8. Supplementary Material

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1 8. Supplementary Material 8.1. Proofs Proof of Proposition 1 Proof. (R2) = (R3): As for both (R2) and (R3) we have µ L and µ ( s) = µ s (u), s X u, we only need to prove that µ s conv(x s ), s L is equivalent to µ M. By inspecting the definition (6) of M we see that it is the intersection of (i) µ u X u, u V and (ii) µ s X s, s L. However, by the equality constraints µ u (s) = µ s (u) constraints (i) are already enforced, in (R2), hence equality follows. (R4) = (R5): Analoguously to (R2) = (R3). Relaxation (R1) is weaker than (R2) and (R3): Relaxations (R2) and (R3) include additional constraints Optimization Subproblems of Procedure 1 In Procedure 1 the two problem in lines 1 and 3 must be solved. Solution of the optimization problem in line 1 was discussed in the main part of the paper. Therefore, it only remains to show how to carry optimization of the problem in line 3 efficiently for all cases that can occur. This is shown in Table 3. Checking validity of the operations in Table 3 for i = u V and i = uv E is straightforward. For i = M and J = V, we prove correctness below. Correctness for i = M and J = L is analoguous. Lemma 2. The reparametrization adjustment problem (9) for i = M and J = V is given by (14). Moreover it is the dual of a minimum cost network flow problem. Proof. Recall from network flow theory [7], that x u X u is optimal for cost θ, iff π R V, ψ R L such that { 0, x θ u (x u ) π(u) + ψ(x u ) u(x u ) = 1 0, x u(x u ) = 0 u V, x u X u. Consider the primal/dual pair min (µu) u V u V θ u, µ u max,π,ψ u V u, δ u u V, x u X u µ u (x u ) = 0 π(u) R l L, { u:l X u µ u (l) = 0 { ψ(l) R δ u (x u ), x u x u 0, x u = x u µ u (x u ) δ u (x u ), x u = x u (x u ) u 0, x u x u µ u (x u ) R θu (x u ) + u (x u ) + π(u) ψ(x u ) = 0 (12) On the right side, the adjustment problem (3) is written down explicitly. The left hand side is a minimum cost flow problem, hence the second part of the claim is proven. The last equality above on the right hand side ensures that u (x u ) = θ u (x u ) π(u) + ψ(x u ). Substituting this everywhere on the right hand side of (12) gives max π,ψ u V π(u) ( x u X u δ u (x u ) ) + l L ψ(l) ( u V:l X u δ u (l) ) s.t. π(u) R ψ(l) R This form matches the format given in (14) Time complexity θ u (x u ) + π(u) ψ(x u ) { 0, 0, x u = x u x u x u The time complexity of running one iteration of message passing is essentially the time to run all required invocations of Algorithm 1 via the routines described in Table 3. Total runtime per iteration for the various algorithms we have proposed can (13)

2 be found in Table 4. We assume that X u = L u V. In sparse assignment problems, where this is not the case, run-time decreases according to sparsity. If we hold the unary potentials θ u, u V in a heap, we can support operation min s J X u θ u (s ) which is required in the third line in Table 3 in time log( L ), since either J X u = X u (sending) or J X u = 1 (receiving). Hence, all our algorithms scale to realistic problem sizes Detailed Experimental Evaluation Plots showing lower bound and primal solution energy per over time can be seen in Figure 2. In Table 5 dataset statistics are given together with final upper and lower bound as well as runtime averaged over all instances in specific datasets are given. A per-instance evaluation of all considered algorithms can be found in Table 6.

3 Algorithm 1 input i V J N G (i) Solution (i,j) j J of (9) i = u V J E {M} L (u,uv) = (θu min xu Xu θu(xu)) / J uv E J (u,m) = (θu min xu Xu θu(xu)) / J (u,s) = (θu(s) min s / J Xu θu(s ))/ J s X u J i = uv E J = {u}, u V (uv,u) (x u) = min {θ uv (x u, x v )} min xuv X uv {θ uv (x uv )} x v X v i = M J = V u (x u ) = θ u (x u ) π (u) + ψ (x u ) (π, ψ u V ) argmax π(u) ( x u X u δ u (x u ) ) π,ψ + l L ψ(l) ( u V:l X u δ u (l) ) { 0, x u = x u s.t. θu (x u ) + π(u) ψ(x u ) 0, x u x u (14) i = M J = L s (u) = θ u (s) π (u) + ψ (s) (π, ψ u V ) argmax π(u) ( x u X u δ xu (u) ) π,ψ + l L ψ(l) ( u V:l X u δ l (u) ) { 0, x u = x u s.t. θu (x u ) + π(u) ψ(x u ) 0, x u x u (15) Table 3. Message computation problems (9) Algorithm Time complexity GM O( L V + L 2 Ê ) AMP O( L V + L 2 E + L 3 ) AMP O( L V + L 2 E + L 2 log L ) AMCF O( L V + L 2 E + MCF ( V, V 2 ) Table 4. Time complexity per iteration for the three proposed algorithms. MCF (n, m) is the time to solve a min-cost-flow problem on a graph with n nodes and m edges: Orlin s algorithm has time complexity O(m 2 log n + m log 2 n) [7]. AMP stores reparametrized unary costs in a heap to accelerate computation of the messages between label factors and unaries.

