}We study (vertex-disjoint) $P_2$-packings in graphs under a parameterized perspective. Starting from a maximal $P_2$-packing $\p$ of size $j$ we use extremal arguments for determining how many vertices of $\p$ appear in some $P_2$-packing of size $(j+1)$. We basically can 'reuse' $2.5j$ vertices. We also present a kernelization algorithm that gives a kernel of size bounded by $7k$. With these two results we build an algorithm which constructs a $P_2$-packing of size $k$ in time $\Oh^*(2.482^{3k})$.
In the past decade, many parameterized algorithms were developed for packing problems. Our goal is to obtain tradeoffs that improve the running times of these algorithms at the cost of computing approximate solutions. Consider a packing problem for which there is no known algorithm with approximation ratio $α$, and a parameter $k$. If the value of an optimal solution is at least $k$, we seek a solution of value at least $αk$; otherwise, we seek an arbitrary solution. Clearly, if the best known parameterized algorithm that finds a solution of value $t$ runs in time $O^*(f(t))$ for some function…
Given a graph $G = (V,E)$, $A \subseteq V$, and integers $k$ and $\ell$, the \textsc{$(A,\ell)$-Path Packing} problem asks to find $k$ vertex-disjoint paths of length $\ell$ that have endpoints in $A$ and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters $|A|$, $\ell$, $k$, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also…
We present three deterministic parameterized algorithms for well-studied packing and matching problems, namely, Weighted q-Dimensional p-Matching ((q,p)-WDM) and Weighted q-Set p-Packing ((q,p)-WSP). More specifically, we present an O*(2.85043^{(q-1)p}) time deterministic algorithm for (q,p)-WDM, an O*(8.04143^p) time deterministic algorithm for the unweighted version of (3,p)-WDM, and an O*((0.56201\cdot 2.85043^q)^p) time deterministic algorithm for (q,p)-WSP. Our algorithms significantly improve the previously best known O* running times in solving (q,p)-WDM and (q,p)-WSP, and the previousl…
Given a directed graph $D$ on $n$ vertices and a positive integer $k$, the Arc-Disjoint Cycle Packing problem is to determine whether $D$ has $k$ arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size $k$. In particular, we show that it can be solved in $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$ time and has a kernel …
We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with $d$ colors in time within a polynomial factor…
We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An $α$-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs $c$-approximate solutions in $O(1)$ time, obtains an $(α\cdot c)$-approximate solution to the considered problem, using calls to the oracle of size at most $f(k)$ for some function $f$ that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth $\ell$ has a $(1+\…
An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed…
The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdős-Pósa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson-Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle …
We study a general family of facility location problems defined on planar graphs and on the 2-dimensional plane. In these problems, a subset of $k$ objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the $n^{O(k)}$ time brute force algorithm of selecting $k$ objects can be improved to $n^{O(\sqrt{k})}$ time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Vorono…
Let integers $r\ge 2$ and $d\ge 3$ be fixed. Let ${\cal G}_d$ be the set of graphs with no induced path on $d$ vertices. We study the problem of packing $k$ vertex-disjoint copies of $K_{1,r}$ ($k\ge 2$) into a graph $G$ from parameterized preprocessing, i.e., kernelization, point of view. We show that every graph $G\in {\cal G}_d$ can be reduced, in polynomial time, to a graph $G'\in {\cal G}_d$ with $O(k)$ vertices such that $G$ has at least $k$ vertex-disjoint copies of $K_{1,r}$ if and only if $G'$ has. Such a result is known for arbitrary graphs $G$ when $r=2$ and we conjecture that it ho…
We consider the problem of assigning agents to resources under the two-sided preference list model where resources specify an upper-quota and a lower-quota, that is, respectively the maximum and minimum number of agents that can be assigned to it. Different notions of optimality including envy-freeness and relaxed stability are investigated for this setting and the goal is to compute a largest optimal matching. Krishnaa et al. [SAGT 2020] show that in this setting, the problem of computing a maximum size envy-free matching (MAXEFM) or a maximum size relaxed stable matching (MAXRSM) is not appr…
A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the Globally Minimal Defensive Alliance…
Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk where every vertex occurs at most $r$ times and the total number of vertex occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial $P$ on $n$ variables and integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing pr…
We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvoß[6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time $|V|^{2^{O(d)}} \cdot enc(I)^{O(1)}$, where $V$ is the set of vertices of the integer knapsack polytope and $enc(I)$ is the encoding length of the bin packin…
We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least …
We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices, (2) a cycle on exactly $k$ vertices, (3) a cycle on at least $k$ vertices, (4) a feedback vertex set of size at most $k$, and (5) a set of $k$ pairwise vertex-disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previou…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
For some years it was believed that for "connectivity" problems such as Hamiltonian Cycle, algorithms running in time 2^{O(tw)}n^{O(1)} -called single-exponential- existed only on planar and other sparse graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [FOCS 2011], Bodlaender et al. [ICALP 2013], and Fomin et al. [SODA 2014], who provided single-exponential algorithms on general graphs for essentially all connectivity problems that were known to be solvable in single-exponential time on sparse graphs. In this article we f…
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of struct…
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.
