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**Abstracts**

**Charlie Colbourn**, Arizona State University

*Graph Decompositions and Optimal Grooming*

Given a multigraph $G=(V,E)$ and a set of graphs ${\cal H} = \{H_1,\dots,H_s\}$,
an $\cal H$-decomposition of $G$ is a partition of its edge set
into subgraphs, each of which is isomorphic to a member of $\cal
H$. Graph decompositions have been widely studied in graph theory
and in combinatorial design theory. Indeed in the very special case
when ${\cal H}$ contains only the complete graph $K_k$ and $G \is
K_v$, a decomposition is a balanced incomplete block design (with
$\lambda=1$). In order to model the assignment of traffic to wavelengths
in an optical ring network, this definition has been extended to
associate a cost with each graph $H_i$, and to ask for a decomposition
of minimum total cost.

Of particular interest is the case when $G \is K_v$ and $\cal H$
contains all simple graphs on at most $C$ edges. Then the decomposition
is called a $C$-grooming. Taking the cost of each $H_i$ to be its
number of vertices of nonzero degree, a $C$-grooming of minimum
cost yields a wavelength assignment in an optical ring that incurs
the least hardware cost.

We first outline the almost complete determination of minimum cost
$C$-groomings for $1 \leq C \leq 6$ using techniques from graph
theory and design theory, and report on recent results for $C \in
\{7,8,9\}$. We examine graph- and design-theoretic constructions
to obtain upper bounds on the cost, and develop lower bounds via
an interesting use of linear programming duality.

A 'two-period' variant of grooming in which a $C$-grooming on all
nodes must induce a $C'$-grooming with $C' < C$ is examined in
the cases when $1\leq C \leq 4$, where complete determinations of
the minimum cost have been completed. The talk focuses on general
design- and graph-theoretic techniques that apply, and no background
in optical networking is assumed.

**Michel X. Goemans**, Massachusetts Institute
of Technology

*Minimum Bounded Degree Spanning Trees*

In this talk, I will consider the minimum spanning tree problem
under the additional restriction that all degrees of the spanning

tree must be at most a given value $k$. I will describe two approaches,
one by myself and one by Mohit Singh and Lap Chi

Lau. These results show that one can efficiently find a spanning
tree of maximum degree at most $k+2$, or even at most $k+1$ for
the second approach, whose cost is at most the cost of the optimum
spanning tree of maximum degree $k$. This is best possible, as the
problem of just deciding whether a graph has a spanning tree of
maximum degree $k$ is NP-complete.

The first approach uses a sequence of simple algebraic, polyhedral
and combinatorial arguments, and relies on uncrossing, polyhedral

characterizations, matroid intersection and graph orientation. The
second approach also uses uncrossing and is a prime example of

iterative relaxation, a new technique extending Jain's iterative
rounding.

**Penny Haxell**, University of Waterloo

*Scarf's Lemma and the Stable Paths Problem*

We address a question in graphs called the stable paths problem,
which is an abstraction of a network routing problem concerning
the Border Gateway Protocol (BGP). The main tool we use is Scarf's
Lemma, an important result from game theory. This talk will describe
Scarf's Lemma and how it is related to other results more familiar
to combinatorialists, and then will explain its implications for
the stable paths problem.

**Bruce Reed**, McGill University

* Graph Colouring a la Chvatal.*

A natural approach to solving combinatorial optimization problems
is to formulate them as Integer Linear Programs, solve the

fractional relaxation, and attempt to use this solution to solve
the Integer Linear Program. We discuss graph colouring from this
perspective. In doing so, we consider a three pronged approach which
combines polyhedral combinatorics, structural decompositions, and
the probabilistic method. I learnt all three techniques as a young
lad from Vasek Chvatal.

**Xingxing Yu** , Georgia Institute of Technology

*On judicious partitions of graphs*

Judicious partition problems ask for partitions of the vertex
set of a graphs so that several quantities are optimized simultaneously.
I will discuss several judicious partition problems of Bollobas
and Scott, and present our recent results on these problems.

This is joint work with Baogang Xu.

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