Fixed-parameter Tractable Algorithm Research Articles

Half-integrality, LP-branching, and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools to fixed-parameter tractable (FPT) algorithms [e.g., Cygan et al., FOCS 2011, 52nd Annual Symposium on Foundations of Computer Science, IEEE, 2011, pp. 150--159; Narayanaswamy et al., STACS 2012, Symposium on Theoretical Aspects of Computer Science, 2012, pp. 338--349]. Although this approach has yielded significant speedups for a range of important problems, it requires the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser--Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover [Nemhauser and Trotter, Math. Program., 8 (1975), pp. 232--248] and Node Multiway Cut [Garg et al., J. Alg., 50 (2004), pp. 49--61]. Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from constraint satisfaction problems [in particular Thapper and Živný, FOCS 2012, 53rd Annual Symposium on Foundations of Computer Science, IEEE, 2012, pp. 669--678] to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties. Our results unify and significantly extend the previously known cases, and yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set where the group is given by oracle access. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. [Algorithmica, 74 (2016), pp. 630--642]. Additionally, we propose a network-flow-based approach to solve several cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.

On $k$-Submodular Relaxation

$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\{0, 1, 2, \dots, k\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, determine whether $f$ is extended to a $k$-submodular function $g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, where $g$ is called a $k$-submodular relaxation of $f$. We give a polymorphic characterization of those functions which admit a $k$-submodular relaxation, and also give a combinatorial $O((k^n)^2)$-time algorithm to find a $k$-submodular relaxation or establish that a $k$-submodular relaxation does not exist. Our algorithm has interesting properties: (1) If the input function is integer valued, then our algorithm outputs a half-integral relaxation, and (2) if the input function is binary, then our algorithm outputs the unique optimal relaxation. We present applications of our algorithm to valued constraint satisfaction problems.

Partially Polynomial Kernels for Set Cover and Test Cover

An instance of the $(n-k)$-Set Cover or the $(n-k)$-Test Cover problems is of the form $(\mathcal{U},\mathcal{S},k)$, where $\mathcal{U}$ is a set with $n$ elements, $\mathcal{S}\subseteq 2^\mathcal{U}$ with $|\mathcal{S}|=m$, and $k$ is the parameter. The instance is a Yes-instance of $(n-k)$-Set Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that every element of $\mathcal{U}$ is contained in some set in $\mathcal{S}'$. Similarly, it is a Yes-instance of $(n-k)$-Test Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that for any pair of elements from $\mathcal{U}$, there exists a set in $\mathcal{S}'$ that contains one of them but not the other. It is known in the literature that both $(n-k)$-Set Cover and $(n-k)$-Test Cover do not admit polynomial kernels (under some well-known complexity theoretic assumptions). However, in this paper we show that they do admit “partially polynomial kernels”: we give polynomial time algorithms that take as input an instance $(\mathcal{U},\mathcal{S},k)$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) and return an equivalent instance $(\tilde{\mathcal{U}},\tilde{\mathcal{S}}, \tilde{k})$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) with $\tilde{k} \leq k$ and $| \tilde{\mathcal{U}}|= \mathcal{O}(k^2)$ (respectively, $|\tilde{\mathcal{U}}|=\mathcal{O}(k^7)$). These results allow us to generalize, improve, and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain “sparsity properties.” Using a part of our partial kernelization algorithm for $(n-k)$-Set Cover, we also get an improved fixed-parameter tractable algorithm for this problem which runs in time $\mathcal{O}(4^kk^{\mathcal{O}(1)}(m+n)+mn)$ improving over the previous best of $\mathcal{O}(8^{k+o(k)}(m+n)^{\mathcal{O}(1)})$. On the other hand, the partially polynomial kernel for $(n-k)$-Test Cover gives an algorithm with running time $\mathcal{O}(2^{\mathcal{O}(k^2)}(m+n)^{\mathcal{O}(1)})$. We believe such an approach could also be useful for other covering problems.

