TY - GEN
T1 - Communication complexity of pairs of graph families with applications
AU - Kolay, Sudeshna
AU - Panolan, Fahad
AU - Saurabh, Saket
N1 - Publisher Copyright:
© Sudeshna Kolay, Fahad Panolan, and Saket Saurabh; licensed under Creative Commons License CC-BY.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - Given a graph G and a pair (F1,F2) of graph families, the function GDISJG,F1,F2 takes as input, two induced subgraphs G1 and G2 of G, such that G1 ϵ F1 and G2 ϵ F2 and returns 1 if V (G1) / V (G2) = theta; and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we obtain nuanced upper bounds on the communication complexity of GDISJG,F1,F2 . A concept related to communication protocols is that of a (F1,F2)-separating family of a graph G. A collection F of subsets of V (G) is called a (F1,F2)-separating family for G, if for any two vertex disjoint induced subgraphs G1 ϵ F1,G2 ϵ F2, there is a set F ϵ F with V (G1)F and V (G2) / F = theta;. Given a graph G on n vertices, for any pair (F1,F2) of hereditary graph families with sublinear communication complexity for GDISJG,F1,F2 , we give an enumeration algorithm that finds a subexponential sized (F1,F2)-separating family. In fact, we give an enumeration algorithm that finds a 2o(k)nO(1) sized (F1,F2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V (G) \ S has a bipartition (V1, V2) with G[V1] ϵ F1 and G[V2] ϵ F2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.
AB - Given a graph G and a pair (F1,F2) of graph families, the function GDISJG,F1,F2 takes as input, two induced subgraphs G1 and G2 of G, such that G1 ϵ F1 and G2 ϵ F2 and returns 1 if V (G1) / V (G2) = theta; and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we obtain nuanced upper bounds on the communication complexity of GDISJG,F1,F2 . A concept related to communication protocols is that of a (F1,F2)-separating family of a graph G. A collection F of subsets of V (G) is called a (F1,F2)-separating family for G, if for any two vertex disjoint induced subgraphs G1 ϵ F1,G2 ϵ F2, there is a set F ϵ F with V (G1)F and V (G2) / F = theta;. Given a graph G on n vertices, for any pair (F1,F2) of hereditary graph families with sublinear communication complexity for GDISJG,F1,F2 , we give an enumeration algorithm that finds a subexponential sized (F1,F2)-separating family. In fact, we give an enumeration algorithm that finds a 2o(k)nO(1) sized (F1,F2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V (G) \ S has a bipartition (V1, V2) with G[V1] ϵ F1 and G[V2] ϵ F2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.
KW - Communication Complexity
KW - FPT algorithms
KW - Separating Family
UR - http://www.scopus.com/inward/record.url?scp=85038412663&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.MFCS.2017.13
DO - 10.4230/LIPIcs.MFCS.2017.13
M3 - Conference contribution
AN - SCOPUS:85038412663
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017
A2 - Larsen, Kim G.
A2 - Raskin, Jean-Francois
A2 - Bodlaender, Hans L.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017
Y2 - 21 August 2017 through 25 August 2017
ER -