9th Cologne-Twente Workshop on Graphs and Combinatorial Optimization Cologne, Germany, May 25-27, 2010 Extended Abstracts Ulrich Faigle, Rainer Schrader, Daniel Herrmann (eds.) hh Table of Contents Pablo Adasme, Abdel Lisser Semidenite Programming for Stochastic Wireless OFDMA Networks 1 Edoardo Amaldi, Claudio Iuliano On finding a minimum weight cycle basis with cycles of bounded length 5 Stephan Dominique Andres The classification of B-perfect graphs 9 Avraham Trahtman, Tomer Bauer, Noam Cohen Linear Visualization of a Road Coloring 13 Frank Baumann, Christoph Buchheim, Frauke Liers Exact Bipartite Crossing Minimization under Tree Constraints 17 Pietro Belotti, Sonia Cafieri, Jon Lee, Leo Liberti On the convergence of feasibility based bounds tightening 21 Andrea Bettinelli, Alberto Ceselli, Giovanni Righini Branch-and-price for the multi-depot pickup and delivery problem with heterogeneous fleet and soft time windows 25 Matthijs Bomhoff, Bodo Manthey Bisimplicial Edges in Bipartite Graphs 29 Valentina Cacchiani, Alberto Caprara, Paolo Toth A Heuristic Algorithm for the Train-Unit Assignment Problem 33 Alberto Ceselli, Roberto Cordone, Yari Melzani, Giovanni Righini Optimization algorithms for the Max Edge Weighted Clique problem with Multiple Choice constraints 37 Irene Charon, Olivier Hudry A branch and bound method for a clique partitioning problem 43 Alberto Costa, Pierre Hansen, Leo Liberti Static symmetry breaking in circle packing 47 Raphael Machado, Celina de Figueiredo, Nicolas Trotignon Chromatic index of chordless graphs 51 I Satoru Fujishige , Britta Peis Lattice Polyhedra and Submodular Flows 55 Ismael Gonzalez Yero, Juan A. Rodriguez-Velazquez, Magdalena Lemanska On the partition dimension of Cartesian product graphs 61 Stefano Gualandi, Federico Malucelli, Domenico Sozzi On the Design of the Fiber To The Home Networks 65 Stefano Gualandi, Francesco Maffioli, Claudio Magni A branch-and-price approach to the k-Clustering Minimum Biclique Completion Problem 69 Guignard Adrien The game chromatic number of 1-caterpillars 73 Jochen Harant On Hamiltonian cycles through prescribed edges of a planar graph 79 Ararat Harutyunyan Some bounds on alliances in trees 83 Johannes Hatzl The Inverse 1-Median Problem in Rd with the Chebyshev-Norm 87 Shahadat Hossain, Trond Steihaug Graph Models and their Efficient Implementation for Sparse Jacobian Matrix Determination 91 Gerold Jaeger An Effective SAT Encoding for Magic Labeling 97 Imed Kacem New Fully Polynomial Time Approximation Scheme for the makespan minimization with positive tails on a single machine with a fixed non-availability interval 101 Enver Kayaaslan On Enumerating All Maximal Bicliques of Bipartite Graphs 105 Walter Kern, Jacob Jan Paulus A tight analysis of Brown-Baker-Katseff sequences for online strip packing 109 Stefanie Kosuch, Marc Letournel, Abdel Lisser On a Stochastic Knapsack Problem 111 II Dmitrii Lozovanu, Stefan Pickl Determining Optimal Stationary Strategies for Discounted Stochastic Optimal Control Problem on Networks 115 Bodo Manthey, Kai Plociennik Approximating Independent Set in Semi-Random Graphs 119 Jannik Matuschke Lattices and maximum flow algorithms in planar graphs 123 Vahan Mkrtchyan, Eckhard Steffen Maximum ∆-edge-colorable subgraphs of class II graphs 129 Petros Petrosyan, Ani Shashikyan, Arman Torosyan Interval total colorings of bipartite graphs 133 Val Pinciu Pixel Guards in Polyominoes 137 Rija Erves, Janez Zerovnik Mixed connectivity of Cartesian graph products and bundles 141 Maja Rotovnik, Janez Zerovnik Wide - sense nonblocking logd(N; 0; p) networks 145 Oliver Schaudt Efficient total domination 149 Ingo Schiermeyer Progress on rainbow connection 153 Joachim Spoerhase An Optimal Algorithm for the Indirect Covering Subtree Problem 157 Cynthia Wyels, Maggy Tomova Radio Labeling Cartesian Graph Products 163 Ping-Ying Tsai Cycle embedding in alternating group graphs with faulty vertices and faulty edges 169 Vera Weil On Reed's Conjecture in Triangle-Free Graphs 173 III Appendix Matjaz Konvalinka, Igor Pak Complexity of O'Hara's algorithm Appendix IV hh hh Semidefinite Programming for Stochastic Wireless OFDMA Networks Pablo Adasme a,1, Abdel Lisser a,2 aLaboratoire de Recherche en Informatique, Universite´ Paris-Sud XI, Baˆtiment 490, 91405, Orsay Cedex France Key words: Risk modeling, stochastic programming, semidefinite programming, resource allocation in OFDMA. 