4 energy 3,000 3,500 AMP AMCF GM HBP DD energy 4,240 4,260 4,280 AMP AMCF GM HBP DD energy AMP AMCF GM HBP DD 4, energy 4, runtime(s) house AMP AMCF GM HBP DD energy 1,500 2,000 2, runtime(s) hotel AMP AMCF GM HBP DD energy runtime(s) car AMP AMCF GM HBP DD 80 3, runtime(s) motor runtime(s) Hassan runtime(s) worms Figure 2. Runtime plots for house, hotel, car, motor, graph flow and worms datasets. Continuous lines denote dual lower bounds and dashed ones primal energies. Values are averaged over all instances of the dataset. The x-axis is logarithmic. hotel house graph flow car motor worms Dataset/Algorithm AMP AMCF GM HBP DD #I 105 LB #V 30 UB #L 30 time(s) #I 105 LB #V 30 UB #L 30 time(s) #I 6 LB #V 126 UB #L 126 time(s) #I 30 LB #V 49 UB #L 49 time(s) #I 20 LB #V 52 UB #L 52 time(s) #I 30 LB #V 605 UB #L 1500 time(s) Table 5. Description of datasets together with averaged algorithm results over all instances. #I means number of instances in dataset, #V the number of nodes per instance, #L the number of labels per instance, LB means lower bound, UB upper bound. Bold numbers indicate highest lower bound, lowest primal energy and smallest runtime.

5 hotel UB energy hotel frame15frame22 LB runtime(s) UB energy hotel frame15frame29 LB runtime(s) UB energy hotel frame15frame36 LB runtime(s) UB energy hotel frame15frame43 LB runtime(s) UB energy hotel frame15frame50 LB runtime(s) UB energy hotel frame15frame57 LB runtime(s) UB energy hotel frame15frame64 LB runtime(s) UB energy hotel frame15frame71 LB runtime(s) UB energy hotel frame15frame78 LB runtime(s) UB energy hotel frame15frame85 LB runtime(s) UB energy hotel frame15frame92 LB runtime(s) UB energy hotel frame15frame99 LB runtime(s) UB energy hotel frame1frame15 LB runtime(s) UB energy hotel frame1frame22 LB runtime(s) UB energy hotel frame1frame29 LB runtime(s) UB energy hotel frame1frame36 LB runtime(s) UB energy hotel frame1frame43 LB runtime(s) UB energy hotel frame1frame50 LB runtime(s) UB energy hotel frame1frame57 LB runtime(s) UB energy hotel frame1frame64 LB runtime(s)

6 UB energy hotel frame1frame71 LB runtime(s) UB energy hotel frame1frame78 LB runtime(s) UB energy hotel frame1frame8 LB runtime(s) UB energy hotel frame1frame85 LB runtime(s) UB energy hotel frame1frame92 LB runtime(s) UB energy hotel frame1frame99 LB runtime(s) UB energy hotel frame22frame29 LB runtime(s) UB energy hotel frame22frame36 LB runtime(s) UB energy hotel frame22frame43 LB runtime(s) UB energy hotel frame22frame50 LB runtime(s) UB energy hotel frame22frame57 LB runtime(s) UB energy hotel frame22frame64 LB runtime(s) UB energy hotel frame22frame71 LB runtime(s) UB energy hotel frame22frame78 LB runtime(s) UB energy hotel frame22frame85 LB runtime(s) UB energy hotel frame22frame92 LB runtime(s) UB energy hotel frame22frame99 LB runtime(s) UB energy hotel frame29frame36 LB runtime(s) UB energy hotel frame29frame43 LB runtime(s) UB energy hotel frame29frame50 LB runtime(s)