Kernelization complexity of possible winner and coalitional manipulation problems in voting

In the Possible winner problem in computational social choice theory, we are given a set of partial preferences and the question is whether a distinguished candidate could be made winner by extending the partial preferences to linear preferences. Previous work has provided, for many common voting rules, fixed parameter tractable algorithms for the Possible winner problem, with number of candidates as the parameter. However, the corresponding kernelization question is still open and in fact, has been mentioned as a key research challenge [10]. In this paper, we settle this open question for many common voting rules.We show that the Possible winner problem for maximin, Copeland, Bucklin, ranked pairs, and a class of scoring rules that includes the Borda voting rule does not admit a polynomial kernel with the number of candidates as the parameter. We show however that the Coalitional manipulation problem which is an important special case of the Possible winner problem does admit a polynomial kernel for maximin, Copeland, ranked pairs, and a class of scoring rules that includes the Borda voting rule, when the number of manipulators is polynomial in the number of candidates. A significant conclusion of our work is that the Possible winner problem is harder than the Coalitional manipulation problem since the Coalitional manipulation problem admits a polynomial kernel whereas the Possible winner problem does not admit a polynomial kernel.

A fixed-parameter algorithm for the maximum agreement forest problem on multifurcating trees

The Maximum Agreement Forest (MAF) problem on two given phylogenetic trees is an important NP-hard problem in the field of computational biology. In this paper, we study the parameterized version of the MAF problem: given two unrooted (multifurcating) phylogenetic trees T1 and T2 with the same leaf-label set L, and a parameter k, either construct an agreement forest of at most k trees for T1 and T2, or report that no such a forest exists. Whether there is a fixed-parameter tractable algorithm for this problem was posed as an open problem several times in the literature. In this paper, we resolve this open problem by presenting a parameterized algorithm of running time O(4kn5) for the problem.

Faster Existential FO Model Checking on Posets

We prove that the model checking problem for the existential fragment of first-order (FO) logic on partially ordered sets is fixed-parameter tractable (FPT) with respect to the formula and the width of a poset (the maximum size of an antichain). While there is a long line of research into FO model checking on graphs, the study of this problem on posets has been initiated just recently by Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon their result in two ways: (1) the runtime of our algorithm is O(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|). n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow. We complement this result by showing that, under a certain complexity-theoretical assumption, the existential FO model checking problem does not have a polynomial kernel.

Tight lower bounds for the Workflow Satisfiability Problem based on the Strong Exponential Time Hypothesis

The Workflow Satisfiability Problem (WSP) asks whether there exists an assignment of authorized users to the steps in a workflow specification, subject to certain constraints on the assignment. The problem is NP-hard even when restricted to just not equal constraints. Since the number of steps k is relatively small in practice, Wang and Li (2010) [21] introduced a parametrisation of WSP by k. Wang and Li (2010) [21] showed that, in general, the WSP is W[1]-hard, i.e., it is unlikely that there exists a fixed-parameter tractable (FPT) algorithm for solving the WSP. Crampton et al. (2013) [10] and Cohen et al. (2014) [6] designed FPT algorithms of running time O⁎(2k) and O⁎(2klog2⁡k) for the WSP with so-called regular and user-independent constraints, respectively. In this note, we show that there are no algorithms of running time O⁎(2ck) and O⁎(2cklog2⁡k) for the two restrictions of WSP, respectively, with any c<1, unless the Strong Exponential Time Hypothesis fails.

Complexity of alignment and decoding problems: restrictions and approximations

We study the computational complexity of the Viterbi alignment and relaxed decoding problems for IBM model 3, focusing on the problem of finding a solution which has significant overlap with an optimal. That is, an approximate solution is considered good if it looks like some optimal solution with a few mistakes, where mistakes can be wrong values (such as a word aligned incorrectly or a wrong word in decoding), as well as insertions and deletions (spurious/missing words in decoding). In this setting, we show that it is computationally hard to find a solution which is correct on more than half (plus an inverse polynomial fraction) of the words. More precisely, if there is a polynomial-time algorithm computing an alignment for IBM model 3 which agrees with some Viterbi alignment on $$l/2+l^\epsilon $$l/2+l∈ words, where l is the length of the English sentence, or producing a decoding with $$l/2+l^\epsilon $$l/2+l∈ correct words, then P $$=$$= NP. We also present a similar structure inapproximability result for phrase-based alignment. As these strong lower bounds are for the general definitions of the Viterbi alignment and decoding problems, we also consider, from a parameterized complexity perspective, which properties of the input make these problems intractable. As a first step in this direction, we show that Viterbi alignment has a fixed-parameter tractable algorithm with respect to limiting the range of words in the target sentence to which a source word can be aligned. We note that by comparison, limiting maximal fertility--even to three--does not affect NP-hardness of the result.

