TY - JOUR

T1 - Communication complexity and graph families

AU - Kolay, Sudeshna

AU - Panolan, Fahad

AU - Saurabh, Saket

N1 - Funding Information:
The research leading to these results has received funding from the European Research Council (ERC) via grants Rigorous Theory of Preprocessing, reference 267959 and PARAPPROX, reference 306992. A preliminary version of this article appeared in Proc. MFCS 2017 Authors’ addresses: S. Kolay, Eindhoven University of Technology, Netherlands, Eindhoven, 5612AZ, Netherlands; email: s.kolay@tue.nl; F. Panolan, University of Bergen, Department of Informatics, Bergen 5020, Norway; email: fahad. panolan@uib.no; S. Saurabh, University of Bergen, Department of Informatics, Bergen 5020, Norway and Institute of Mathematical Sciences, Chennai, 600113, India; email: saket@imsc.res.in. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2019 Association for Computing Machinery. 1942-3454/2019/03-ART11 $15.00 https://doi.org/10.1145/3313234
Funding Information:
The research leading to these results has received funding from the European Research Council (ERC) via grants Rigorous Theory of Preprocessing, reference 267959 and PARAPPROX, reference 306992. A preliminary version of this article appeared in Proc. MFCS 2017
Publisher Copyright:
© 2019 Association for Computing Machinery.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - Given a graph G and a pair (F1, F2) of graph families, the function GDISJG, F1, F2 takes as input, two induced subgraphsG1 andG2 ofG, such thatG1 ∈ F1 andG2 ∈ F2 and returns 1 ifV(G1) ∩ V(G2) = ∅ 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 ∈ F1,G2 ∈ F2, there is a set F ∈ F with V(G1) ⊆ F andV(G2) ∩ F = ∅. Given a graphG onn 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] ∈ F1 and G[V2] ∈ 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 subgraphsG1 andG2 ofG, such thatG1 ∈ F1 andG2 ∈ F2 and returns 1 ifV(G1) ∩ V(G2) = ∅ 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 ∈ F1,G2 ∈ F2, there is a set F ∈ F with V(G1) ⊆ F andV(G2) ∩ F = ∅. Given a graphG onn 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] ∈ F1 and G[V2] ∈ 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=85065755671&partnerID=8YFLogxK

U2 - 10.1145/3313234

DO - 10.1145/3313234

M3 - Article

AN - SCOPUS:85065755671

SN - 1942-3454

VL - 11

JO - ACM Transactions on Computation Theory

JF - ACM Transactions on Computation Theory

IS - 2

M1 - 11

ER -