7 UB energy hotel frame29frame57 LB runtime(s) UB energy hotel frame29frame64 LB runtime(s) UB energy hotel frame29frame71 LB runtime(s) UB energy hotel frame29frame78 LB runtime(s) UB energy hotel frame29frame85 LB runtime(s) UB energy hotel frame29frame92 LB runtime(s) UB energy hotel frame29frame99 LB runtime(s) UB energy hotel frame36frame43 LB runtime(s) UB energy hotel frame36frame50 LB runtime(s) UB energy hotel frame36frame57 LB runtime(s) UB energy hotel frame36frame64 LB runtime(s) UB energy hotel frame36frame71 LB runtime(s) UB energy hotel frame36frame78 LB runtime(s) UB energy hotel frame36frame85 LB runtime(s) UB energy hotel frame36frame92 LB runtime(s) UB energy hotel frame36frame99 LB runtime(s) UB energy hotel frame43frame50 LB runtime(s) UB energy hotel frame43frame57 LB runtime(s) UB energy hotel frame43frame64 LB runtime(s) UB energy hotel frame43frame71 LB runtime(s)

8 UB energy hotel frame43frame78 LB runtime(s) UB energy hotel frame43frame85 LB runtime(s) UB energy hotel frame43frame92 LB runtime(s) UB energy hotel frame43frame99 LB runtime(s) UB energy hotel frame50frame57 LB runtime(s) UB energy hotel frame50frame64 LB runtime(s) UB energy hotel frame50frame71 LB runtime(s) UB energy hotel frame50frame78 LB runtime(s) UB energy hotel frame50frame85 LB runtime(s) UB energy hotel frame50frame92 LB runtime(s) UB energy hotel frame50frame99 LB runtime(s) UB energy hotel frame57frame64 LB runtime(s) UB energy hotel frame57frame71 LB runtime(s) UB energy hotel frame57frame78 LB runtime(s) UB energy hotel frame57frame85 LB runtime(s) UB energy hotel frame57frame92 LB runtime(s) UB energy hotel frame57frame99 LB runtime(s) UB energy hotel frame64frame71 LB runtime(s) UB energy hotel frame64frame78 LB runtime(s) UB energy hotel frame64frame85 LB runtime(s)

9 UB energy hotel frame64frame92 LB runtime(s) UB energy hotel frame64frame99 LB runtime(s) UB energy hotel frame71frame78 LB runtime(s) UB energy hotel frame71frame85 LB runtime(s) UB energy hotel frame71frame92 LB runtime(s) UB energy hotel frame71frame99 LB runtime(s) UB energy hotel frame78frame85 LB runtime(s) UB energy hotel frame78frame92 LB runtime(s) UB energy hotel frame78frame99 LB runtime(s) UB energy hotel frame85frame92 LB runtime(s) UB energy hotel frame85frame99 LB runtime(s) UB energy hotel frame8frame15 LB runtime(s) UB energy hotel frame8frame22 LB runtime(s) UB energy hotel frame8frame29 LB runtime(s) UB energy hotel frame8frame36 LB runtime(s) UB energy hotel frame8frame43 LB runtime(s) UB energy hotel frame8frame50 LB runtime(s) UB energy hotel frame8frame57 LB runtime(s) UB energy hotel frame8frame64 LB runtime(s) UB energy hotel frame8frame71 LB runtime(s)

10 UB energy hotel frame8frame78 LB runtime(s) UB energy hotel frame8frame85 LB runtime(s) UB energy hotel frame8frame92 LB runtime(s) UB energy hotel frame8frame99 LB runtime(s) UB energy hotel frame92frame99 LB runtime(s) house UB energy house frame10frame100 LB runtime(s) UB energy house frame10frame95 LB runtime(s) UB energy house frame10frame96 LB runtime(s) UB energy house frame10frame97 LB runtime(s) UB energy house frame10frame98 LB runtime(s) UB energy house frame10frame99 LB runtime(s) UB energy house frame11frame100 LB runtime(s) UB energy house frame11frame101 LB runtime(s) UB energy house frame11frame96 LB runtime(s) UB energy house frame11frame97 LB runtime(s) UB energy house frame11frame98 LB runtime(s) UB energy house frame11frame99 LB runtime(s) UB energy house frame12frame100 LB runtime(s) UB energy house frame12frame101 LB runtime(s) UB energy house frame12frame102 LB runtime(s)

