**David F. Manlove ^{1}
Department of Computing Science,
University of Glasgow, Glasgow G12 8QQ, Scotland.
Email: davidm@dcs.gla.ac.uk.**

We study a class of optimisation problems called *minimaximal* and
*maximinimal* optimisation problems. Such a problem
may
be obtained from an optimisation problem ,
called the *source*
optimisation problem, by defining a
partial order
on the set of feasible solutions
for a
given instance *x* of .
If the objective of
is to maximise
(respectively minimise) the measure over
,
then the objective of is to minimise (maximise) the measure over all elements of
that are
maximal (minimal) with respect to .
Many optimisation problems
occurring in the literature are minimaximal or maximinimal optimisation
problems.

In this thesis, we present the first unifying framework for formulating
minimaximal and maximinimal optimisation problems, based on this partial
order concept. To accompany this framework, we define a variety of partial
orders, an important example being the partial order of set inclusion.
By considering various source optimisation
problems from the literature, and partial orders from our collection, we use
our framework to obtain a range of minimaximal and maximinimal optimisation
problems. We study these individual examples mainly from the point of view
of algorithmic complexity.

The series of examples begins with the source optimisation problem
CHROMATIC NUMBER, whose objective is to minimise the number of colours
over all proper colourings of a given graph.
A related problem is ACHROMATIC NUMBER, in which we
seek to maximise the number of colours over all proper colourings of a given
graph *G* such that each pair of distinct colours occurs at the endpoints of
some edge of *G*. We show that ACHROMATIC NUMBER is a maximinimal
counterpart of CHROMATIC NUMBER, by defining a partition-related
partial order on the set of all proper colourings of *G*. A natural refinement
of this partial order gives rise to an additional maximinimal optimisation
problem concerned with graph colouring,
which we call B-CHROMATIC NUMBER. The objective of this problem is
to maximise the number of colours over all proper colourings of *G* such that
for each colour *i*, there is a distinguished vertex of colour *i* that is
adjacent to a vertex of every colour .
Whilst the ACHROMATIC
NUMBER problem has been studied for over thirty years, the B-CHROMATIC
NUMBER problem is new. We show that the decision version of B-CHROMATIC
NUMBER is NP-complete in arbitrary graphs, and also bipartite graphs.
However, we prove that B-CHROMATIC NUMBER is solvable in polynomial time
for trees, in contrast with ACHROMATIC NUMBER.

Many other examples of minimaximal and maximinimal optimisation problems relating to graph parameters have already been studied in the literature. We focus on graph-theoretic minimaximal and maximinimal optimisation problems that may be obtained from source optimisation problems using the partial order of set inclusion. In particular, we investigate source optimisation problems and their minimaximal or maximinimal counterparts relating to vertex, edge and total covers, and independent sets, matchings and total matchings in a graph. Each of the twelve optimisation problems implied by the previous sentence has been studied in some form in the literature. We review existing complexity results and obtain a number of new results, namely NP-completeness proofs for the decision versions of MAXIMUM MINIMAL TOTAL COVER in planar graphs, MINIMUM MAXIMAL TOTAL MATCHING in bipartite and chordal graphs, and MINIMUM INDEPENDENT DOMINATING SET (or MINIMUM MAXIMAL INDEPENDENT SET) in planar cubic graphs.

We also study source optimisation problems and their minimaximal or maximinimal
counterparts relating to strong stable sets, cliques, dominating sets,
total dominating sets, edge dominating sets and irredundant sets in a graph.
Again, we survey existing algorithmic results, if any, for the twelve implicit
optimisation problems, and obtain several new results, namely
NP-completeness proofs for the decision versions
of MINIMUM MAXIMAL STRONG STABLE SET in
planar graphs of maximum degree 3, MINIMUM MAXIMAL CLIQUE in general
graphs, and MINIMUM TOTAL DOMINATING SET in planar cubic graphs.

For a graph *G*=(*V*,*E*), a refinement of the notion of an independent set that
is maximal with respect to the partial order of set inclusion has been
considered in the literature. An independent set
is *k**-maximal* ()
if the removal of any *r*-1 vertices from *S*,
together with the addition of any *r* vertices from
(for any ), results in a non-independent set. We consider minimaximal
optimisation problems related to finding minimum cardinality *k*-maximal
independent sets in a graph. We focus mainly on the case *k*=2, and prove
that the decision version of the problem of determining the minimum size of
a 2-maximal independent set is NP-complete, even for planar graphs of maximum
degree 3. However, for trees, we give a linear-time algorithm.

