39 research outputs found
Strong Parameterized Deletion: Bipartite Graphs
The purpose of this article is two fold: (a) to formally introduce a stronger version of graph deletion problems; and (b) to study this version in the context of bipartite graphs. Given a family of graphs F, a typical instance of parameterized graph deletion problem consists of an undirected graph G and a positive integer k and the objective is to check whether we can delete at most k vertices (or k edges) such that the resulting graph belongs to F. Another version that has been recently studied is the one where the input contains two integers k and l and the objective is to check whether we can delete at most k vertices and l edges such that the resulting graph belongs to F. In this paper, we propose and initiate the study of a more general version which we call strong deletion. In this problem, given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be transformed into a graph in F by deleting at most l edges. In this paper we study this stronger version of deletion problems for the class of bipartite graphs. In particular, we study Strong Bipartite Deletion, where given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be made bipartite by deleting at most l edges. While fixed-parameter tractability when parameterizing by k or l alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and l
Faster FPT Algorithm for 5-Path Vertex Cover
The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time
On the Tree Conjecture for the Network Creation Game
Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC\u2703] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees.
We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees
An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams
We present an algorithm for the maximum matching problem in dynamic
(insertion-deletions) streams with *asymptotically optimal* space complexity:
for any -vertex graph, our algorithm with high probability outputs an
-approximate matching in a single pass using bits of
space.
A long line of work on the dynamic streaming matching problem has reduced the
gap between space upper and lower bounds first to factors
[Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to
factors [Dark-Konrad; CCC 2020]. Our upper bound now
matches the Dark-Konrad lower bound up to factors, thus completing this
research direction.
Our approach consists of two main steps: we first (provably) identify a
family of graphs, similar to the instances used in prior work to establish the
lower bounds for this problem, as the only "hard" instances to focus on. These
graphs include an induced subgraph which is both sparse and contains a large
matching. We then design a dynamic streaming algorithm for this family of
graphs which is more efficient than prior work. The key to this efficiency is a
novel sketching method, which bypasses the typical loss of
-factors in space compared to standard -sampling
primitives, and can be of independent interest in designing optimal algorithms
for other streaming problems.Comment: Full version of the paper accepted to ITCS 2022. 42 pages, 5 Figure