11 UB energy house frame12frame97 LB runtime(s) UB energy house frame12frame98 LB runtime(s) UB energy house frame12frame99 LB runtime(s) UB energy house frame13frame100 LB runtime(s) UB energy house frame13frame101 LB runtime(s) UB energy house frame13frame102 LB runtime(s) UB energy house frame13frame103 LB runtime(s) UB energy house frame13frame98 LB runtime(s) UB energy house frame13frame99 LB runtime(s) UB energy house frame14frame100 LB runtime(s) UB energy house frame14frame101 LB runtime(s) UB energy house frame14frame102 LB runtime(s) UB energy house frame14frame103 LB runtime(s) UB energy house frame14frame104 LB runtime(s) UB energy house frame14frame99 LB runtime(s) UB energy house frame15frame100 LB runtime(s) UB energy house frame15frame101 LB runtime(s) UB energy house frame15frame102 LB runtime(s) UB energy house frame15frame103 LB runtime(s) UB energy house frame15frame104 LB runtime(s)

12 UB energy house frame15frame105 LB runtime(s) UB energy house frame16frame101 LB runtime(s) UB energy house frame16frame102 LB runtime(s) UB energy house frame16frame103 LB runtime(s) UB energy house frame16frame104 LB runtime(s) UB energy house frame16frame105 LB runtime(s) UB energy house frame17frame102 LB runtime(s) UB energy house frame17frame103 LB runtime(s) UB energy house frame17frame104 LB runtime(s) UB energy house frame17frame105 LB runtime(s) UB energy house frame18frame103 LB runtime(s) UB energy house frame18frame104 LB runtime(s) UB energy house frame18frame105 LB runtime(s) UB energy house frame19frame104 LB runtime(s) UB energy house frame19frame105 LB runtime(s) UB energy house frame1frame86 LB runtime(s) UB energy house frame1frame87 LB runtime(s) UB energy house frame1frame88 LB runtime(s) UB energy house frame1frame89 LB runtime(s) UB energy house frame1frame90 LB runtime(s)

13 UB energy house frame1frame91 LB runtime(s) UB energy house frame20frame105 LB runtime(s) UB energy house frame2frame87 LB runtime(s) UB energy house frame2frame88 LB runtime(s) UB energy house frame2frame89 LB runtime(s) UB energy house frame2frame90 LB runtime(s) UB energy house frame2frame91 LB runtime(s) UB energy house frame2frame92 LB runtime(s) UB energy house frame3frame88 LB runtime(s) UB energy house frame3frame89 LB runtime(s) UB energy house frame3frame90 LB runtime(s) UB energy house frame3frame91 LB runtime(s) UB energy house frame3frame92 LB runtime(s) UB energy house frame3frame93 LB runtime(s) UB energy house frame4frame89 LB runtime(s) UB energy house frame4frame90 LB runtime(s) UB energy house frame4frame91 LB runtime(s) UB energy house frame4frame92 LB runtime(s) UB energy house frame4frame93 LB runtime(s) UB energy house frame4frame94 LB runtime(s)

14 UB energy house frame5frame90 LB runtime(s) UB energy house frame5frame91 LB runtime(s) UB energy house frame5frame92 LB runtime(s) UB energy house frame5frame93 LB runtime(s) UB energy house frame5frame94 LB runtime(s) UB energy house frame5frame95 LB runtime(s) UB energy house frame6frame91 LB runtime(s) UB energy house frame6frame92 LB runtime(s) UB energy house frame6frame93 LB runtime(s) UB energy house frame6frame94 LB runtime(s) UB energy house frame6frame95 LB runtime(s) UB energy house frame6frame96 LB runtime(s) UB energy house frame7frame92 LB runtime(s) UB energy house frame7frame93 LB runtime(s) UB energy house frame7frame94 LB runtime(s) UB energy house frame7frame95 LB runtime(s) UB energy house frame7frame96 LB runtime(s) UB energy house frame7frame97 LB runtime(s) UB energy house frame8frame93 LB runtime(s) UB energy house frame8frame94 LB runtime(s)

15 UB energy house frame8frame95 LB runtime(s) UB energy house frame8frame96 LB runtime(s) UB energy house frame8frame97 LB runtime(s) UB energy house frame8frame98 LB runtime(s) UB energy house frame9frame94 LB runtime(s) UB energy house frame9frame95 LB runtime(s) UB energy house frame9frame96 LB runtime(s) UB energy house frame9frame97 LB runtime(s) UB energy house frame9frame98 LB runtime(s) UB energy house frame9frame99 LB runtime(s) Hassan UB board torresani LB runtime(s) UB books torresani LB runtime(s) UB hammer torresani LB runtime(s) UB party torresani LB runtime(s) UB table torresani LB runtime(s) UB walking torresani LB runtime(s) car UB car1 LB runtime(s) UB car10 LB runtime(s) UB car11 LB runtime(s)

