6=∅, E(P₂)6=∅, E(P1)∩E(P2) = ∅ and P1∪P2 = P, and for all φ1 ∈ IP₁ andφ2 ∈ IP₂ we proceed as follows. Suppose that for all pathsP⁰ containingm⁰ < medges and allφ⁰ ∈ IP⁰ we have computed the partial solutions in the dynamic programming table at (P⁰, φ⁰). If there exist partial solutionsS1 andS2 at (P1, φ1) and (P2, φ2) respectively, we call the merging process to merge S₁andS₂. Suppose that the merging process returns a partial solutionS at (P, φ) for someφ∈ IP. If there is no partial solution stored currently at (P, φ) then we storeS at that location. Otherwise if there there exists a partial solutionS⁰ stored at (P, φ) and the cost ofS is smaller than the cost ofS⁰ then we replaceS⁰ with S.

12.2 Analysis

LetW be the collection of walks given by Lemma 56. LetW⁰ be the shadow ofW. LetF be the forest obtained by Lemma 57. For every connected componentT of F pick some v_T ∈V(T) and considerT to be rooted atv_T.

Let G~⁰ be the planar piece ofG, that is~ G~⁰=G~\(V(H~)\(F~)). Fix some planar drawing ψofG⁰. LetDbe the disk with∂D=ψ(F) withψ(G)~ ⊂ D.

Let P∈ P with endpointsu,v, and letT be a subtree of some tree inF. We say thatP coversT if for allD∈V(T) we haveV(D)∩V(F)⊆V(P). We say thatP avoids T if for allD∈V(T) we haveV(D)∩V(P)⊆ {u, v}.

IDefinition 60(Basic path). LetP ∈ P. Letu, v∈V(P) be the endpoints ofP. We say thatP isbasic(w.r.t.W) if eitherP\ {u, v} does not intersect any of the walks inW (in this case we callP empty basic) or the following holds. There exists some treeT inF and someD∈V(T), with childrenD1, . . . , Dk, intersectingD in this order along a traversal of D, such that the following conditions are satisfied (see Figure 1):

1. For anyi∈ {1, . . . , k}, letTD_i be the subtree ofT rooted atDiand letTD be the subtree ofT rooted at D. Then at least one of the following two conditions is satisfied:

1.1. P coversTD and avoidsT \ TD.

1.2. There exists j ∈ {1, . . . , k} such that for all l ≤ j, P covers TD_l and P avoids TD\Sj

m=1TD_m.

2. LetT⁰ be a tree in F withT⁰6=T. Then eitherP coversT⁰ or P avoids T⁰.

IDefinition 61(Facial restriction). LetP ∈ P be basic. LetW⁰ be the collection of walks obtained by restricting everyW ∈ W onHP. IfP is empty, we say thatW⁰ is theP-facial restriction ofW. Suppose thatP is not empty. LetT,D,TD,kandj be as in Definition 60.

Figure 2Part of the collection of walks inW(left) and the correspondingP-facial restriction of Wdepicted in bold (right).

LetF⁰ ={T⁰ ∈ F :P coversT⁰} and let R=S

T⁰∈F⁰V(T⁰). IfP covers TD and avoids T \ TD, we say thatR ∪ W⁰∪V(TD) is theP-facial restriction ofW. Otherwise, we say that R ∪ W⁰∪ {D} ∪Sj

i=1V(TD_i) is the P-facial restriction ofW. Figure 2 depicts an example of aP-facial restriction.

IDefinition 62 (Important walk). Let P ∈ P be a basic path. Let W⁰ be the P-facial restriction ofW. LetW⁰⁰ be the shadow ofW⁰. LetQbe a walk inG[W⁰⁰]. We say thatQ isP-important (w.r.to.W) if the following conditions hold:

1. Both endpoints ofQare in V(P).

2. Qis the concatenation of walksQ1, . . . , Q`such that for eachi∈ {1, . . . , `}there exists someW_i ∈ W such thatQ_i is a sub-walk ofW_i, and for eachj∈ {1, . . . , `−1}we have {Wj, Wj+1} ∈E(F).

IProposition 63. For anyu, v∈V(P), there is at most one P-important walk from uto v.

Proof. It follows immediately by the fact thatF is a forest. J I Lemma 64. Let P ∈ P be basic w.r.t. W with endpoints u, v ∈ V(F), where u = v if P is closed. Let WP be the P-facial restriction of W. Let Γ = S

W~∈WP

W~ . Let C be the partition of Bu ∪Bv that corresponds to the weakly-connected components of Γ.

