TY - GEN
T1 - A new holant dichotomy inspired by quantum computation
AU - Backens, Miriam
PY - 2017/7/14
Y1 - 2017/7/14
N2 - Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problems into those which are in FP and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before. Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant+. This family sits in between the known families of Holant∗, for which a full dichotomy is known, and Holantc, for which only a restricted dichotomy is known. Using knowledge from entanglement theory - both previously existing work and new results of our own - we prove a full dichotomy theorem for Holant+, which is very similar to the restricted Holantc dichotomy and may thus be a stepping stone to a full dichotomy for that family.
AB - Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problems into those which are in FP and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before. Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant+. This family sits in between the known families of Holant∗, for which a full dichotomy is known, and Holantc, for which only a restricted dichotomy is known. Using knowledge from entanglement theory - both previously existing work and new results of our own - we prove a full dichotomy theorem for Holant+, which is very similar to the restricted Holantc dichotomy and may thus be a stepping stone to a full dichotomy for that family.
KW - Computational complexity
KW - Counting complexity
KW - Dichotomy
KW - Entanglement
KW - Holant
UR - http://www.scopus.com/inward/record.url?scp=85027253728&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2017.16
DO - 10.4230/LIPIcs.ICALP.2017.16
M3 - Conference contribution
AN - SCOPUS:85027253728
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
A2 - Muscholl, Anca
A2 - Indyk, Piotr
A2 - Kuhn, Fabian
A2 - Chatzigiannakis, Ioannis
PB - Schloss Dagstuhl
T2 - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Y2 - 10 July 2017 through 14 July 2017
ER -