16 UB car12 LB runtime(s) UB car13 LB runtime(s) UB car14 LB runtime(s) UB car15 LB runtime(s) UB car16 LB runtime(s) UB car17 LB runtime(s) UB car18 LB runtime(s) UB car19 LB runtime(s) UB car2 LB runtime(s) UB car20 LB runtime(s) UB car21 LB runtime(s) UB car22 LB runtime(s) UB car23 LB runtime(s) UB car24 LB runtime(s) UB car25 LB runtime(s) UB car26 LB runtime(s) UB car27 LB runtime(s) UB car28 LB runtime(s) UB car29 LB runtime(s) UB car3 LB runtime(s)

17 UB car30 LB runtime(s) UB car4 LB runtime(s) UB car5 LB runtime(s) UB car6 LB runtime(s) UB car7 LB runtime(s) UB car8 LB runtime(s) UB car9 LB runtime(s) motor UB motor1 LB runtime(s) UB motor10 LB runtime(s) UB motor11 LB runtime(s) UB motor12 LB runtime(s) UB motor13 LB runtime(s) UB motor14 LB runtime(s) UB motor15 LB runtime(s) UB motor16 LB runtime(s) UB motor17 LB runtime(s) UB motor18 LB runtime(s) UB motor19 LB runtime(s) UB motor2 LB runtime(s) UB motor20 LB runtime(s)

18 UB motor3 LB runtime(s) UB motor4 LB runtime(s) UB motor5 LB runtime(s) UB motor6 LB runtime(s) UB motor7 LB runtime(s) UB motor8 LB runtime(s) UB motor9 LB runtime(s) worms UB inf C18G1 2L1 1-lowThresh-more-hyp.surf LB runtime(s) UB inf cnd1threel lowThresh-more-hyp.surf LB runtime(s) UB inf cnd1threel lowThresh-more-hyp.surf LB runtime(s) UB inf cnd1threel lowThresh-more-hyp.surf LB runtime(s) UB inf cnd1threel lowThresh-more-hyp.surf LB runtime(s) UB inf cnd1threel lowThresh-more-hyp.surf LB runtime(s) UB inf eft3rw10035l lowThresh-more-hyp.surf LB runtime(s) UB inf eft3rw10035l lowThresh-more-hyp.surf LB runtime(s) UB inf eft3rw10035l lowThresh-more-hyp.surf LB runtime(s) UB inf egl5l lowThresh-more-hyp.surf LB runtime(s) UB inf elt3l lowThresh-more-hyp.surf LB runtime(s) UB inf elt3l lowThresh-more-hyp.surf LB runtime(s) UB inf elt3l lowThresh-more-hyp.surf LB runtime(s)

19 UB inf hlh1fourl lowThresh-more-hyp.surf LB runtime(s) UB inf hlh1fourl lowThresh-more-hyp.surf LB runtime(s) UB inf hlh1fourl lowThresh-more-hyp.surf LB runtime(s) UB inf hlh1fourl lowThresh-more-hyp.surf LB runtime(s) UB inf hlh1fourl lowThresh-more-hyp.surf LB runtime(s) UB inf mir61l lowThresh-more-hyp.surf LB runtime(s) UB inf mir61l lowThresh-more-hyp.surf LB runtime(s) UB inf mir61l lowThresh-more-hyp.surf LB runtime(s) UB inf pha4a7l lowThresh-more-hyp.surf LB runtime(s) UB inf pha4a7l lowThresh-more-hyp.surf LB runtime(s) UB inf pha4a7l lowThresh-more-hyp.surf LB runtime(s) UB inf pha4b2l lowThresh-more-hyp.surf LB runtime(s) UB inf pha4i2l lowThresh-more-hyp.surf LB runtime(s) UB pha4i2l lowThresh-more-hyp.surf LB runtime(s) UB pha4i2l lowThresh-more-hyp.surf LB runtime(s) UB inf unc54l lowThresh-more-hyp.surf LB runtime(s) UB inf unc54l lowThresh-more-hyp.surf LB runtime(s)

Supplementary file related to the paper titled On the Design and Deployment of RFID Assisted Navigation Systems for VANET

Supplementary file related to the paper titled On the Design and Deployment of RFID Assisted Navigation Systems for VANET SUPPLEMENTARY FILE RELATED TO SECTION 3: RFID ASSISTED NAVIGATION SYS- TEM MODEL