For any x∈ B_u∪B_v let fⁱⁿ(x) =in-degree_W

P(x) and f^out(x) =out-degree_W

P(x). Then there exists some a ∈ A ∪(A × A)∪nil and l, r, p ∈(V(G)~ ∪nil) such that the dynamic programming table contains some partial solution S at location (P,(C, fⁱⁿ, f^out, a, l, r, p)), such thatcost_G~(S)≤cost_G~(WP).

Proof. Let us first assume thatF contains only one tree. We will deal with the more general case later on. First suppose thatP is an empty basic path. LetP =x1, x2, . . . , xm, where x₁=uandxm=v. We will prove the assertion by induction onm. For the base case, suppose thatm= 2, and thusP contains only one edge. Leta=l=r=p=nil. In this case, a partial solutionS at location (P,(C, fⁱⁿ, f^out, a, l, r, p)) is computed in the initialization step of the dynamic programming table and clearly we havecost_G~(S)≤cost_G~(WP) and we are done. Now suppose thatm >2 and we have proved the assertion for allm⁰< m. We first decomposeP into two edge-disjoint pathsP₁andP₂, such thatV(P₁)∩V(P₂) =wfor some 1< j < mand w=xj. Fori∈ {1,2} letWP_i be thePi-facial restriction of W~OPT and let Γi=S

W~∈W_PiW~ . LetC₁be the partition ofBu∪Bwthat corresponds to the weakly-connected components of Γ₁ and letC2be the partition ofBw∪Bvthat corresponds to the weakly-connected components

of Γ2. For anyx∈Bu∪Bw let f₁ⁱⁿ(x) =in-degree_WP

1(x) andf₁^out(x) =out-degree_WP

1(x).

Also for any x ∈ Bw∪Bv let f₂ⁱⁿ(x) = in-degree_WP

2(x) and f₂^out(x) = out-degree_WP

2(x).

By the induction hypothesis, there exists partial solutionsS1 andS2 that are compatible with (P₁,(C₁, f₁ⁱⁿ, f₁^out,nil,nil,nil,nil)) and (P₂,(C₂, f₂ⁱⁿ, f^out,nil,nil,nil,nil)) respectively, and we have costG~(S1) ≤ costG~(WP₁), costG~(S2) ≤ costG~(WP₂). The dynamic program will mergeS₁ andS₂ to get the desired S. Note that by the construction, for everyx∈B_wwe havef₁ⁱⁿ(x) =f₂^out(x) andf₂ⁱⁿ(x) =f₁^out(x). Therefore, by the first merging phase, we can merge walks in S₁andS₂. Also, we leta=l=r=p=niland we proceed the second phase of merging. Finally, by the construction and definition ofP-facial restriction, (T5) holds and the merging process returns a partial solutionS compatible with (P,(C, fⁱⁿ, f^out, a, l, r, p)) withcost_G~(S)≤cost_G~(WP), as desired.

Now suppose thatP is not empty basic. LetT, D, TD,kandj be as in Definition 60.

We will prove the assertion by induction on T. For the base case, suppose thatDis a leaf of T. Suppose thatP =x₁, x₂, . . . , x_mfor somem >0, wherex₁=uandx_m=v, andD is a (possibly closed) walk from xi ∈V(P) to xj ∈V(P). We may assume w.l.o.g. thati≤j.

There are some possible cases here based oniandj. First suppose thati >1,j < mandj− i≥3. In this case, letP1=x1, . . . , xi,P2=xi, xi+1,P3=xi+1, . . . , xj−1,P4=xj−1, xj and P5=xj, . . . , xm. LetP6=P1∪P2,P7=P6∪P3andP8=P7∪P4. Fori∈ {1,2,3,4,5,6,7,8}

letWPi be theP_i-facial restriction of W~_OPT and let Γ_i =S

W~∈W_PiW~ . We also define C_i, f_iⁱⁿ andf_i^out as in the previous case. Note that P1,P3 andP5 are empty basic paths. We leta1 =a3=a5=nil, l1 =l3 =l5 =nil, r1=r3=r5 =nil, p1 =p3 =p5 =niland thus we can find partial solutionsS1,S3 andS5 compatible with (P1,(C1, f₁ⁱⁿ, f₁^out, a1, l1, r1, p1)), (P₃,(C₃, f₃ⁱⁿ, f₃^out, a₃, l₃, r₃, p₃)) and (P₅,(C₅, f₅ⁱⁿ, f₅^out, a₅, l₅, r₅, p₅)) respectively. We also have

