Kernelization for Graph Packing Problems via Rainbow Matching

We introduce a new kernelization tool, called rainbow matching technique, that is appropriate for the design of polynomial kernels for packing problems. Our technique capitalizes on the powerful combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two (di)graph packing problems, namely the Triangle-Packing in Tournament problem (\TPT), where we ask for a directed triangle packing in a tournament, and the Induced 2-Path-Packing (\IPP) where we ask for a packing of $k$ induced paths of length two in a graph. The existence of a sub-quadratic kernels for these problems was proven for the first time in [Fomin, Le, Lokshtanov, Saurabh, Thomass\'e, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of $\mathcal{O}(k^{3/2})$ vertices and $\mathcal{O}(k^{5/3})$ vertices respectively. In the same paper it was questioned whether these bounds can be (optimally) improved to linear ones. Motivated by this question, we apply the rainbow matching technique and prove that \TPT admits an (almost linear) kernel of $k^{1+\frac{\mathcal{O}(1)}{\sqrt{\log{k}}}}$ vertices and that \IPP admits kernel of $\mathcal{O}(k)$ vertices

Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as input a graph $G$ and integers $l$ and $p$, and the objective is to find a subgraph $H$ of $G$ and a $p$-edge-coloring of $H$, such that $|E(H)| \geq l$. We study the above problem from the viewpoint of Parameterized Complexity. We obtain \FPT\ algorithms when parameterized by: $(1)$ the vertex cover number of $G$, by using {\sc Integer Linear Programming}, and $(2)$ $l$, a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters $p+k$, where $k$ is one of the following: $(1)$ the solution size, $l$, $(2)$ the vertex cover number of $G$, and $(3)$ l - {\mm}(G), where {\mm}(G) is the size of a maximum matching in $G$; we show that the (decision version of the) problem admits a kernel with $\mathcal{O}(k \cdot p)$ vertices. Furthermore, we show that there is no kernel of size $\mathcal{O}(k^{1-\epsilon} \cdot f(p))$, for any $\epsilon > 0$ and computable function $f$, unless \NP \subseteq \CONPpoly

Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ &gt; 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p

Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ &gt; 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p

Solution discovery via reconfiguration for problems in P

In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of $k$ tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number $b$ of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) $k$ and transformation budget (number of steps) $b$. Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints

Finding Diverse Trees, Paths, and More

Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding $r$ solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as {\sc Vertex Cover}, {\sc Feedback Vertex Set}, {\sc $d$-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph $G$ and an integer $r$, the problem of computing $r$ spanning trees of $G$ maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.Comment: 15 page

Fixed-Parameter Algorithms for Fair Hitting Set Problems

Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}. In the classical \textsc{Hitting Set} problem, the input is a universe $\mathcal{U}$, a family $\mathcal{F}$ of subsets of $\mathcal{U}$, and a non-negative integer $k$. The goal is to determine whether there exists a subset $S \subseteq \mathcal{U}$ of size $k$ that \emph{hits} (i.e., intersects) every set in $\mathcal{F}$. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family $\mathcal{B}$ of subsets of $\mathcal{U}$, where each subset in $\mathcal{B}$ can be thought of as the group of elements of the same \emph{type}. We want to find a set $S \subseteq \mathcal{U}$ of size $k$ that (i) hits all sets of $\mathcal{F}$, and (ii) does not contain \emph{too many} elements of each type. We call this problem \textsc{Fair Hitting Set}, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for \textsc{Hitting Set}