12 research outputs found
Hitting Meets Packing: How Hard Can it Be?
We study a general family of problems that form a common generalization of
classic hitting (also referred to as covering or transversal) and packing
problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of
a graph G prevent us from packing vertex-disjoint objects of type X?
This problem captures a spectrum of problems with standard hitting and packing
on opposite ends. Our main motivating question is whether the combination
X-HitPack can be significantly harder than these two base problems. Already for
a particular choice of X, this question can be posed for many different
complexity notions, leading to a large, so-far unexplored domain in the
intersection of the areas of hitting and packing problems.
On a high-level, we present two case studies: (1) X being all cycles, and (2)
X being all copies of a fixed graph H. In each, we explore the classical
complexity, as well as the parameterized complexity with the natural parameters
k+l and treewidth. We observe that the combined problem can be drastically
harder than the base problems: for cycles or for H being a connected graph with
at least 3 vertices, the problem is \Sigma_2^P-complete and requires
double-exponential dependence on the treewidth of the graph (assuming the
Exponential-Time Hypothesis). In contrast, the combined problem admits
qualitatively similar running times as the base problems in some cases,
although significant novel ideas are required. For example, for X being all
cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching
method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover
and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on
graphs of treewidth tw. The key step enabling this running time relies on a
combinatorial bound obtained from an algebraic (linear delta-matroid)
representation of possible matchings
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum