Wave propagation in frequency domain: the Helmholtz equation. Inspired by both Chopard’s ParFlow method [1] and the MR-FDPF method developed at INSA Lyon [3], the Helmholtz equation predicts the radio wave propagation in a deterministic manner, considering physical effects such as diffraction, self-interference or corridor effect. To begin with, we worked with a 2D environment. To do so, consider - as done in [3] - the classical propagation equation of a scalar wave: ∆u(x, y, t)−àǫ∂t2u(x, y, t) = −s(x, y, t), where s(x, y, t) is a source term, ǫ(x, y) is the local electric permittivity andà(x, y) the local permeability. The physical values ǫ andàdepend on the material : they represent the architecture of the floor.
Applying the Fourier transform to the wave equation eliminates the time differential, and adding a diffusive term permits not to overestimate the reflections on the walls and on the boundary (σ 6= 0, whereσ is the electric conductivity). Eventually, it yields in the following complex Helmholtz equation : ∆Ψ + (ω2àǫ−iωàσ)Ψ =−S(x, y, ω)
Finite difference scheme: The Helmholtz equation is simulated through a finite difference scheme which consists in discretizing the rectangle [0, L]×[0, l] in a grid [|0, Nx|]×[|0, Ny|], using the same step ∆x for both dimensions. ΨjNx+i is an approximation of Ψ(i∆x, j∆x). The classic discretization of the Laplacian leads to the following sparse linear system :
∀k∈[1, NxNy],Ψk+1+ Ψk−1+ Ψk+Nx+ Ψk−Nx+ (β2n2k−4−i(∆x)2ωαk)Ψk =Fk with the following conventions: FjNx+i = −(∆x)2S(i∆x, j∆x, ω), β = ω∆c0x, αk = àkσk, ∀i /∈ [1, NxNy]Ψi = 0. This sparse linear system can be efficiently solved by using an LU factoriza- tion of the system matrix. In practice, the Python library SuperLU [2] was used.
Time complexity: Thanks to theSuperLU library our simple discretization method achieves the same complexity as the MR-FDPF method [4]. For a given 2D map, a pre-computation is required in time O(Nx3) to factorize the system. The computation of the field created by a source in any given point is inO(Nx2log(Nx)).
From 2D to 3D: To make the model fit reality, it is crucial to model indoor radio wave propagation in 3D environment. Trivially increasing the number of voxels in the methodol- ogy sketched above would yield an excessive complexity increase. Thus, the 2.5D empirical approach presented in [4] is more relevant to deal with 3D, it relies on the projections of the field in the floork to compute the field in the floork+ 1, using on of these alternatives :
• Field Projectingmodels the 3D propagation by projecting the field map through the roof with an attenuation coefficient depending on the nature of the ceiling.
• Source Projecting consists in projecting the source (of the floork) in the floork+ 1 with an attenuation factor and then in computing the 2D propagation in the floork+ 1 from this virtual source.
• A combination of the two latter alternatives.
3 Mathematical Programming formulation
The data from the simulation enabled to build a mixed-integer linear program - with stochastic constraints - which ensures wireless connection all over the building at minimum cost and takes into account wireless demand at each point of the building. To build this network, our model considers two types of equipment: wired access points (AP) and wireless repeaters. Both types of devices have different costs and capacities. Considering a 2D or 3D grid,Vclients defines a set representing the points of the grid to cover and Vcand a set representing eligible positions for APs or repeaters. A point which has to to be covered and which is also a potential AP position is duplicated, therefore the union of both sets is empty. V =Vclients∪Vcand. V∗ = V ∪ {r} represents all the vertices of the grid includingr the root of the graph, to which all APs have to be connected.