1 Introduction Resource allocation such as maximizing link capacity or minimizing power con- sumption in a wireless OFDMA network are commonly formulated as mathe- matical programs [1]. These programs usually involve random variables in the input data. In this paper, we propose a (0-1) stochastic quadratic formulation. The study is made on the basis of an OFDMA quadratic model [4] in which a probabilistic constraint based approach is considered [2]. Then, a semidefinite programming (SDP) relaxation is derived to solve the stochastic quadratic model. The paper is organized as follows: Section 2 presents the stochastic quadratic formulation. Section 3 presents the SDP relaxation. Finally, section 4 concludes the paper. 2 Probabilistic formulation We consider an OFDMA network composed by a base station (BS) and several mobile users. The BS has to assign a set of N sub-carriers to a set of K users using a modulation size of c 1,...,M bits in each sub-carrier. The goal ∈ { } is to minimize the total power consumption in the network. We consider the following probabilistic constrained model [4]: 1 Email: [email protected] 2 Email: [email protected] CTW2010, University of Cologne, Germany. May 25-27, 2010 K N M c SQIP0: min xk,n,yn,c k=1 n=1 c=1 Pk,nxk,nyn,c (1) { } N M st: P n=1 xk,n Pc=1 c Pyn,c P Rk (1 αk), k (2) { · ≥ } ≥ − ∀ K h i kP=1 xk,n 1P n (3) M ≤ ∀ P yn,c 1 n (4) c=1 ≤ ∀ x ,y 0, 1 (5) Pk,n n,c ∈{ } Here, the objective function represents the total power consumption. The first constraint corresponds to a chance constraint where αk is the risk to be taken for each user k. In this model, we consider separated chance constraints. The second constraint imposes that each sub-carrier should be assigned to only one user at a time while the third one imposes that each sub-carrier must use one integer modulation size. The decision variables are given by xk,n and yn,c, respectively. We assume that Rk are random variables with joint probability distribution H. Let us consider the case where H is concentrated in the fi- nite number of points also called scenarios Rk = (rk,1, ..., rk,l, ..., rk,Lk ) with probabilities p such that Lk p =1,p 0, k. k,l l=1 k,l k,l ≥ ∀ P Then, the problem (1)-(5) can be reformulated as follows [2]: K N M c SQIP1: min xk,n,yn,c k=1 n=1 c=1 Pk,nxk,nyn,c (6) { } st: N x M Pc y P Pr l Γ , k (7) n=1 k,n c=1 · n,c ≥ k,l ∀ ∈ k ∀ h i Pl Γ pk,l P1 αk k (8) ∈ k K ≥ − ∀ Pk=1 xk,n 1 n (9) M ≤ ∀ P yn,c 1 n (10) c=1 ≤ ∀ x ,y 0, 1 (11) Pk,n n,c ∈{ } Constraints (8) mean that we have to choose a subset Γk of scenarios such that the sum of the probabilities of this subset is greater than (1 αk). For this subset, the bit rate constraints will be active and valid, whereas− for the scenarios not in this subset, the constraints are not activated. This problem can be reformulated by introducing the auxiliary binary variable ϕ for each observation l = 1 : L , k as follows: k,l k ∀ 0 if l Γk ϕk,l = ∈ (12) 1 otherwise This yields the following problem: K N M c SQIP2: min xk,n,yn,c,ϕk,l k=1 n=1 c=1 Pk,nxk,nyn,c (13) { } P P P 2 st: N x M c y r ϕ k, l = 1 : L (14) n=1 k,n c=1 · n,c ≥ k,l −M k,l ∀ k PLk hP i l=1 pk,lϕk,l αk k (15) K ≤ ∀ Pk=1 xk,n 1 n (16) M ≤ ∀ P yn,c 1 n (17) c=1 ≤ ∀ x ,y ,ϕ 0, 1 (18) Pk,n n,c k,l ∈{ } where is an arbitrary number such that M max rk,l +1 (19) M ≥ k,l { } This problem is a quadratic optimization problem with binary variables. This quadratic problem is NP-hard, and so is its stochastic formulation. In this case, we seek lower bounds using strong relaxations, namely SDP relaxations. 3 Semidefinite relaxation In order to write a SDP relaxation for SQIP2, we define the (0-1) vector zT = (x1,1, , x1,N , , xK,1, , xK,N ,y1,1, ,y1,M , ,yN,1, ,yN,M ,ϕ1,1, , ϕ ,··· ,ϕ ,··· ,ϕ ···). Then, let Z···be a symmetric··· positive··· semidefinite··· 1,L1 ··· K,1 ··· K,Lk matrix defined as: zzT z Z = 0 (20) zT 1 We can construct symmetric matrices for the objective function in (13), U P k,l for constraints in (14) and Vk for constraints in (15). We propose the following SDP relaxation for SQIP2: SSDP2 : min T race( Z) (21) Z P st: T race(U Z) r k, l = 1 : L (22) k,l ≥ k,l ∀ k T race(V Z) α k (23) k ≤ k ∀ T race([ex ][ex ]T Z) 1 n (24) n n ≤ ∀ T race([ey ][ey ]T Z) 1 n (25) n n ≤ ∀ T race(ζc Z) 0 k,n,c (26) k,n ≥ ∀ diag(zzT )= z (27) Z 0 (28) 3 In this model, [exn] and [eyn] are coefficient vectors for constraints in (16) and (17) according to vector z. Thus, the rank-1 matrices we construct with these vectors are used to strength our SDP relaxation [3]. The symmetric matrices ζc for all k,n,c are used to have positive values in matrix Z only in the k,n { } positions where i,j,i<j is positive, this is, in the entries of where we {P c } P put the elements Pk,n,k,n,c from (13).