Algorithmic graph theory and its applications

Tınaz Ekim Algorithmic graph theory and its applications Tınaz Ekim tinaz.ekim@epfl.ch Swiss Federal Institute of Technology Recherche Opérationnelle Sud Est (ROSE)

Tınaz Ekim Contents PART I (Applications of graph theory) • Introduction to coloring problems • Some examples of applications • Biprocessor tasks • Frequency assignment • Aircraft scheduling • University timetabling • Sports scheduling • 3D image reconstruction PART II (Generalized coloring problems) • Application : automated storage system • Related results

Tınaz Ekim PART I Coloring problems Stable set : set of vertices pairwise non-adjacent • Min (vertex) Coloring: Coloring vertices of a graph with a minimum number of colors ((G)) such that adjacent vertices don’t receive the same color.  Min number of stable sets covering all vertices • Min (edge) Coloring: Coloring edges of a graph with a minimum number of colors ('(G)) such that adjacent egdes don’t receive the same color. Clique : set of vertices pairwise adjacent

Tınaz Ekim PART I Complexity of coloring problems Min (vertex) Coloring  NP-hard [Karp, 72] Min (edge) Coloring  NP-hard[Holyer, 81] NP-hard : No polynomial time exact algorithm is known, and most probably, there is no such an algorithm ! P=NP or PNP ? (1.000.000 $ question)

Tınaz Ekim PART I Methods to cope with NP-hard problems: • Restriction to particular cases • becomes polynomially solvable (give an exact algorithm), or • remains NP-hard (give the proof) • Approximation with performance guarantee • approximation ratio   quality measure • -approximation algo. |A|  |OPT| (1 if min) • Heuristics to obtain relatively «good» solutions in a reasonable time

Tınaz Ekim i j PART I Biprocessor tasks • n tasks • Each task executed on 2 predefined processors simultaneously • A processor can not work on 2 tasks simultaneously • Min # of periods to execute all the tasks vertex i ↔ processor i ij linked ↔  task to be processed by i and j simultaneously edge color k ↔ set of tasks to be executed at period k '(G)= min # of periods NP-hard but well approximated

Tınaz Ekim PART I Frequency assignment • n base stations • same frequency for close stations  interference • Min  of frequencies vertex i ↔ station i ij linked ↔ possible interference between i and j color ↔ set of stations that can have the same frequency (G) = min # of frequencies G disk unit graph  3-approximation

Tınaz Ekim j i PART I Aircraft scheduling • n flights • Time interval of flight i is (ai,bi) • Min number of aircrafts to be assigned vertex i ↔ flight i ij linked ↔ (ai,bi) overlaps (aj,bj) color ↔ set of flights scheduled to the same aircraft (G)= min number of aircrafts G interval graph  polynomial

Tınaz Ekim 2 1 2 1 3 PART I Aircraft scheduling b a a b e d c e c d • Sort vertices with respect to non-decreasing right endpoints • Greedy coloring with this ordering gives optimal coloring!

Tınaz Ekim PART I Various constraints • Cardinality constraint: each color set can have at most t vertices Limited resource (machine) in scheduling problems Aircrafts: each aircraft can fly at most t times / period → becomes NP-hard in interval graphs [de Werra, Kobler, 03] • Precoloring extension: some vertices are already assigned to some colors Certain jobs are already assigned Aircrafts: some flights are already assigned to some aircrafts → becomes NP-hard in interval graphs [Biro, Hujter, Tuza, 92] • List coloring: restricted set of colors for each vertex Job that can be processed only in certain time periods, or only by certain machines Aircrafts: some flights can only be executed by some aircrafts → at least as difficult as precoloring extension

Tınaz Ekim PART I University timetabling • Students choose courses in a list (time slot + prof.) • Probability pi for course i to be opened • pi = 1 if i is a mandatory course • Objective: min  of rooms • vertex i ↔ course i • ij linked ↔ i and j can not be in the same room • color ↔ set of courses assigned to the same room • Min expected # of rooms: Min Ck [1-(iCk)(1-pi)] • NP-hard even in bipartite graphs [Murat, Paschos, 2006]

Tınaz Ekim S1 1 3 1 3 1 3 S1 S2 S2 S1 2 4 2 4 2 4 S2 Day 2 Day 3 Day 1 PART I Sports scheduling 1 3 2n=4 teams vertex  team edge  match Edge color  matches played at 1 day 2 4 Constraint: each team plays exactly twice in each of the n stadiums(except 1 team) Construction for 2n=2k[de Werra, Ekim, Raess, 05]

Tınaz Ekim PART I 3D image reconstruction • [R. Zenklusen, Master’s thesis, 2005 (OR + Computer Vision)] • Images of the surface  3D reconstruction problem • Usually reduced to 1D problem  oversimplified • 3D reconstruction using graph flows[Roy, 1999]

Tınaz Ekim PART I 3D image reconstruction Approximation of the 3D image  energy minimization Min cut Smoothness + light intensity + consistency  energy function Capacities on the arcs

Tınaz Ekim PART I 3D image reconstruction Max flow = Min s-t cut  3D image

Tınaz Ekim PART II Generalized coloring problems • Min Cocoloring: z(G) = min (p+k such that the vertices of G can be partitioned into p cliques and k stable sets) [Lesniak,Straight 77] • Min Split-coloring: S(G) min (k : such that the vertices of G can be partitioned into k cliques and k stable sets)[Ekim, de Werra, 05] • G is a split graph if its vertex set can be partitioned into a stable set and a clique. z(G)=2 S(G)=2

Tınaz Ekim clique = decreasing subsequence 2 7 3 6 stable set= increasing subsequence 4 1 5 PART II Permutation graphs Given a permutation (N) where N=1, … ,n the permutation graph G=(V,E) corresponding to  is defined as follows: V= 1, … ,n and ijE iff i < j and (i) > (j) 5 1 3 7 6 2 4

7 2 Tınaz Ekim PART II 3 6 label  1/size 4 1 5 Storage Area Storage Area 5 1 3 7 6 2 4 • No return before unloading the charge • Decreasing sizes of items  Increasing labels • Min # of trips along the corridor

7 2 Tınaz Ekim PART II 3 6 Min Split-coloring 4 1 5 Storage Area Storage Area 5 1 3 7 6 2 4

7 2 Tınaz Ekim PART II 3 6 Min Cocoloring 4 1 5 5 1 3 7 6 2 4 Storage Area

Tınaz Ekim PART II Complexity results • Min Split-coloring and Min Cocoloring NP-hard,  differen-tial approximation scheme ((1-)-approximation, for all >0)[Demange, Ekim, de Werra, 06] • Min CocoloringNP-hard in permutation gr. [Wagner, 84] • Min Split-coloringNP-hard in permutation graphs but 2-approximable [Demange, Ekim, de Werra, 06] • Min l-modal NP-hard,  differential approximation scheme • Min Split-coloring and Min Cocoloring P in cacti [Ekim, de Werra, 05], cographs [Demange, Ekim, de Werra, 05], in chordal graphs [Hell et al. 04]. • Min Cocoloring P in L(Bipartite), L(line-perfect graph), Min Split-coloring NP-hard in L(Bipartite) [Demange, Ekim, de Werra, 05]

Tınaz Ekim Conclusions • Several open questions related to permutations • On-line models for robotics problems • Telecommunication (network design, network security), production systems (scheduling, stock management), transportation (distribution, routing), computer vision, robotics, etc.  Combinatorial optimization problems • New challenges for both theoretical and algorithmic aspects

S(G)=2

Algorithmic graph theory and its applications