Parameters:
• The power gain matrixP = (pi,j)(i,j)∈V2 computed by the data generation model;
• (CiA)i∈Vcand such that CiA >0 is the installation cost of an AP ati;
• (CiR)i∈Vcand such that CiR >0 is the installation cost of a repeater ati. CiR < CiA;
• γA > 0 and γR >0 are the maximum communication rate that an AP and a repeater can handle;
• The vector (Di)i∈Vclients with (Di≥0) represents the bandwith demand at pointi. These random variables are not independent. Moreover we set : ∀i∈Vcand, Di= 0;
• ∀j∈V, pjmax=maxi∈Vcandpi,j
• nmax>0 a maximal noise level which is the only control parameter;
• We define a capacity matrix W = (wi,j)(i,j)∈(V∗)2 : – ∀(i, j)∈V ×Vcand, wi,j=Blog(1 + npi,j
max)>0 : maximal data rate from i to j.
– ∀(i, j)∈(V ×Vclients)∪({r} ×V∗)∪(Vclients× {r}), wi,j = 0
– ∀(i, j)∈Vcand× {r}, wi,j=M whereM a real number such thatPi∈VclientsDi ≤M almost surely;
Thus we consider the capacited oriented graph G= (V∗,(V∗)2, W), and the purpose is to build a flow on Gfrom the client points to the root by selecting relay nodes.
Decision variables:
• (Ai)i∈Vcand ∈ {0,1}Vcand : indicates the presence of an AP at i;
• (Aci)i∈Vcand,c∈C ∈ {0,1}Vcand×C : indicates the presence of an AP at i emitting on the channel c;
• (Ri)i∈Vcand ∈ {0,1}Vcand : indicates the presence of a repeater at i;
• (Rci)i∈Vcand,c∈C ∈ {0,1}Vcand×C : indicates the presence of a repeater ati emitting on the channel c;
• (fi,j)(i6=j)∈(V∗)2 ∈R(V+∗)2 : the packet flow fromi to j.
• (fi,jc )(i6=j,c)∈V2×C ∈RV+2×C : the packet flow from i toj on channelc.
Objective: minimize the installation cost: Pi∈VcandAiCiA+RiCiR Constraints:
• Link capacity: ∀(i6=j)∈(V∗)2, fi,j ≤wi,j;
• Kirchhoff law : ∀i∈V,Pj∈V∗\{i}fi,j−fj,i =Di
• AP or repeater: ∀i∈Vcand, Ai+Ri ≤1;
• Only APs can be directly wired to the root: ∀i∈Vcand, fi,r ≤M Ai ;
• Machine capacity: ∀j ∈Vcand,Pi∈V fi,j ≤AjγA+RjγR;
• Unique emission channel: ∀i∈Vcand, Ai=Pc∈CAci, Ri=Pc∈CRci;
• Flow decomposition : ∀(i6=j)∈V2, fi,j=Pc∈Cfi,jc
• Channel selection between client and candidate: ∀(i, j, c) ∈ Vclients×Vcand× C, fi,jc ≤ wi,j(Acj+Rcj);
• Channel selection between candidates: ∀(i6=j, c)∈Vcand×Vcand×C, fi,jc ≤wi,j(Aci+Rci)
• Noise constraint between a client and a candidate: ∀(i, j, c)∈Vclients×Vcand× C: X
k∈Vcand\{j}
pk,i(Ack +Rck)≤(pimax+nmax)(1− fi,jc
wi,j) +nmaxfi,jc wi,j
• Noise constraints between candidates: ∀(i, j, c)∈Vcand×Vcand× C: X
k∈Vcand\{i,j}
pk,i(Ack+Rkc)≤(pimax+nmax)(1− fj,ic
wj,i) +nmaxfj,ic wj,i
4 Solution and perspectives
Simulation’s performance TAB.1 below contains computation times obtained with a pro- cessor Intel(R) Xeon(R) CPU E3-1271 v3 @ 3.60GHz for several simulations. The length and the width of the building correspond to a discretization step of 3cm, equivalent to the quarter of the wavelength for the WiFi 2.4GHz standard.