costG~(S1)≤costG~(WP₁), costG~(S3)≤costG~(WP₃) andcostG~(S5)≤costG~(WP₅). Leta2= a₄=a₆=a₇=a₈= (x_i, x_j). IfDdoes not have a parent inT, then we letl₂=l₄=l₆= l7 =l8 = r2 =r4 =r6 =r7 =r8 =p2 =p4 =p6 =p7 =p8 ∈V(D) to be an arbitrary vertex ofD. Otherwise, suppose that Dhas a parentD⁰ in T. IfD⁰ does not have a parent inT, then we let l₂ =l₄=l₆=l₇ =l₈=r₂=r₄=r₆ =r₇ =r₈ =p₂=p₄=p₆ =p₇= p8∈ V(D)∩V(D⁰). Otherwise, suppose thatD⁰ has a parentD⁰⁰ inT. In this case, we letl₂ =l₄ =l₆ =l₇ =l₈ =r₂ =r₄ =r₆=r₇ =r₈ ∈V(D)∩V(D⁰) andp₂=p₄=p₆= p7=p8∈V(D⁰)∩V(D⁰⁰). Therefore, by computing the initialization step, we can find a partial solution S₂ compatible with (P₂,(C₂, f₂ⁱⁿ, f₂^out, a₂, l₂, r₂, p₂)) and a partial solution S4 compatible with (P4,(C4, f₄ⁱⁿ, f₄^out, a4, l4, r4, p4)), and we have costG~(S2) ≤costG~(WP₂) and cost_G~(S4) ≤ cost_G~(WP₄). Now by merging S1 and S2, we get a partial solution S6

compatible with (P₆,(C₆, f₆ⁱⁿ, f₆^out, a₆, l₆, r₆, p₆)). By merging S6 andS3, we get a partial solutionS7compatible with (P7,(C7, f₇ⁱⁿ, f₇^out, a7, l7, r7, p7)). By mergingS7 andS4, we get a partial solutionS8compatible with (P₈,(C₈, f₈ⁱⁿ, f₈^out, a₈, l₈, r₈, p₈)), and finally by merging S8 andS5, we get the desired partial solutionS compatible with (P,(C, fⁱⁿ, f^out, a, l, r, p)) withcost_G~(S)≤cost_G~(W_P). Ifi= 1 or j=m, we will follow a similar approach. The only different is that instead of dividingP into five paths, we divide it into four paths. Finally, the last case is whenj−i <3. In this case, ifi6=j, we define the same subpathsP1,P2,P4

and P₅, and we follow a similar approach. Otherwise, suppose thati=j. In this case, we letP1=x1, . . . , xi andP2=xi, . . . , xm and by following the same approach by mering two partial solutions, we get the desiredS.

Now suppose thatD∈V(T) is non-leaf. In this case, we prove the assertion by induction on j, wherej comes from Definition 60. Note that we perform a second induction inside the first induction. For the base case, suppose thatj = 1. In this case,D₁ is a child ofD andP covers TD1 and avoidsTD\ TD1. Therefore, by using the first induction hypothesis

onD1, there existsa ∈ A ∪(A × A)∪niland l, r, p∈ V(G)~ ∪nil such that the dynamic programming table contains some partial solution S at location (P,(C, fⁱⁿ, f^out, a, l, r, p)) withcost_G~(S)≤cost_G~(WP). Now for the samea,l,r,pandS, we have thatS is compatible we can apply the second merging phase. Finally, by the construction (T5) holds and we get a partial solutionS₄ compatible with (P₄,(C₄, f₄ⁱⁿ, f₄^out, a₄, l₄, r₄, p₄)). Now, we merge

then we have a = a2, l = l2, r = r2 andp = p2, and thus this case is also immediate.