In this paper we study a maximization version of the classical Feedback Vertex Set (FVS) problem, namely, the Max Min FVS problem, in the realm of parameterized complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal feedback vertex set of size at least $k$. We obtain following results for Max Min FVS. 1) We first design a fixed parameter tractable (FPT) algorithm for Max Min FVS running in time $10^kn^{\mathcal{O}(1)}$. 2) Next, we consider the problem parameterized by the vertex cover number of the input graph (de…
The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this …
Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is particularly hard -- it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present …
Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for …
In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most $p$ into a given universe generalizes the Maximum Matching problem, which is recovered by taking $p=2$. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problem --- …
The Maximum Agreement Forest problem has been extensively studied in phylogenetics. Most previous work is on two binary phylogenetic trees. In this paper, we study a generalized version of the problem: the Maximum Agreement Forest problem on multiple rooted multifurcating phylogenetic trees, from the perspective of fixed-parameter algorithms. By taking advantage of a new branch-and-bound strategy, two parameterized algorithms, with running times $O(2.42^k m^3 n^4)$ and $O(2.74^k m^3 n^5)$, respectively, are presented for the hard version and the soft version of the problem, which correspond to…
EF(X) Orientations: A Parameterized Complexity Perspective
National Technical University of Athens（雅典国立技术大学）；Archimedes, Athena Research Center（阿基米德，雅典研究中心）；Université Paris-Dauphine, PSL University（巴黎达芬大学，PSL大学）；Athens University of Economics and Business（雅典经济与商业大学）
Indian Institute of Technology, Kharagpur（印度理工学院克勒格布尔分校）；Indian Institute of Technology, Gandhinagar（印度理工学院甘地讷格尔分校）；Indian Institute of Technology, Kanpur（印度理工学院坎普尔分校）
We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers $k_a$ and $k_d$ corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most $k_d$ voter groups such th…
We study the parameterized complexity of a variant of the $F$-free Editing problem: Given a graph $G$ and a natural number $k$, is it possible to modify at most $k$ edges in $G$ so that the resulting graph contains no induced subgraph isomorphic to $F$? In our variant, the input additionally contains a vertex-disjoint packing $\mathcal{H}$ of induced subgraphs of $G$, which provides a lower bound $h(\mathcal{H})$ on the number of edge modifications required to transform $G$ into an $F$-free graph. While earlier works used the number $k$ as parameter or structural parameters of the input graph …
Learning Small Decision Trees with Few Outliers: A Parameterized Perspective
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙特-奥弗涅大学、法国国家科学研究中心、克莱蒙特-奥弗涅理工学院、圣艾蒂安矿业学院、LIMOS实验室）；Ben-Gurion University of the Negev（内盖夫本·古里安大学）
Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisin…
We propose an iterated local search based on several classes of local and large neighborhoods for the bin packing problem with conflicts. This problem, which combines the characteristics of both bin packing and vertex coloring, arises in various application contexts such as logistics and transportation, timetabling, and resource allocation for cloud computing. We introduce $O(1)$ evaluation procedures for classical local-search moves, polynomial variants of ejection chains and assignment neighborhoods, an adaptive set covering-based neighborhood, and finally a controlled use of 0-cost moves to…
Given a family of graphs $\mathcal{F}$, a graph $G$, and a positive integer $k$, the $\mathcal{F}$-Deletion problem asks whether we can delete at most $k$ vertices from $G$ to obtain a graph in $\mathcal{F}$. $\mathcal{F}$-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph $G = (V, \cup_{i=1}^α E_{i})$, where the edge set of $G$ is partitioned into $α$ color classes, is called an $α$-edge-colored graph. A natural extension of the $\mathcal{F}$-Deletion problem to edge-colored graphs is the $α$-Simultaneous $\mathcal{…
Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$. The \textsc{$(p,r,\mathcal{F})$-Packing} problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $…
Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new…
In this paper we consider Simultaneous Feedback Edge Set (Sim-FES) problem. In this problem, the input is an $n$-vertex graph $G$, an integer $k$ and a coloring function ${\sf col}: E(G) \rightarrow 2^{[α]}$ and the objective is to check whether there is an edge subset $S$ of cardinality at most $k$ in $G$ such that for all $i \in [α]$, $G_i - S$ is acyclic. Here, $G_i=(V(G), \{e\in E(G) \mid i \in {\sf col}(e)\})$ and $[α]=\{1,\ldots,α\}$. When $α=1$, the problem is polynomial time solvable. We show that for $α=3$ Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic graphs. The…
For the vast majority of local graph problems standard dynamic programming techniques give c^tw V^O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw^O(tw) V^O(1) time. We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Feedback Vertex Set and Connected Dominating Set, consequently answerin…
We design the first efficient sensitivity oracles and dynamic algorithms for a variety of parameterized problems. Our main approach is to modify the algebraic coding technique from static parameterized algorithm design, which had not previously been used in a dynamic context. We particularly build off of the `extensor coding' method of Brand, Dell and Husfeldt [STOC'18], employing properties of the exterior algebra over different fields. For the $k$-Path detection problem for directed graphs, it is known that no efficient dynamic algorithm exists (under popular assumptions from fine-grained co…
