Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers \(k_\ell \) and \(k_r\), and a family \(\mathcal F \) of \(\ell \) sets over U, the Gen-RBSC problem is to decide whether there is a subfamily \(\mathcal F '\subseteq \mathcal F \) of size at most \(k_\ell \) that covers all blue elements, but at most \(k_r\) of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting, when parameterized by \(k_\ell +k_r\). In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for the parameters \(k_\ell , k_r\), and \(k_\ell +k_r\). For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter \(k_\ell +k_r\), for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless \(\text { co-NP}\subseteq \text { NP}/\mathrm{poly}\). Further, we show that the FPT algorithm does not generalize to higher dimensions.

A classical result by Edwards states that every connected graph G on n vertices and m edges has a cut of size at least \(\frac{m}{2}+\frac{n-1}{4}\). We generalize this result to r-hypergraphs, with a suitable notion of connectivity that coincides with the notion of connectivity on graphs for r = 2. More precisely, we show that for every “partition connected” r-hypergraph (every hyperedge is of size at most r) H over a vertex set V(H), and edge set E(H) = {e 1,e 2,…e m }, there always exists a 2-coloring of V(H) with {1, − 1} such that the number of hyperedges that have a vertex assigned 1 as well as a vertex assigned − 1 (or get “split”) is at least \(\mu_H+\frac{n-1}{r2^{r-1}}\). Here \(\mu_H=\sum_{i=1}^{m}(1- 2/2^{|e_i|})=\sum_{i=1}^{m}(1- 2^{1-|e_i|})\). We use our result to show that a version of r -Set Splitting, namely, Above Average r -Set Splitting (AA- r -SS), is fixed parameter tractable (FPT). Observe that a random 2-coloring that sets each vertex of the hypergraph H to 1 or − 1 with equal probability always splits at least μ H hyperedges. In AA- r -SS, we are given an r-hypergraph H and a positive integer k and the question is whether there exists a 2-coloring of V(H) that splits at least μ H + k hyperedges. We give an algorithm for AA- r -SS that runs in time f(k)n O(1), showing that it is FPT, even when r = c 1 logn, for every fixed constant c 1 < 1. Prior to our work AA- r -SS was known to be FPT only for constant r. We also complement our algorithmic result by showing that unless NP ⊆ DTIME(n loglogn), AA-⌈logn ⌉-SS is not in XP.KeywordsBoolean FunctionConnected GraphReduction RulePrimal GraphOrdinary GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with \(\mathcal {O}(2^{\mathcal {O}(t \log t)} n^{\mathcal {O}(1)})\) running time. Several structural parameterizations for Cycle Packing have been studied in the literature and our FPT algorithm fills a gap in the ecology of such parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm.

Balanced stable marriage: How close is close enough?

The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines.While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k. Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that “almost have the consecutive-ones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row.For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k ·n O(1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply.

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring \(f:V(G) \rightarrow \{r,b\}\) of G such that for every \(i \in \{r,b\}\), there are at least k edges with both end vertices colored i. It is known that the problem is \(\mathsf {NP} \)-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time \(f(w)n^{o(w)}\), unless ETH fails and it can be solved in time \(n^{O(w)}\), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is \(\mathsf {NP} \)-complete even on bipartite graphs and split graphs.

Long directed detours: Reduction to 2-Disjoint Paths

Given a graph G , a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer ℓ , Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least ℓ . In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP ⊆ coNP/poly.

Fixed-parameter tractability for subset feedback set problems with parity constraints

We study the parameterized complexity of the following Euler subgraph problems: (a) Large Euler Subgraph: For a given graph $G$ and integer parameter $k$, does $G$ contain an induced Eulerian subgraph with at least $k$ vertices? (b) Long Circuit: For a given graph $G$ and integer parameter $k$, does $G$ contain an Eulerian subgraph with at least $k$ edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. The complexity of the problem changes drastically on directed graphs, and we obtain the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed $k>3$ and is solvable in polynomial time for $k\leq 3$. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs.

Comparing genomes in terms of gene order is a classical combinatorial optimization problem in computational biology. Some of the popular distances include translocation, reversal, and double-cut-and-join (abbreviated as DCJ), which have been extensively used while comparing two genomes. Let \(d_x\), \(x\in \{\)translocation, reversal, DCJ\(\}\), be the distance between two genomes such that one can be sorted/converted into the other using the minimum number of x-operations. All these problems are NP-hard when the genomes are unsigned. Computing \(d_x\), \(x\in \{\)translocation, reversal, DCJ\(\}\), between two unsigned genomes involves computing a proper alternating cycle decomposition of its breakpoint graph, which becomes the bottleneck for computing the genomic distance under almost all types of genome rearrangement operations and prohibits to obtain approximation factors better than 1.375 in polynomial time. In this paper, we devise an FPT (fixed-parameter tractable) approximation algorithm for computing the DCJ and translocation distances with an approximation factor 4/3+\(\varepsilon \), and the running time is \(O^*(2^{d^*})\), where \(d^*\) represents the optimal DCJ or translocation distance. The algorithm is randomized and it succeeds with a high probability. This technique is based on a new randomized method to generate approximate maximum alternating cycle decomposition.

• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} →V such that for every edge uv ∈ E, |β − 1(u) − β − 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time \(2^{{\mathcal{O}}(k)}n^{k+1}\). In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form \(f(k)n^{{\mathcal{O}}(1)}\) for any function f even for trees, unless the entire W-hierarchy collapses.We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time \(f(k)n^{{\mathcal{O}}(1)}\) for some function f. In this paper we present an algorithm with running time \(2^{{\mathcal O}(k \log k)} n^2\) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.KeywordsInterval GraphGraph ClassHair LengthPermutation GraphVertex PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.