Length Width Nx Ny Factorisation time Resolution time (one source)
18m 12m 600 400 5s 0.07s
30m 10,5m 1000 350 8s 0.15s
30m 24m 1000 800 46s 0.4s
60m 30m 2000 1000 141s 1.2s
TAB. 1: Building dimensions, grid dimensions and computation times
First attempt to solve the problem For this first attempt we limit the analysis to a deterministic demand : the vectorD is constant and thus we get a classic MILP formulation.
We encoded it with AMPL and solved it for different instances using CPLEX solver. Below are the computation times obtained with a processor Intel(R) Xeon(R) CPU E3-1271 v3 @ 3.60GHz for several instance sizes :
|Vclients| |Vcand| Computation time
2 6 1s
5 10 6s
10 20 40s
20 60 4500s
TAB. 2: Number of clients and candidates, computation time
Perspectives Our current implementation considers a deterministic WiFi demand on the building, yet a statistical approach would give a more relevant deployment for real instances.
In that case, we would need to choose between robust or stochastic optimization.
5 Figures
FIG. 1: 2.4GHz WiFi field in a building with source at different positions (red dot).
References
[1] B. Chopard, P. O. Luthi, and J. F. Wagen. Lattice boltzmann method for wave propagation in urban microcells. IEE Proceedings - Microwaves, Antennas and Propagation, 144(4):251–
255, Aug 1997.
[2] James W. Demmel, Stanley C. Eisenstat, John R. Gilbert, Xiaoye S. Li, and Joseph W. H.
Liu. A supernodal approach to sparse partial pivoting. SIAM J. Matrix Analysis and Applications, 20(3):720–755, 1999.
[3] Jean-Marie Gorce, Katia Jaffrès-Runser, and Guillaume De La Roche. The Adaptive Multi- Resolution Frequency-Domain ParFlow (MR-FDPF) Method for Indoor Radio Wave Prop- agation Simulation. Part I : Theory and Algorithms. Technical Report RR-5740, INRIA, November 2005.
[4] Guillaume De La Roche. Simulation de la propagation des ondes radio en environnement multi-trajets pour l’etude des reseaux sans fil. PhD thesis, INSA Lyon, 2007.
Influence Maximization in Social Networks under Deterministic Linear Threshold Model
Furkan Gursoy1, Dilek Gunnec2
1 Dept. of Management Information Systems, Bogazici University, 34342, Istanbul, Turkey furkan.gursoy@boun.edu.tr
2 Dept. of Industrial Engineering, Ozyegin University, 34794, Istanbul, Turkey dilek.gunnec@ozyegin.edu.tr
Abstract
We define the new Targeted and Budgeted Influence Maximization under Determin- istic Linear Threshold Model problem by extending the original influence maximization problem to a targeted version where nodes might carry heterogeneous profit values, and to a budgeted version where nodes might carry heterogeneous costs for becoming seed nodes.
As a solution to this problem, we develop a novel and scalable general algorithm which utilizes a set of alternative methods for different operations: TArgeted and BUdgeted Potential Greedy (TABU-PG) algorithm.
TABU-PG works in an iterative and greedy fashion where nodes are compared at each iteration and the best one(s) are chosen as seed. The main idea behind TABU-PG is to invest in potential future gains which are hoped to be materialized at later iterations.
Alternative methods are provided for calculating potential gain, and for comparing nodes.
In comparing nodes, we propose a hybrid model which considers both gain and effi- ciency. In calculating potential gains, we propose methods which dynamically assign suitable weights to potential gains based on remaining budget. We also propose a new method which ignores the potential gains which are results of partial influences under a parameterized ratio. Moreover, we equip TABU-PG with novel scalability methods which reduces runtime by limiting the seed node candidate pool, or by selecting more nodes at each iteration; trading-off between runtime and spread performance. In addition, we suggest new data generation methods for influence weights on links; and threshold, profit, and cost values for nodes which better mimics the real world dynamics.