Many integer-valued graph parameters have fractional counterparts in the
literature. We define the concept of fractional graph optimisation problems,
relating to fractional graph parameters, and show how minimaximal and
maximinimal versions may be defined, using a
partial order on functions. We formulate several examples of such problems
using the framework, and by considering appropriate linear programming
constructions, we show that the optimal measure (the solution to the
evaluation version of the minimaximal or maximinimal fractional graph
optimisation problem concerned) is computable, has rational values, and is
attained by some function of compact representation which
satisfies the feasibility constraint for the
minimaximal or maximinimal fractional graph optimisation problem concerned.
These three issues have implications for the algorithmic behaviour of a
minimaximal or maximinimal fractional graph optimisation problem. We also
survey complexity results relating to the source fractional graph
optimisation problems and their minimaximal or maximinimal counterparts
that we define.

Minimaximal and maximinimal optimisation problems that relate to areas other
than the domain of graph theory are also studied in this thesis. By considering
source optimisation problems belonging to the Garey and Johnson
[*Computers and Intractability*, Freeman, 1979] problem categories
of Network Design, Sets and Partitions, Data Storage, Compression and
Representation, Mathematical Programming, and Logic, and by imposing
various partial orders, we formulate a range of natural minimaximal and
maximinimal optimisation problems. These interesting individual examples,
most of which are new, are minimaximal or maximinimal versions of the
following problems (Garey and Johnson problem numbers in brackets):
LONGEST PATH (ND29), 3 D-MATCHING (SP1), MINIMUM TEST SET (SP6),
BIN PACKING (SR1), KNAPSACK (MP9),
MAXIMUM 2 -SAT (LO5),
ONE-IN-THREE 3 SAT (LO4), LONGEST COMMON
SUBSEQUENCE (SR10), SHORTEST COMMON SUPERSEQUENCE (SR8),
LONGEST COMMON SUBSTRING (SR10) and SHORTEST COMMON SUPERSTRING
(SR9). We survey complexity results for the source optimisation problem
in each case, and also those, if any, for the minimaximal or maximinimal
counterpart(s) of .
We obtain new NP-completeness results for minimaximal
or maximinimal versions of the first seven problems in the above list, and
construct a polynomial-time algorithm for a minimaximal counterpart of
LONGEST COMMON SUBSTRING.

We also consider the computational complexity of minimaximal and maximinimal
optimisation problems from a more general viewpoint.
We present conditions under which a Turing reduction from an
optimisation problem, ,
to another, ,
is also a Turing
reduction from
to ,
where
is a minimaximal or
maximinimal version of
(*i*=1,2). We call Turing reductions
satisfying these additional constraints *MM-reductions*.
Some examples of MM-reductions are given, involving several partial orders,
and involving source optimisation problems from a variety of domains.

Additionally, we discuss the question of how, in general, we may test a feasible solution of an optimisation problem for maximality or minimality, and how we may find feasible solutions that are maximal or minimal, with respect to a partial order defined on the feasible solutions of for a given instance. In particular, we examine this question when is BIN PACKING and when is CHROMATIC NUMBER.

We show that, for a natural partition-related partial order
defined on the feasible solutions for a given instance *x* of BIN
PACKING, the problem of testing a feasible solution for
-minimality is NP-hard, but perhaps surprisingly, the problem of
finding a feasible solution that is -minimal is
polynomial time solvable.

In the case of CHROMATIC NUMBER, we consider two families of
partition-related partial orders defined on the set of all proper colourings
of a given graph *G*.
We investigate the problems of testing a proper colouring for minimality, and
finding proper graph colourings that are minimal, with respect to these
partial orders. We also consider the complexity of the associated maximinimal
optimisation problems in each case. Our algorithmic results for testing and
finding show where the thresholds between polynomial-time solvability and
NP-hardness lie, within the hierarchy of problems corresponding to the two
partial order families. These developments imply possible local search
strategies for approximating the chromatic number in certain graph classes.

Finally, we revisit the general framework for minimaximal and maximinimal
optimisation problems, considering an alternative definition in which
the partial order is defined on the set of all possible (not necessarily
feasible) solutions. We show that in some, but not all, cases,
the revised definition gives rise to the same maximal and minimal solutions
as would be obtained by defining the partial order concerned on the feasible
solutions. These observations yield a greater insight into the structure of
minimaximal and maximinimal optimisation problems.

**Thesis supervisor:** Dr. Rob Irving, Senior Lecturer in Computing
Science.