Hull number: [formula omitted]-free graphs and reduction rules

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u,v∈V of a connected undirected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is (geodesically) convex if I[S]=S. Given a subset S⊆V, the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G).First, we show a polynomial-time algorithm to compute the hull number of any P5-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

Campaign Management Under Approval-Driven Voting Rules

Approval-like voting rules, such as Sincere-Strategy Preference-Based Approval voting (SP-AV), the Bucklin rule (an adaptive variant of k-Approval voting), and the Fallback rule (an adaptive variant of SP-AV) have many desirable properties: for example, they are easy to understand and encourage the candidates to choose electoral platforms that have a broad appeal. In this paper, we investigate both classic and parameterized computational complexity of electoral campaign management under such rules. We focus on two methods that can be used to promote a given candidate: asking voters to move this candidate upwards in their preference order or asking them to change the number of candidates they approve of. We show that finding an optimal campaign management strategy of the first type is easy for both Bucklin and Fallback. In contrast, the second method is computationally hard even if the degree to which we need to affect the votes is small. Nevertheless, we identify a large class of scenarios that admit a fixed-parameter tractable algorithm.

Computing the Fréchet distance between folded polygons

Computing the Fréchet distance for surfaces is a surprisingly hard problem and the only known polynomial-time algorithm is limited to computing it between flat surfaces. We study the problem of computing the Fréchet distance for a class of non-flat surfaces called folded polygons. We present a fixed-parameter tractable algorithm for this problem. Next, we present a polynomial-time approximation algorithm. Finally, we present a restricted class of folded polygons for which we can compute the Fréchet distance for L∞ in polynomial time.

Algorithms for the workflow satisfiability problem engineered for counting constraints

The workflow satisfiability problem (WSP) asks whether there exists an assignment of authorized users to the steps in a workflow specification that satisfies the constraints in the specification. The problem is NP-hard in general, but several subclasses of the problem are known to be fixed-parameter tractable (FPT) when parameterized by the number of steps in the specification. In this paper, we consider the WSP with user-independent counting constraints, a large class of constraints for which the WSP is known to be FPT. We describe an efficient implementation of an FPT algorithm for solving this subclass of the WSP and an experimental evaluation of this algorithm. The algorithm iteratively generates all equivalence classes of possible partial solutions until, whenever possible, it finds a complete solution to the problem. We also provide a reduction from a WSP instance to a pseudo-Boolean SAT instance. We apply this reduction to the instances used in our experiments and solve the resulting PB SAT problems using SAT4J, a PB SAT solver. We compare the performance of our algorithm with that of SAT4J and discuss which of the two approaches would be more effective in practice.

Variable-Deletion Backdoors to Planning

Backdoors are a powerful tool to obtain efficient algorithms for hard problems. Recently, two new notions of backdoors to planning were introduced. However, for one of the new notions (i.e., variable-deletion) only hardness results are known so far. In this work we improve the situation by defining a new type of variable-deletion backdoors based on the extended causal graph of a planning instance. For this notion of backdoors several fixed-parameter tractable algorithms are identified. Furthermore, we explore the capabilities of polynomial time preprocessing, i.e., we check whether there exists a polynomial kernel. Our results also show the close connection between planning and verification problems such as Vector Addition System with States (VASS).

Parallel color-coding

New shared- and distributed-memory parallelizations of the color-coding method.Bioinformatics applications: motif detection, signaling pathway discovery in protein interaction networks.Parallel performance analysis of optimized color-coding implementations. We present new parallelization and memory-reducing strategies for the graph-theoretic color-coding approximation technique, with applications to biological network analysis. Color-coding is a technique that gives fixed parameter tractable algorithms for several well-known NP-hard optimization problems. In this work, by efficiently parallelizing steps in color-coding, we create two new biological protein interaction network analysis tools: Fascia for subgraph counting and motif finding and FastPath for signaling pathway detection. We demonstrate considerable speedup over prior work, and the optimizations introduced in this paper can also be used for other problems where color-coding is applicable.