Extensive computational experiments with 8 different dataset on 4 real-life networks (Epinions, Acedemia, Pokec, and Inploid) show that TABU-PG heuristics perform signif- icantly better than benchmark heuristics. Moreover, runtime can be reduced with very limited reduction in final influence spread.
Keywords: Influence Maximization, Social Networks, Diffusion Models, Targeted Market- ing, Greedy Algorithm.
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From Proofs to Programs, Graphs and Dynamics. Geometric perspectives on computational complexity
Thomas Seiller
Laboratoire d’Informatique de Paris Nord, Université de Paris 13 and Sorbonne Paris Cité CNRS (UMR 7538), 93430 Villetaneuse, France
seiller@lipn.fr
Abstract
The current state of the art in the field of complexity theory is a demonstrated lack of proof methods against problems still open. The combination of three separate results, called barriers (Relativisation, Natural Proofs and Algebrization), implies that none of the currently known proof methods for separation will successfully settle the remaining open problems. A single research program – Geometric Complexity Theory (GCT) – is considered viable by the community. However, according to its initiator and major contributor K. Mulmuley, GCT will not provide new results within our lifetimes; recent results have moreover closed the easiest path to GCT. As a consequence, complexity theory is in dire need of new tools and methods as such advances should require “funda- mentally new methods” to paraphrase S. Aaronson and A. Widgerson. This talk will be about how such methods may be founded upon some recent developments in logic, and more precisely some specific models of proofs introduced under the name “Interaction Graphs”.
The interplay between logic and computational complexity has been the subject of research for more than 50 years, but it has arguably failed to provide insights on the clas- sification problem. Nevertheless, it has shown how logic is tightly bound to computation, clearly circumscribing the limits of the different approaches. The framework of Interac- tion Graphs, although taking its roots in logic, offers a mathematical model of programs that bypasses these limits and accounts for subtle aspects of computation. Moreover, it unveils deep connections with methods from geometry and dynamical systems that one may hope to exploit to enable potent proof methods from mathematics to be used by researchers against open problems in complexity theory.
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On some tractable constraints on paths in graphs and in proofs
Lê Thành Dũng Nguyễn12
1 École normale supérieure, Paris Sciences et Lettres, Paris, France le.thanh.dung.nguyen@ens.fr
2 LIPN, UMR 7030 CNRS, Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France
Abstract
We show that trails avoiding forbidden transitions and rainbow paths for complete multipartite color classes can be found in linear time, whereas finding rainbow paths is NP-complete for any other restriction on color classes. For the tractable cases, we also state new structural properties equivalent to Kotzig’s theorem on bridges in unique perfect matchings. Finally, we mention some connections with proof nets in linear logic and combinatorial proofs (“proofs without syntax”) for classical propositional logic.
Keywords : Perfect matchings, forbidden transitions, properly colored paths, rainbow paths.
1 Introduction
Many problems which consist of finding a path or trail1 under some constraints between two given vertices are equivalent to the augmenting path problem for matchings, and thus tractable. Some of these problems have associated “structure from acyclicity” theorems which were shown [13] to be equivalent to Kotzig’s theorem on the existence of bridges in unique2 perfect matchings (cf. [13, Theorem 1]): the absence of constrained cycles or closed trails entails the positive existence of some structure in the graph.
Our results here consist of finding new members of this family of constraints on paths which are equivalent in a certain sense, and excluding other constraints through NP-hardness results.
We also bring to attention the fact that this family has a representative in proof theory.
Edge-colored graphs From an assignment of colors to the edges of a graph, one can define either local or global constraints:
• In a properly colored(PC) path (see [2, Chapter 16]) or trail (see [1]), consecutive edges must have different colors. Both can be found in linear time by reduction to augmenting paths, and conversely augmenting paths are a special case of both these problems. The structural result for PC cycles is Yeo’s theorem on cut vertices separating colors [2, §16.3].