Suppose thata2= (u^∗₂, v₂^∗)6=nil,a4= (u^∗₄, v₄^∗)6=nil,l4=r4 andp1=p2, anda= (u^∗, v^∗), whereu^∗∈ {u^∗₂, u^∗₄} andv^∗∈ {v^∗₂, v₄^∗}. We may assume w.l.o.g thata= (u^∗₂, v^∗₄). We have that l = l2, r =r4 and p=p2. For i ∈ {2,4} let Q^∗_i be the grip of Si, and letW~i ∈ Si

that contains Q^∗_i as a sub-walk. LetY₁ be the shortest-path in G~ fromu^∗₂ tol₂. Let Y₂ be the shortest-path in G~ from l2 to l4. Let Y3 be the shortest-path inG~ from l4 tov^∗₄. Let R^∗₁ be the path inG~ from u^∗₂ tov₄^∗ obtained by concatenation of Y₁, Y₂ andY₃. Let Y₁⁰ be the shortest-path in G~ from u^∗₄ to r4. Let Y₂⁰ be the shortest-path in G~ from r4

to l2. Let Y₃⁰ be the shortest-path in G~ from l2 to v₂^∗. Let R^∗₂ be the path in G~ from u^∗₄ tov₂^∗ obtained by concatenation ofY₁⁰,Y₂⁰ andY₃⁰. Now by the construction, we have thatcostG~(S) =costG~(S2) +costG~(S4) +costG~(R^∗₁) +costG~(R^∗₂)−costG~(Q^∗₂)−costG~(Q^∗₄).

Note that by the construction, there exists aP-important walk from u^∗₂ tov₄^∗ (w.r.t.W) and a P-important walk from u^∗₄ to v₂^∗ (w.r.t. W). By Proposition 63 these walks are unique. Let R⁰₁ be the P-important walk from u^∗₂ to v^∗₄ (w.r.t. W) and let R⁰₂ be the P-important walk fromu^∗₄ tov₂^∗ (w.r.t.W). Also, there exists aP2-important walk from u^∗₂ to v₂^∗ (w.r.t. W) and a P4-important walk from u^∗₄ to v₄^∗ (w.r.t. W), and thus by Proposition 63 these walks are unique. Let Q⁰₂ be the P₂-important walk from u^∗₂ to v₂^∗ (w.r.t. W) and let Q⁰₄ be the P4-important walk from u^∗₄ to v₄^∗ (w.r.t. W). By the definition of important walks, we have that cost_G~(R^∗₁)≤cost_G~(R⁰₁),cost_G~(R^∗₂)≤cost_G~(R⁰₂), costG~(Q^∗₂)≤costG~(Q⁰₂) andcostG~(Q^∗₄)≤costG~(Q⁰₄). Also by the construction, we have that cost_G~(R⁰₁) +cost_G~(R⁰₂) =cost_G~(Q⁰₂) +cost_G~(Q⁰₄). Therefore, we have

costG~(S) =costG~(S2) +costG~(S4) +costG~(R^∗₁) +costG~(R^∗₂)−costG~(Q^∗₂)−costG~(Q^∗₄)

≤costG~(S2) +costG~(S4) +costG~(R⁰₁) +costG~(R⁰₂)−costG~(Q^∗₂)−costG~(Q^∗₄)

= (costG~(S2)−costG~(Q^∗₄)) + (costG~(S4)−costG~(Q^∗₂)) +costG~(R⁰₁) +costG~(R⁰₂)

≤costG~(WP₁) +costG~(WP₂)

=costG~(WP).

Now suppose that D is an open walk. If a2 =nil anda4 ∈ (A × A), or a4 =nil and a₂∈(A × A), then this is immediate. Ifa₂= (u^∗₂, v₂^∗)∈ A,a₄= (u^∗₄, v₄^∗)∈ A,l₂=r₂,l₄=r₄ andp2=p4, then we havea= ((u⁰, v⁰),(u⁰⁰, v⁰⁰)) wherev⁰∈ {v₂^∗, v₄^∗}andu⁰⁰∈ {u^∗₂, u^∗₄}. We may assume w.l.o.g thatv⁰ =v₂^∗andu⁰⁰=u^∗₄. Then we havel=l₂,r=r₂andp=p₂. LetR₁ be the shortest-path inG~ fromu⁰tol2. LetR2be the shortest-path inG~ froml1tov⁰. LetR⁰ be the path fromu⁰tov⁰ obtained by the concatenation ofR₁andR₂. LetY₁be the shortest path inG~ fromu⁰⁰ tol4. LetY2 be the shortest path inG~ froml4tov⁰⁰. LetY⁰ be the path fromu⁰⁰ tov⁰⁰ obtained by the concatenation ofY1 andY2. LetZ1 be the shortest path inG~ fromu^∗₂tol₂. LetZ₂be the shortest path inG~ froml₂tol₄. LetZ₃be the shortest path inG~ froml4tov₄^∗. LetZ⁰be the path fromu^∗₂ tov₄^∗ obtained by the concatenation ofZ1,Z2 and Z₃. LetQ^∗₂ be the grip ofS₂and letQ^∗₄ be the grip ofS₄. By the construction, we have that costG~(S) =costG~(S2)+costG~(S4)+costG~(R⁰)+costG~(Y⁰)+costG~(Z⁰)−costG~(Q^∗₂)−costG~(Q^∗₄).

