Time 2O Research Articles

Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms

Let M =( E , I ) be a matroid and let S ={ S 1 , ċ , S t } be a family of subsets of E of size p . A subfamily Ŝ ⊆ S is q - representative for S if for every set Y ⊆ E of size at most q , if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I , then there is a set Xˆ ∈ Ŝ disjoint from Y with Xˆ ∪ Y ∈ I . By the classic result of Bollobás, in a uniform matroid, every family of sets of size p has a q -representative family with at most ( p + q p ) sets. In his famous “two families theorem” from 1977, Lovász proved that the same bound also holds for any matroid representable over a field F. We give an efficient construction of a q -representative family of size at most ( p + q p ) in time bounded by a polynomial in ( p + q p ), t , and the time required for field operations. We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include the following: —In the L ong D irected C ycle problem, the input is a directed n -vertex graph G and the positive integer k . The task is to find a directed cycle of length at least k in G , if such a cycle exists. As a consequence of our 6.75 k + o ( k ) n O (1) time algorithm, we have that a directed cycle of length at least log n , if such a cycle exists, can be found in polynomial time. —In the M inimum E quivalent G raph (MEG) problem, we are seeking a spanning subdigraph D ′ of a given n -vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D . —We provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many “connectivity” problems such as H amiltonian C ycle or S teiner T ree can be solved in time 2 O ( t ) n on n -vertex graphs of treewidth at most t . For the special case of uniform matroids on n elements, we give a faster algorithm to compute a representative family. We use this algorithm to provide the fastest known deterministic parameterized algorithms for k -P ath , k -T ree , and, more generally, k -S ubgraph I somorphism , where the k -vertex pattern graph is of constant treewidth.

A tight lower bound for Vertex Planarization on graphs of bounded treewidth

In the Vertex Planarization problem one asks to delete the minimum possible number of vertices from an input graph to obtain a planar graph. The parameterized complexity of this problem, parameterized by the solution size (the number of deleted vertices) has recently attracted significant attention. The state-of-the-art algorithm of Jansen et al. (2014) runs in time 2O(klogk)⋅n on an n-vertex graph with a solution of size k. It remains open if one can obtain a single-exponential dependency on k in the running time bound.One of the core technical contributions of the work of Jansen, Lokshtanov, and Saurabh is an algorithm that solves a weighted variant of Vertex Planarization in time 2O(wlogw)⋅n on graphs of treewidth w. In this short note we prove that the running time of this routine is tight under the Exponential Time Hypothesis, even in unweighted graphs and when parameterizing by treedepth. Consequently, it is unlikely that a potential single-exponential algorithm for Vertex Planarization parameterized by the solution size can be obtained by merely improving upon the aforementioned bounded treewidth subroutine.

Reducing rank of the adjacency matrix by graph modification

The main topic of this article is to study a class of graph modification problems. A typical graph modification problem takes as input a graph G, a positive integer k and the objective is to add/delete k vertices (edges) so that the resulting graph belongs to a particular family, F, of graphs. In general the family F is defined by forbidden subgraph/minor characterization. In this paper rather than taking a structural route to define F, we take algebraic route. More formally, given a fixed positive integer r, we define Fr as the family of graphs where for each G∈Fr, the rank of the adjacency matrix of G is at most r. Using the family Fr we initiate algorithmic study, both in classical and parameterized complexity, of following graph modification problems: r-Rank Vertex Deletion, r-Rank Edge Deletion and r-Rank Editing. These problems generalize the classical Vertex Cover problem and a variant of the d-Cluster Editing problem. We first show that all the three problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time 2O(klog⁡r)nO(1) for r-Rank Vertex Deletion, and an algorithm for r-Rank Edge Deletion and r-Rank Editing running in time 2O(f(r)klog⁡k)nO(1). We complement our FPT result by designing polynomial kernels for these problems.

Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V ( G ), called a treewidth-modulator , such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition , is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k ) that has a finite integer index and such that Y es -instances have a treewidth-modulator of size O ( k ) admits a linear kernel on the class of H -topological-minor-free graphs, for any fixed graph H . This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H -minor-free graphs. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n -vertex graph G and a non-negative integer k , P lanar - F -D eletion asks whether G has a set X ⊆ V ( G ) such that | X | ⩽ k and G − X is H -minor-free for every H ϵ F . As our second application, we present the first single-exponential algorithm to solve P lanar - F -D eletion . Namely, our algorithm runs in time 2 O ( k ) · n 2 , which is asymptotically optimal with respect to k . So far, single-exponential algorithms were only known for special cases of the family F .

Exploring the Subexponential Complexity of Completion Problems

Let F be a family of graphs. In the F -C ompletion problem, we are given an n -vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It was shown recently that two special cases of F -C ompletion , namely, (i) the problem of completing into a chordal graph known as M inimum F ill-in (SIAM J. Comput. 2013), which corresponds to the case of F ={ C 4 , C 5 , C 6 , …}, and (ii) the problem of completing into a split graph (Algorithmica 2015), that is, the case of F ={ C 4 , 2 K 2 , C 5 }, are solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . The exploration of this phenomenon is the main motivation for our research on F -C ompletion . In this article, we prove that completions into several well-studied classes of graphs without long induced cycles and paths also admit parameterized subexponential time algorithms by showing that: —The problem T rivially P erfect C ompletion , which is F - C ompletion for F ={ C 4 , P 4 }, a cycle and a path on four vertices, is solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . —The problems known in the literature as P seudosplit C ompletion , the case in which F{2 K 2 , C 4 }, and T hreshold C ompletion , in which F =2 K 2 , P 4 , C 4 }, are also solvable in time 2 O (√ k log k ) n O }(1) . We complement our algorithms for F -C ompletion with the following lower bounds: —For F ={2 K 2 }, F = { C 4 }, F ={ P o 4 }, and F ={2 K 2 , P 4 }, F -C ompletion cannot be solved in time 2 o(k) n O (1) unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F -C ompletion problems for any F ⊆ {2 K 2 , C 4 , P 4 }.

Obtaining split graphs by edge contraction

We study the parameterized complexity of the following Split Contraction problem: Given a graph G, and an integer k as parameter, determine whether G can be modified into a split graph by contracting at most k edges. We show that Split Contraction can be solved in FPT time 2O(k2)n5, but admits no polynomial kernel unless NP⊆coNP/poly.

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Given a graph G and an integer k , the Feedback Vertex Set (FVS) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph. The first fixed-parameter algorithm for FVS in undirected graphs appeared in a monograph of Mehlhorn in 1984. The fixed-parameter tractability (FPT) status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC ’08, JACM ’08) showed that it is fixed-parameter tractable by giving a 4 k k ! · n O (1) time algorithm. There are two subset versions of this problems: We are given an additional subset S of vertices (resp., edges), and we want to hit all cycles passing through a vertex of S (resp., an edge of S ); the two variants are known to be equivalent in the parameterized sense. Recently, the Subset FVS problem in undirected graphs was shown to be FPT by Cygan et al. (ICALP’11, SIDMA’13) and independently by Kakimura et al. (SODA ’12). We generalize the result of Chen et al. (STOC ’08, JACM ’08) by showing that a Subset FVS in directed graphs can be solved in time 2 O ( k 3 ) ċ n O (1) (i.e., FPT parameterized by size k of the solution). By our result, we complete the picture for FVS problems and their subset versions in undirected and directed graphs. The technique of random sampling of important separators was used by Marx and Razgon (STOC ’11, SICOMP ’14) to show that Undirected Multicut is FPT, and it was generalized by Chitnis et al. (SODA ’12, SICOMP ’13) to directed graphs to show that Directed Multiway Cut is FPT. In addition to proving the FPT of a Directed Subset FVS, we reformulate the random sampling of important separators technique in an abstract way that can be used with a general family of transversal problems. We believe this general approach will be useful for showing the FPT of other problems in directed graphs. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from 2 2 O ( k ) to 2 O ( k 2 ) in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA ’12, SICOMP ’13).

