Title: Pure Pairs. V. Excluding Some Long Subdivision
A “pure pair” in a graph G is a pair A, B of disjoint subsets of V(G) such that A is complete or anticomplete to B. Jacob Fox showed that for all ε>0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with |A|,|B|≥εn. He also proved that for all c>0 there exists ε>0 such that for every comparability graph G with n>1 vertices, there is a pure pair A, B with |A|,|B|≥εn1−c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c>0, and all ℓ1,ℓ2≥4/c+9, there exists ε>0 such that, if G is an (n>1)-vertex graph with no hole of length exactly ℓ1 and no antihole of length exactly ℓ2, then there is a pure pair A, B in G with |A|≥εn and |B|≥εn1−c. This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph. more »« less
Balogh, József; Clemen, Felix Christian; Lavrov, Mikhail; Lidický, Bernard; Pfender, Florian
(, Combinatorics, Probability and Computing)
null
(Ed.)
Abstract The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+ 1 -free graph on n vertices with e ( G ) > ex( n , K r +1 )– α n 2 , then one can remove εn 2 edges from G to obtain an r -partite graph. Füredi gave a short proof that one can choose α = ε . We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.
Abstract Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447 ). In this new setting, each vertex v in some subset of V ( G ) has a request for a certain color r ( v ) in its list of colors L ( v ). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant $$\varepsilon >0$$ ε > 0 such that any graph G in some graph class $$\mathscr {C}$$ C satisfies at least $$\varepsilon$$ ε proportion of the requests. More formally, for $$k > 0$$ k > 0 the goal is to prove that for any graph $$G \in \mathscr {C}$$ G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every $$R \subseteq V$$ R ⊆ V and a request vector $$(r(v): v\in R, ~r(v) \in L(v))$$ ( r ( v ) : v ∈ R , r ( v ) ∈ L ( v ) ) , there exists an L -coloring of G satisfying at least $$\varepsilon |R|$$ ε | R | requests. If this is true, then $$\mathscr {C}$$ C is called $$\varepsilon$$ ε - flexible for lists of size k . Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021 ) introduced the notion of weak flexibility , where $$R = V$$ R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists $$\varepsilon (b)>0$$ ε ( b ) > 0 so that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_b$$ K 4 , C 5 , C 6 , C 7 , B b is weakly $$\varepsilon (b)$$ ε ( b ) -flexible for lists of size 4 (here $$K_n$$ K n , $$C_n$$ C n and $$B_n$$ B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_5$$ K 4 , C 5 , C 6 , C 7 , B 5 is $$\varepsilon$$ ε -flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.
Inamdar, Tanmay; Lokshtanov, Daniel; Saurabh, Saket; Surianarayanan, Vaishali
(, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz
(Ed.)
In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth.
Chudnovsky, Maria; Gartland, Peter; Hajebi, Sepehr; Lokshtanov, Daniel; Spirkl, Sophie
(, Society for Industrial and Applied Mathematics - Symposium on Discrete Algorithms)
We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c > 0 such that for every integer n > 1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most clog10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasipolynomial time if the input graph is even-hole-free. The quasi-polynomial complexity will remain the same even if the exponent of the logarithm is reduced to 1 (which would be asymptotically best possible).
Chudnovsky, Maria; Abrishami, Tara; Hajebi, Sepehr; Spirkl, Sophie
(, Advances in Combinatorics)
A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-_free_ if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $$c$$ for which every (theta, triangle)-free graph $$G$$ has treewidth at most $$c\log (|V(G)|)$$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $$G$$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $$k$$-Coloring (for fixed $$k$$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.
Scott, Alex, Seymour, Paul, and Spirkl, Sophie. Pure Pairs. V. Excluding Some Long Subdivision. Retrieved from https://par.nsf.gov/biblio/10426953. Combinatorica . Web. doi:10.1007/s00493-023-00025-8.
Scott, Alex, Seymour, Paul, & Spirkl, Sophie. Pure Pairs. V. Excluding Some Long Subdivision. Combinatorica, (). Retrieved from https://par.nsf.gov/biblio/10426953. https://doi.org/10.1007/s00493-023-00025-8
@article{osti_10426953,
place = {Country unknown/Code not available},
title = {Pure Pairs. V. Excluding Some Long Subdivision},
url = {https://par.nsf.gov/biblio/10426953},
DOI = {10.1007/s00493-023-00025-8},
abstractNote = {A “pure pair” in a graph G is a pair A, B of disjoint subsets of V(G) such that A is complete or anticomplete to B. Jacob Fox showed that for all ε>0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with |A|,|B|≥εn. He also proved that for all c>0 there exists ε>0 such that for every comparability graph G with n>1 vertices, there is a pure pair A, B with |A|,|B|≥εn1−c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c>0, and all ℓ1,ℓ2≥4/c+9, there exists ε>0 such that, if G is an (n>1)-vertex graph with no hole of length exactly ℓ1 and no antihole of length exactly ℓ2, then there is a pure pair A, B in G with |A|≥εn and |B|≥εn1−c. This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph.},
journal = {Combinatorica},
author = {Scott, Alex and Seymour, Paul and Spirkl, Sophie},
}
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