By the construction, there exists a P-important walk from u^∗₂ tov₄^∗, a P-important walk fromu⁰ tov^∗₂, and aP-important walk fromu^∗₄tov⁰⁰. Therefore by Proposition 63, these

important walks are unique. LetR⁰⁰,Y⁰⁰ andZ⁰⁰ be the important walks fromu⁰ tov⁰,u⁰⁰ to v⁰⁰, andu^∗₂ tov₄^∗respectively. Therefore, we have that

costG~(S) =costG~(S2) +costG~(S4) +costG~(R⁰) +costG~(Y⁰) +costG~(Z⁰)

−costG~(Q^∗₂)−costG~(Q^∗₄)

≤costG~(S2) +costG~(S4) +costG~(R⁰⁰) +costG~(Y⁰⁰) +costG~(Z⁰⁰)

−costG~(Q^∗₂)−costG~(Q^∗₄)

≤costG~(WP₁) +costG~(WP₂) =costG~(WP).

If a4= ((u4, v4),(u⁰₄, v⁰₄))∈(A × A), a2 = (u2, v2)∈ A, l2 =r2 andp2 =p4, then we havea= ((u₄, v₄),(u₂, v₄⁰)),l=l₄,r=r₂ andp=p₂. Let (Q^∗₄, Q^∗∗₄ ) be the grip of S4, and letQ^∗₂ be the grip ofS2. We may assume w.l.o.g thatQ^∗₄ is a path fromu4tov4, andQ^∗∗₄ is a path fromu⁰₄ tov⁰₄. LetY₁ be the shortest path inG~ fromu₂tol₂. LetY₂be the shortest path inG~ froml2 tov₄⁰. LetY⁰ be the path fromu2 tov₄⁰ obtained by the concatenation of Y₁ andY₂. LetZ₁ be the shortest path inG~ fromu⁰₄ tor₄. LetZ₂ be the shortest path in G~ fromr4tol2. LetZ3 be the shortest path inG~ froml2tov2. LetZ⁰ be the path fromu⁰₄ tov2 obtained by the concatenation ofZ1,Z2 andZ3. By the construction, we have that cost_G~(S) =cost_G~(S2) +cost_G~(S4) +cost_G~(Y⁰) +cost_G~(Z⁰)−cost_G~(Q^∗₂)−cost_G~(Q^∗∗₄ ). By the construction, there exists aP-important walk fromu⁰₄ tov2, and aP-important walk fromu₂ tov⁰₄. Therefore by Proposition 63, these important walks are unique. LetY⁰⁰ and Z⁰⁰be the important walks fromu2to v₄⁰, andu⁰₄ tov2respectively. Therefore we have

costG~(S) =costG~(S2) +costG~(S4) +costG~(Y⁰) +costG~(Z⁰)−costG~(Q^∗₂)−costG~(Q^∗∗₄ )

≤costG~(S2) +costG~(S4) +costG~(Y⁰⁰) +costG~(Z⁰⁰)−costG~(Q^∗₂)−costG~(Q^∗∗₄ )

≤costG~(WP1) +costG~(WP2) =costG~(WP).

Now suppose that F contains more than one tree. Let A = {T1, . . . ,Tm} be the set of all trees in F. For every T ∈ A we define the level of T, L(T), as follows. Let D be the root of T. We set L(T) = 0, if there exists a basic path P⁰ that covers T and avoids allT⁰ ∈ F \ {T }. We call a minimal such path, acorresponding basic path for T and we denote it byPT; it is immediate that there is a unique such minimal path. Let F0 = {T ∈ F : L(T) = 0}. Now for i≥0, suppose that we have defined trees of level i andFi. Suppose thatL(T)∈ {0, . . . , i}. We set/ L(T) =i+ 1 if there exists a basic path P⁰ that coversT such that for allT⁰ ∈Si

j=0Fj, P⁰ either avoids or coversT⁰, and for all T⁰⁰∈(F \ {T })\Si

j=0F_j,P⁰ avoidsT⁰⁰. We also call a minimal such path corresponding basic forT and we denote it byP_T. LetFi+1={T ∈ F :L(T) =i+ 1}.

