For digraphs G and H, a homomorphism of G to H is a mapping $f:\
V(G)\dom V(H)suchthatuv\in A(G)impliesf(u)f(v)\in A(H).If,moreover,eachvertexu \in V(G)isassociatedwithcostsc_i(u), i \in V(H),thenthecostofahomomorphismfis\sum_{u\in V(G)}c_{f(u)}(u).ForeachfixeddigraphH,theminimumcosthomomorphismproblemforH,denotedMinHOM(H),canbeformulatedasfollows:GivenaninputdigraphG,togetherwithcostsc_i(u),u\in V(G),i\in V(H),decidewhetherthereexistsahomomorphismofGtoH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
For digraphs G and H, a homomorphism of G to H is a mapping $f:\
V(G)\dom V(H)suchthatuv\in A(G)impliesf(u)f(v)\in A(H).Ifmoreovereachvertexu \in V(G)isassociatedwithcostsc_i(u), i \in V(H),thenthecostofahomomorphismfis\sum_{u\in V(G)}c_{f(u)}(u).ForeachfixeddigraphH, the {\em minimum cost homomorphism problem} for H,denotedMinHOM(H),isthefollowingproblem.GivenaninputdigraphG,togetherwithcostsc_i(u),u\in V(G),i\in V(H),andanintegerk,decideifGadmitsahomomorphismtoHofcostnotexceedingk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs H(everyvertexofHhasaloop).ItisknownthattheproblemMinHOM(H)ispolynomialtimesolvableifthedigraphH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <sothati<j, s<randir, js
\in A(H)implythatis \in A(H)andjr \in A(H).WegiveaforbiddeninducedsubgraphcharacterizationofreflexivedigraphswithaMin−Maxordering;ourcharacterizationimpliesapolynomialtimetestfortheexistenceofaMin−Maxordering.Usingthischaracterization,weshowthatforareflexivedigraphH$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd