Fig 1.
An instance of the MWEDP problem over a 3-regular random graph of V = 20 and M = 6: examples of solutions of the unconstrained (left) and optimal (right) MWEDP problem are displayed.
In the latter, the purple communication is redirected along a longer path to avoid edge-overlap. The yellow one has two shortest paths of equal length (degeneracy) in the unconstrained case, but once the edge-disjointness is enforced this degeneracy is broken and only one of the two is optimal (right).
Fig 2.
The modified cavity graph G[ij].
Fig 3.
Mapping into a weighted matching problem.
Left: intermediate step where is built. On the leftmost part we show an example of several communications passing along (ij) and exiting along the remaining neighbors k ∈ ∂i \ j. Right: the final step where
is built; the best configuration around node i when the blue current passes through (ij) is given by the minimum weighted matching on the complete auxiliary graph
. Edges red and green represent the best matching, i.e. the configuration where two other communications enter/exit neighbors of i \ j.
Fig 4.
Number of iterations to reach convergence as a function of reinforcement parameter ρ on BRITE graphs AS-BA217 (V = 100) with M = 25,40, blrand1 (V = 500) with M = 125 and mesh 15x15 with M = 22. Inset: the number of accommodated paths Macc is substantially unchanged in the range of parameter values under study.
Fig 5.
Left: Fraction of instances in which convergence standard MP fails (reinforced MP always converged in our experiments). Right: number of iterations for convergence for standard MP and reinforced MP (ρ = 0.002) in case of random graphs of V = 1000 and 〈k〉 = 3 as a function of M/V. Notice how the reinforcement term, besides ensuring convergence, greatly improves the convergence time.
Fig 6.
We plot the performance in terms of Macc/M for (from top to bottom) regular, RER, ER and SF graphs of fixed size V = 103 and average degree 〈k〉 = 3. Error bars are smaller than the size of the symbols. A fast reinforcement parameter ρ = 0.002 for MP reinf was used.
Fig 7.
We plot (left) the relative performance of MSG over MP in terms of total length of the solution paths: y = 100(Lg/LMP − 1). Here Lg and LMP denote the total path lengths calculated with MSG and MP respectively. We use Reg, RER, ER and SF graphs of fixed size V = 103 and average degree 〈k〉 = 3,5,7 (from top to bottom). On the right we report the number of instances where the two algorithms find the same solution in term of Macc/M over 100 realizations.
Fig 8.
We plot 1 − Macc/M (top) and the total length per node L/V (bottom) for Reg (left) snd ER (right) graphs as a function of the scaling variable . We can notice the finite-size effects decreasing with system size leading to the curves corresponding to the biggest graphs V = 8000,10000 to almost superimpose. Note that in the SAT phase the total length grows linearly in log V for all system sizes as expected but in the UNSAT phase the graphs split. Error bars are smaller than point size.