On Computing the Maximum Parsimony Score of a Phylogenetic Network

Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as maximum parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of maximum parsimony on networks, “hardwired” and “softwired,” and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multiterminal Cut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available

Fixed-Parameter Algorithms for Finding Agreement Supertrees

We study the agreement supertree approach for combining rooted phylogenetic trees when the input trees do not fully agree on the relative positions of the taxa. We consider two ways to deal with such conflict. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. We show that this problem is NP-complete and give a fixed-parameter tractable (FPT) algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. Guillemot and Berry (2010) gave an FPT algorithm for this problem when the input trees are all binary. We give an FPT algorithm for the more general case where the input trees are allowed to have arbitrary degree.

Independent dominating set problem revisited

An independent dominating set of a graph G is a subset D of V such that every vertex not in D is adjacent to at least one vertex of D and no two vertices in D are adjacent. The independent dominating set (IDS) problem asks for an independent dominating set with minimum cardinality. First, we show that the independent dominating set problem and the dominating set problem on cubic bipartite graphs are both NP-complete. As an additional result, we give an alternative and more direct proof for the NP-completeness of both the independent dominating set problem and the dominating set problem on at-most-cubic grid graphs. Next, we show that there are fixed-parameter tractable algorithms for the independent dominating set problem and the dominating set problem on at-most-cubic graphs, which run in O(3.3028k+n) and O(4.2361k+n) time, respectively. Moreover, we consider the weighted independent dominating set problem on (k,ℓ)-graphs. We show that the problem on (2,1)-graphs is NP-complete. We also show that the problem can be solved in linear time for (1,1)-graphs and in polynomial time for (1,ℓ)-graphs for constant ℓ, respectively.

Perfect Phylogeny Problems with Missing Values.

The perfect phylogeny problem is of central importance to both evolutionary biology and population genetics. Missing values are a common occurrence in both sequence and genotype data, but they make the problem of finding a perfect phylogeny NPhard even for binary characters. We introduce new and efficient perfect phylogeny algorithms for broad classes of binary and multistate data with missing values. Specifically, we address binary missing data consistent with the rich data hypothesis (RDH) introduced by Halperin and Karp and give an efficient algorithm for enumerating phylogenies. This algorithm is useful for computing the probability of data with missing values under the coalescent model. In addition, we use the partition intersection (PI) graph and chordal graph theory to generalize the RDH to multi-state characters with missing values. For a bounded number of states, we provide a fixed parameter tractable algorithm for the perfect phylogeny problem with missing data. Utilizing the PI graph, we are able to show that under multiple biologically motivated models for character data, our generalized RDH holds with high probability, and we evaluate our results with extensive empirical analysis.

Some algorithmic results for [2]-sumset covers

Let X={xi:1≤i≤n}⊂N+, and h∈N+. The h-iterated sumset of X, denoted hX, is the set {x1+x2+…+xh:x1,x2,…,xh∈X}, and the [h]-sumset of X, denoted [h]X, is the set ⋃i=1hiX. A [h]-sumset cover of S⊂N+ is a set X⊂N+ such that S⊆[h]X. In this paper, we focus on the case h=2, and study the APX-hard problem of computing a minimum cardinality [2]-sumset cover X of S (i.e. computing a minimum cardinality set X⊂N+ such that every element of S is either an element of X, or the sum of two – non-necessarily distinct – elements of X). We propose two new algorithmic results. First, we give a fixed-parameter tractable (FPT) algorithm that decides the existence of a [2]-sumset cover of size at most k of a given set S. Our algorithm runs in O(2(3log⁡k−1.4)kpoly(k)) time, and thus outperforms the O(5k2(k+3)2k2log⁡(k)) time FPT result presented in Fagnot et al. (2009) [5]. Second, we show that deciding whether a set S has a smaller [2]-sumset cover than itself is NP-hard.

Backdoors to Planning

Backdoors measure the distance to tractable fragments and have become an important tool to find fixed-parameter tractable (fpt) algorithms. Despite their success, backdoors have not been used for planning, a central problem in AI that has a high computational complexity. In this work, we introduce two notions of backdoors building upon the causal graph. We analyze the complexity of finding a small backdoor (detection) and using the backdoor to solve the problem (evaluation) in the light of planning with (un)bounded plan length/domain of the variables. For each setting we present either an fpt-result or rule out the existence thereof by showing parameterized intractability. In three cases we achieve the most desirable outcome: detection and evaluation are fpt.