The role of planarity in connectivity problems parameterized by treewidth

For some years it was believed that for “connectivity” problems such as Hamiltonian Cycle, algorithms running in time 2O(tw)⋅nO(1) – called single-exponential – existed only on planar and other topologically constrained graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [3, FOCS 2011], Bodlaender et al. [1, ICALP 2013], and Fomin et al. [11, SODA 2014], who provided single-exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single-exponential time on topologically constrained graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like Cycle Packing, that cannot be solved in time 2o(twlog⁡tw)⋅nO(1) on general graphs but that can be solved in time 2O(tw)⋅nO(1) when restricted to planar graphs. Our main contribution is to show that there exist natural problems that can be solved in time 2O(twlog⁡tw)⋅nO(1) on general graphs but that cannot be solved in time 2o(twlog⁡tw)⋅nO(1) even when restricted to planar graphs. Furthermore, we prove that Planar Cycle Packing and Planar Disjoint Pathscannot be solved in time 2o(tw)⋅nO(1). The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited.

Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in time 2O(tw)|V|O(1) for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than twO(tw)|V|O(1). Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time ctw|V|O(1) for a small constant c, e.g., for Hamiltonian Cycle and Steiner Tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in the runtime in terms of the weights).We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic ctw|V|O(1) time algorithms, also for weighted and counting versions. For example, in this time we can solve Traveling Salesman or count the number of Hamiltonian cycles. The rank based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying “small” sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the Matrix Tree Theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

On finding the longest antisymmetric path in directed acyclic graphs

Given a directed acyclic graph and a set of vertex pairs, an antisymmetric path is a directed path that does not contain vertices from the same pair in the set. The goal of the Longest Antisymmetric Path problem is to determine the longest antisymmetric path that connects two given vertices in the graph. The problem has important applications in software testing and bioinformatics. The problem is known to be NP-hard. In this paper, we study the computational lower bound of the problem, we show that the problem cannot be solved in time 2o(n13) unless 3SAT can be solved in subexponential time. In addition, we study the inapproximability of the problem and show that the problem cannot be approximated within a ratio of (n−2)2 in polynomial time unless P=NP. Our technique also suggests that both the lower bound and inapproximability also hold for directed acyclic graphs of degree at most 6.

Isomorphism testing of Boolean functions computable by constant-depth circuits

Given two n-variable Boolean functions f and g, we study the problem of computing an ε-approximate isomorphism between them. An ε-approximate isomorphism is a permutation π of the n Boolean variables such that f(x1,x2,…,xn) and g(xπ(1),xπ(2),…,xπ(n)) differ on at most an ε fraction of all Boolean inputs {0,1}n. We give a randomized 2O(nlog⁡(n/ε)O(d)) time algorithm that computes an ε-approximate isomorphism between two isomorphic Boolean functions f and g that are given by depth d circuits of poly(n) size, where d is a constant independent of n, for any positive ε. In contrast, the best known algorithm for computing an exact isomorphism between n-ary Boolean functions has running time 2O(n)[12] even for functions computed by poly(n) size DNF formulas. Our algorithm is based on a result for hypergraph isomorphism with bounded edge size [4] and the classical Linial–Mansour–Nisan result on approximating small depth and size Boolean circuits by small degree polynomials using Fourier analysis [11].

A linear edge kernel for two-layer crossing minimization

The two-layer crossing minimization problem (TLCM), given a bipartite graph G with n vertices and a positive integer k, asks whether G has a two-layer drawing with at most k crossings. We consider a simple generalization of TLCM called leaf-edge-weighted TLCM (LEW-TLCM), where we allow positive weights on edges incident to leaves, and show that this problem admits a kernel with O(k) edges provided that the given graph is connected. As a straightforward consequence, LEW-TLCM (and hence TLCM) has a fixed parameter algorithm that runs in 2O(klog⁡k)+nO(1) time which improves on the previously best known algorithm with running time 2O(k3)n.