We say that some T ∈ F is outer-most if there is noT⁰ ∈ F withL(T⁰)> L(T) and P_T ⊂P_T⁰.

Let us first suppose that there exists only one outer-most treeT ∈ F, such thatPT ⊆P. We will deal with the more general case later. Also, suppose thatT ∈ Fmfor somem≥0.

We will prove the assertion by induction onm. We also prove that for this case, we have a=l =r=p=nil. For the base case, ifm= 0, then by the construction,T is the only tree with a corresponding basic pathP_T ⊆P. For this case, we have already established the assertion and we are done. Now suppose that we have proved the assertion for all m⁰< m. LetF⁰=F \ {T }. LetT₁, . . . ,T_t∈ F⁰ be all outer-most trees inF⁰, such that for j∈ {1, . . . , t} we havePT_j ⊆P, and they intersectF in this order. By the construction, for everyj∈ {1, . . . , t} we haveL(Tj)< m. For everyj∈ {1, . . . , t} letP_Tj be a corresponding basic path forTj. By the induction hypothesis, for everyj∈ {1, . . . , t} there exists a partial

solutionSj forPTj. Now, we can apply the same argument when we had only one treeT ∈ F forT. Note that for eachj∈ {1, . . . , t}, we haveaj=lj=rj=pj =nil. The only difference is that the intermediate basic paths here are not necessarily empty basic paths, and eachTj

appears as an intermediate basic path, with a partial solutionSj. Therefore, by following a similar approach to the previous cases and merging appropriately we get a partial solutionS, as desired.

Now, suppose that there exist more than one outer-most treesT ∈ F, whereP_T ⊆P. Let B={T₁, . . . ,T_t} be the set of all such trees, where they intersect consecutive subpaths of P in this order. In this case, we prove the assertion by induction on t. For the base case wheret= 1, we have already proved the assertion. Now suppose that we have proved the assertion for allt⁰< t. LetB⁰={T1, . . . ,Tt−1}. LetP⁰ ⊆P be the subpath ofP such that B⁰ is the set of all outer-most trees, withP_Tj ⊆P⁰ for all 1≤j≤t−1. By the induction hypothesis, there exists a partial solutionS⁰ forP⁰. LetP⁰⁰=P ⊆P⁰. By the construction, P⁰⁰is a corresponding basic path forTt, and thus by the induction hypothesis, there exists a partial solutionS⁰⁰ forP⁰⁰. Therefore, by mergingS⁰ andS⁰⁰ we get a partial solutionS for

P, as desired. J

ITheorem 65. Let G~ be an n-vertex (0,0,1, p)-nearly embeddable graph (that is, planar with a single vortex) and letH~ be the vortex of G. Then there exists an algorithm which~ computes a walkW~ visiting all vertices in V(H)~ of total length OPTG~(V(H~)) in timen^O(p). Proof. We have F ∈ P and it is immediate to check that F is basic. Since F is a cycle, both the endpoitns ofF are some vertexv^◦. It follows by Lemma 64 that there exists some φ∈ I_F such that the dynamic programming table contains a partial solution S at location (F, φ). Let φ = (C, fⁱⁿ, f^out, a, l, r, p). It follows by condition (T4) that for all x ∈ Bv^◦

we havein-degree_S(x) = out-degree_S(x). Thus by repeatedly merging pairs of walks that respectively terminate to and start from the same vertex, we obtain a collectionS⁰ of closed walks with costG~(S⁰) = costG~(S) that visit all vertices inV(H~). By Lemma 57 we can assume thatC is the trivial partition containing only one cluster, that isC={{B_v^◦}}. Since C is trivial, it follows by (T5) that all vertices inV(H~) are in the same weakly-connected component of S

W∈SW. Thus all vertices in V(H~) are in the same strongly-connected component of S

W∈S⁰W. Since all walks in S⁰ are closed, by repeatedly shortcutting pairs of intersecting walks, we obtain a walk W~ with that visits all vertices inV(H~) with cost_G~(W~ ) =cost_G~(S⁰) =cost_G~(S)≤cost_G~(W~_OPT) =OPT_G~.

The running time is polynomial in the size of the dynamic programming table, which is at most|P| ·maxP∈P|IP|=O(n²)·p^O(p)·n^O(p)·O(n²) =n^O(p). J

13 The dynamic program for traversing a vortex in a bounded genus

In document Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs∗ (Pldal 29-36)