• In a rainbow (also called heterochromatic or multicolored) path,all edges have different colors. The subject of rainbow connectivity has been an active area of research recently, but the problem is NP-complete [4] in the general case.
For rainbow paths, we investigate whether restrictions on the shape of the color classes – that is, the subgraphs induced by all edges of a given color – make the problem tractable, and we establish that there is a single case which is not NP-hard:
1Following a common usage (see e.g. [2, Section 1.4]), a path is a walk without repeating vertices and a trailis a walk without repeatingedges; acycle(resp.closed trail) is a closed walk without repeating vertices (resp. edges). Paths (resp. cycles) are trails (resp. closed trails), but the converse does not always hold.
2This is indeed an acyclicity condition: recall that a perfect matching is unique if and only if it admits no alternating cycle.
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Theorem 1. Let A be a class of graphs without isolated vertices3. The rainbow path problem for graphs whose color classes are all in A can be solved in linear time if all graphs in A are complete multipartite, and is NP-complete otherwise.
The first case is part of our family of equivalent constraints, and the associated structural theorem is as follows:
Theorem 2. Let Gbe an edge-colored graph whose color classes are complete multipartite. If G has no rainbow cycle, then there exists a color c such that for all c-colored edges (u, v), u and v are in different connected components after removing the color class of c.
Forbidden transitions A very general notion of local constraints is to simply forbid some pairs of edges from occuring consecutively in a path. We take the following definition from [12].
Definition 1. Let G = (V, E) be a multigraph. A transition graph for a vertex v ∈ V is a graph whose vertices are the edges incident to v. A transition system on G is a family T = (T(v))v∈V of transition graphs.
A path (resp. trail) v1, e1, v2. . . , ek−1, vk is said to be compatible (or avoiding forbidden transitions) if fori= 1, . . . , k−14, ei andei+1 are adjacent inT(vi+1).
That is, the edges of the transition graphs specify the allowed transitions. Finding a com- patible path has been proven to be NP-complete [12]. However, the question for compatible trails does not seem to have been asked before in its full generality. We show that:
Theorem 3. Finding a compatible trail can be done with a time complexitylinearin the number of allowed transitions (thus, in at most quadratic time in the size of the graph).
Theorem 4 (“Structure from acyclicity”). Let G be a multigraph with transition system T, with at least one edge. If, for all vertices v in G, the transition graph T(v) is connected, and Ghas no closed trail compatible with T, then Ghas a bridge.
Corollary 1(New5 proof of [1, Theorem 2.4]). LetGbe an edge-colored graph such that every vertex of G is incident with at least two differently colored edges. Then, if Gdoes not have a PC closed trail, then Ghas a bridge.
2 The edge-colored line graph
A key ingredient in the aforementioned results is a kind of line graph construction mapping graphs with forbidden transitions to edge-colored graphs.
Definition 2. Let G = (V, E) be a multigraph and T be a transition system on G. The EC-line graph LEC(G, T) is formed by taking the line graph of G, coloring its edges so that the clique corresponding tov is given the colorv (using the vertices ofGas the set of colors), and deleting the edges corresponding to forbidden transitions.
Formally,LEC(G, T) is defined as the graph with vertex setE and edge setF =⊔v∈V T(v), equipped with an edge coloring c : F → V with values in V: for f ∈ F, c(f) is the unique vertex such thatf ∈T(c(f)).
Proposition 1. Let Gbe a multigraph with transition system T, and s6=t be vertices of G.
The compatible paths betweensand tcorrespond bijectively torainbow paths inLEC(G, T) between some vertex of∂(s)and some vertex of∂(t)which do not cross edges with color sort.
Similarly, the compatible trailsbetweensandtwhere neithersnortappear as intermediate vertices correspond bijectively toPC pathsinLEC(G, T)between some vertex of∂(s)and some vertex of∂(t) which do not cross any edge with color sor t.