A single-exponential FPT algorithm for the [formula omitted]-minor cover problem

Given a graph G and a parameter k∈N, the parameterized K4-minor cover problem asks whether at most k vertices can be deleted to turn G into a K4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-tVertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While single-exponential FPT algorithms, i.e. running in 2O(k)⋅nO(1) time, are known for these two latter problems, it was open whether the K4-minor cover problem could be solved in single-exponential FPT time. This paper answers this question in the affirmative. Observe that it is known to be unlikely that Treewidth-tVertex Deletion can be solved in time 2o(k)⋅nO(1).

Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters

In the Cluster Editing problem, also known as Correlation Clustering, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential-time parameterized algorithm that in time 2O(pk)+nO(1) decides whether G can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies. Our algorithmic findings are complemented by the following tight lower bound on the asymptotic behavior of our algorithm. We show that unless ETH fails, for any constant 0<σ≤1, there is p=Θ(kσ) such that there is no algorithm deciding in time 2o(pk)⋅nO(1) whether G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.

Faster algorithms for vertex partitioning problems parameterized by clique-width

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a k-expression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k−1)-expression via rank-width and then solving Dominating Set using the (23k−1)-expression, the above algorithm will only give a runtime of 423knO(1). There have been results which overcome this exponential jump; the best known algorithm can solve Dominating Set in time 2O(k2)nO(1) by avoiding constructing a k-expression Bui-Xuan et al. (2013) [7]. We improve this to 2O(klogk)nO(1). Indeed, we show that for a graph of clique-width k, a large class of domination and partitioning problems (LC-VSP), including Dominating Set, can be solved in 2O(klogk)nO(1).Our main tool is a variant of rank-width using the rank of a 0–1 matrix over the rational field instead of the binary field.

Max-Cut Parameterized Above the Edwards-Erdős Bound

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut parameterized above the Edwards-Erd?s bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size $$ \frac{m}{2} + \frac{n-1}{4} + k $$ in time 2O(k)?n4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years. Our algorithm has asymptotically optimal running time, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.

Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs

In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, k-Leaf Out-Branching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely k-Internal Out-Branching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that on these classes of graphs, the k-Directed Path problem is solvable in time O((1+ε)knf(ε)), for any ε>0, where f is some function of ε.Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem-specific combinatorial structures, and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

Satisfying more than half of a system of linear equations over GF(2): A multivariate approach

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x1,…,xn consisting of equations of the form ∏i∈Ixi=b, where xi,b∈{−1,1} and I⊆[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0). We show that MaxLin2-AA[k] has a kernel with at most O(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove that Max-r-Lin2-AA[k,r] has a kernel with at most (2k−1)r variables.

On the computational complexity of finding hard tautologies

It is well known (cf. Krajíček and Pudlák [‘Propositional proof systems, the consistency of first order theories and the complexity of computations’, J. Symbolic Logic 54 (1989) 1063–1079]) that a polynomial time algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1(k) and outputs a tautology τk of size at least k such that P is not p-bounded on the set of all formulas τk. We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP≠coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT∩{0, 1}k. In Find, given 1(k) and a tautology α of size at most k c 0 , one should output a size k tautology β that has no size k c 1 P-proof from substitution instances of α. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2O(k) algorithm. Using a stronger hypothesis about the proof complexity of the Nisan–Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan–Wigderson generator as a proof system.

Fixed-parameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing via iterative localization

Given a permutation π of {1,…,n} and a positive integer k, can π be partitioned into at most k subsequences, each of which is either increasing or decreasing? We give an algorithm with running time 2O(k2logk)nO(1) that solves this problem, thereby showing that it is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number of a given permutation graph on n vertices is at most k. Our algorithm solves in fact a more general problem: within the mentioned running time, it decides whether the cochromatic number of a given perfect graph on n vertices is at most k.To obtain our result we use a combination of two well-known techniques within parameterized algorithms: iterative compression and greedy localization. Consequently we name this combination “iterative localization”. We further demonstrate the power of this combination by giving an algorithm with running time 2O(k2logk)nlogn that decides whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of a given set of horizontal and vertical lines.