ARCHEOLOGICAL SERIATION AND INTERVAL GRAPHS By Pranathi Reddy Tetali

Outline Problem statement Graph Construction Relation to graph problem NP-Hard problem Special Properties Depicting graph solution Comments

Problem Statement Archeology seriation is the attempt to place a set of items in their proper chronological order. The problem also called sequence dating is to figure out the time relationships between set of artifacts, found in graves and the time intervals during which they were in use.

Problem statement It involves mapping all the artifacts found in each grave to the corresponding time duration. This problem has much in common with interval graphs and consecutive 1s property of incidence matrices.

Graph Construction Assumptions: If two different artifacts occurred together in the same grave, then their time periods must have overlapped. Since number of graves was large, if time periods overlapped then the artifacts appeared together in some graves.

Graph Construction Consider 6 artifacts: a,b,c,d,e,f The adjacency matrix tells which pairs of artifacts occurred together in graves. 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑎 0 1 0 0 0 0 𝑏 1 0 1 1 1 0 𝑐 0 1 0 1 1 0 𝑑 0 1 1 0 1 0 𝑒 0 1 1 1 0 1 𝑓 0 0 0 0 1 0

Graph Construction The problem now is to represent them in chronological order. This can be done by permuting adjacency matrix to incidence matrix with consecutive 1’s property. However, this method produces many correct permutations. To limit the number of correct orders we will use graph theory of interval graphs.

Graph Construction Let G be a graph whose vertices represent artifacts and edges correspond to pairs of artifacts that appear together in same grave.

Relation to a graph problem This Real world problem is converted to interval graph problem. The problem in graphical terms can be described as- “To obtain an interval model with all the adjacent vertices intersecting while the non adjacent vertices are apart.

Relation to a graph problem From the graph we construct a set of intervals on the real line corresponding to time periods during which the artefacts were in use. Artefacts correspond to overlapping intervals and sets of artifacts correspond to overlapping intervals.

Relation to a graph problem The interval model obtained from the graph:

An NP- Hard problem It takes many years to determine all possible permutations and obtain correct order. The problem is solvable in polynomial time on interval graphs that is NP-complete while it is NP-Hard in general case.

Special Properties The clique matrix of an undirected graph is an incidence matrix having maximal cliques as rows and vertices as columns. Corollary: An undirected graph G is an interval graph if and only if the clique matrix of G has the consecutive ones property for columns.

Special Properties Given a finite set X and a collection F of subsets of X, the consecutive arrangement problem is to determine whether or not there exists a permutation π of X in which the elements of each subset S ∈ F appear as a consecutive subsequence of π. X is the set of maximal cliques of G. F = {S (v)│v ∈ V}, S(v) is set of maximal cliques of G.

Special Properties Algorithm calculates 𝜋 (F): 1: procedure consecutive (X ,F, 𝜋) 2: let 𝜋 be the set of all permutations of X 3: for all S ∈ F do 4: remove from 𝜋 those permutations in which the elements of S do not occur as a subsequence 5: end procedure Alternatively we can use PQ-Tree representation.

Special Properties Theorem: Interval graphs can be recognized in O(n+m) time. Moreover, if G is an interval graph, then there is an algorithm taking O(n+m) time to construct a proper PQ-tree T such that consistent(T) is the set of orderings of the maximal cliques of G in which, for every vertex v of G, the maximal cliques containing vertex v occur consecutively.

Special Properties Some other properties that define interval graphs: It is chordal and its complement G is a comparability graph. It contains no induced 𝐶 4 and G is transitively orientable. It is chordal and contains no asteroidal triple (AT).

Depicting Graph Solution The interval model directly displays the chronological order. From the interval graph we get, the following intervals

Comments The interval graph is used to optimize the seriation process. It is not simple in practice as few different arrangements of intervals are possible. Additional information is required to exactly determine one order from the few permutations.

References Kendall, D. (1969). INCIDENCE MATRICES, INTERVAL GRAPHS AND SERIATION IN ARCHAEOLOGY. PACIFIC JOURNAL OF MATHEMATICS, 28(3), 565- 570. Retrieved October 7, 2014, from http://projecteuclid.org/download/pdf_1/e uclid.pjm/1102983306

References Interval Graph Isomorphism. (n.d.). Retrieved October 7, 2014, from http://www.lsi.upc.edu/~valiente/graph-00- 01-d.pdf Mertzios, G. (2008). A matrix characterization of interval and proper interval graphs. Applied Mathematics Letters, 21, 332- 337. Retrieved October 7, 2014, from https://community.dur.ac.uk/george.mertzios/ papers/Jour/Jour_NIR_SNIR.pdf

Thank You!!!

Any Queries?