3Indeed, a color class, which is an edge-induced graph, cannot have isolated vertices.
4For a cycle (resp. closed trail), we must also requireek−1 ande1 to be adjacent inT(v1) =T(vk).
5The original proof applies Yeo’s theorem to a construction which does not generalize to forbidden transi- tions, but provides a trail-finding algorithm in linear timein the size of the graph.
Theorems 3 and 4 immediately follow from the second half of this proposition together with the known results on PC paths. However, to get the hardness result for rainbow paths, in addition to the EC-line graph, we need to reuse the proof techniques from [12] and [4], in particular a characterization of complete multipartite graphs by excluded vertex-induced subgraphs [12, Lemma 7]. As for the first half of Theorem 1, it uses the fact that one can retrieve the vertex partition of a complete multipartite graph in linear time, for instance by computing its cotree [5].
3 Constrained cycles in logic
In a recent work [9], we showed that thecorrectnessof aproof net – a graph-like representation of a proof in linear logic[6] – is equivalent to the uniqueness of a given perfect matching, and is therefore part of our family of equivalent problems. Thus, it can be decided in linear time, and the associated structural property is the key lemma in the proof of the “sequentialization theorem”, an inductive characterization of the set of correct proof nets which mirrors exactly the inference rules of linear logic.
One direction of the equivalence, from proof nets to perfect matchings, had been established previously by Retoré [11, §1]6. His reduction can be understooda posteriori as a composition of constructions on edge-colored graphs: it amounts to equipping a proof net with a transition system, taking the EC-line graph introduced above, and applying a known reduction from edge-colored graphs with chromatic degree ≤2to perfect matchings [8]7.
Let us give a rough presentation of proof nets in graph-theoretic terms. A proof net may be seen as the syntax tree of a propositional formula, with ∧ and ∨nodes and literals at the leaves, together with additional edges between the leaves pairing together opposite literals.
The syntax tree may be interpreted as the cotree of a cograph whose vertices are the literals, as usual, see e.g. [3]. This leads to a restatement of correctness, also due to Retoré [11, §2].
Definition 3. A cographic proof is an pair of graphs (G, M), G being a cograph and M a 1-regular graph, with the same set of vertices.
Avicious circlein (G, M) is achordlesscycle in8 G∪M which alternates between edges in Gand edges inM. A cographic proof iscorrect if it contains no vicious circle.
A proof net is correct if and only if the corresponding cographic proof (with the 1-regular graph representing the pairing of the leaves) is correct in the sense above. Note that vicious circles are not merely properly colored cycles for the natural 2-edge-coloring of the cographic proof, because of the additional chordlessness condition.
Finally, let us mention that cographic proofs also have applications outside of linear logic.
Indeed, they have been used to define “proofs without syntax” for classical propositional logic:
Hughes’scombinatorial proofs[7] are graph homomorphisms (with additional properties) from some correct cographic proof to the cograph of the classical formula being proven, and this gives a sound and complete proof system. The tractability of our family of constraints on cycles ensures that proofs are checkable in polynomial time.
6This was the first indication of a connection between linear logic and unique perfect matchings. Let us mention as well that in an earlier attempt to connect linear logic with graph theory [10, Chapter 2], Retoré proved a weaker version of the structural theorem for rainbow acyclic graphs (it requires the color classes to be completebipartiteinstead of complete multipartite).
7This paper only defines the reduction for 2-edge-colored graphs, but the required generalization is straight- forward. Note also that the two last steps give a direct reduction from compatible trails to perfect match- ings. Although we have not managed to find it in the literature, there is at least one other place where it occurs implicitly, which also inspired us: a solution to an algorithmic puzzle by Christoph Dürr, see http://tryalgo.org/en/matching/2016/07/16/mirror-maze/.
8ByG∪M we mean the graph whose edges are the union of those inGandM, on the common vertex set.
This union may result in a multigraph with parallel edges.