Polynomial Time For Trees Research Articles

Reconstructing tumor evolutionary histories and clone trees in polynomial-time with SubMARine.

Tumors contain multiple subpopulations of genetically distinct cancer cells. Reconstructing their evolutionary history can improve our understanding of how cancers develop and respond to treatment. Subclonal reconstruction methods cluster mutations into groups that co-occur within the same subpopulations, estimate the frequency of cells belonging to each subpopulation, and infer the ancestral relationships among the subpopulations by constructing a clone tree. However, often multiple clone trees are consistent with the data and current methods do not efficiently capture this uncertainty; nor can these methods scale to clone trees with a large number of subclonal populations. Here, we formalize the notion of a partially-defined clone tree (partial clone tree for short) that defines a subset of the pairwise ancestral relationships in a clone tree, thereby implicitly representing the set of all clone trees that have these defined pairwise relationships. Also, we introduce a special partial clone tree, the Maximally-Constrained Ancestral Reconstruction (MAR), which summarizes all clone trees fitting the input data equally well. Finally, we extend commonly used clone tree validity conditions to apply to partial clone trees and describe SubMARine, a polynomial-time algorithm producing the subMAR, which approximates the MAR and guarantees that its defined relationships are a subset of those present in the MAR. We also extend SubMARine to work with subclonal copy number aberrations and define equivalence constraints for this purpose. Further, we extend SubMARine to permit noise in the estimates of the subclonal frequencies while retaining its validity conditions and guarantees. In contrast to other clone tree reconstruction methods, SubMARine runs in time and space that scale polynomially in the number of subclones. We show through extensive noise-free simulation, a large lung cancer dataset and a prostate cancer dataset that the subMAR equals the MAR in all cases where only a single clone tree exists and that it is a perfect match to the MAR in most of the other cases. Notably, SubMARine runs in less than 70 seconds on a single thread with less than one Gb of memory on all datasets presented in this paper, including ones with 50 nodes in a clone tree. On the real-world data, SubMARine almost perfectly recovers the previously reported trees and identifies minor errors made in the expert-driven reconstructions of those trees. The freely-available open-source code implementing SubMARine can be downloaded at https://github.com/morrislab/submarine.

Sequential Metric Dimension

In the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph $G$. At every step, one vertex $v$ of $G$ can be probed which results in the knowledge of the distance between $v$ and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location. We address the generalization of this game where $k\geq 1$ vertices can be probed at every step. Our game also generalizes the notion of the {\it metric dimension} of a graph. Precisely, given a graph $G$ and two integers $k,\ell \geq 1$, the {\sc Localization} problem asks whether there exists a strategy to locate a target hidden in $G$ in at most $\ell$ steps and probing at most $k$ vertices per step. We first show that, in general, this problem is \textsf{NP}-complete for every fixed $k \geq 1$ (resp., $\ell \geq 1$). We then focus on the class of trees. On the negative side, we prove that the \Localization problem is \textsf{NP}-complete in trees when $k$ and $\ell$ are part of the input. On the positive side, we design a $(+1)$-approximation algorithm for the problem in $n$-node trees, {\it i.e.}, an algorithm that computes in time $O(n \log n)$ (independent of $k$) a strategy to locate the target in at most one more step than an optimal strategy. This algorithm can be used to solve the \Localization problem in trees in polynomial time if $k$ is fixed. We also consider some of these questions in the context where, upon probing the vertices, the relative distances to the target are retrieved. This variant of the problem generalizes the notion of the {\it centroidal dimension} of a graph.

Learning tractable Bayesian networks in the space of elimination orders

The computational complexity of inference is now one of the most relevant topics in the field of Bayesian networks. Although the literature contains approaches that learn Bayesian networks from high dimensional datasets, traditional methods do not bound the inference complexity of the learned models, often producing models where exact inference is intractable. This paper focuses on learning tractable Bayesian networks from data. To address this problem, we propose strategies for learning Bayesian networks in the space of elimination orders. In this manner, we can efficiently bound the inference complexity of the networks during the learning process. Searching in the combined space of directed acyclic graphs and elimination orders can be extremely computationally demanding. We demonstrate that one type of elimination trees, which we define as valid, can be used as an equivalence class of directed acyclic graphs and elimination orders, removing redundancy. We propose methods for incrementally compiling local changes made to directed acyclic graphs in elimination trees and for searching for elimination trees of low width. Using these methods, we can move through the space of valid elimination trees in polynomial time with respect to the number of network variables and in linear time with respect to treewidth. Experimental results show that our approach successfully bounds the inference complexity of the learned models, while it is competitive with other state-of-the-art methods in terms of fitting to data.

OCTAL: Optimal Completion of gene trees in polynomial time

BackgroundFor a combination of reasons (including data generation protocols, approaches to taxon and gene sampling, and gene birth and loss), estimated gene trees are often incomplete, meaning that they do not contain all of the species of interest. As incomplete gene trees can impact downstream analyses, accurate completion of gene trees is desirable.ResultsWe introduce the Optimal Tree Completion problem, a general optimization problem that involves completing an unrooted binary tree (i.e., adding missing leaves) so as to minimize its distance from a reference tree on a superset of the leaves. We present OCTAL, an algorithm that finds an optimal solution to this problem when the distance between trees is defined using the Robinson–Foulds (RF) distance, and we prove that OCTAL runs in O(n^2) time, where n is the total number of species. We report on a simulation study in which gene trees can differ from the species tree due to incomplete lineage sorting, and estimated gene trees are completed using OCTAL with a reference tree based on a species tree estimated from the multi-locus dataset. OCTAL produces completed gene trees that are closer to the true gene trees than an existing heuristic approach in ASTRAL-II, but the accuracy of a completed gene tree computed by OCTAL depends on how topologically similar the reference tree (typically an estimated species tree) is to the true gene tree.ConclusionsOCTAL is a useful technique for adding missing taxa to incomplete gene trees and provides good accuracy under a wide range of model conditions. However, results show that OCTAL’s accuracy can be reduced when incomplete lineage sorting is high, as the reference tree can be far from the true gene tree. Hence, this study suggests that OCTAL would benefit from using other types of reference trees instead of species trees when there are large topological distances between true gene trees and species trees.

Computing the maximal canonical form for trees in polynomial time

Known algorithms computing a canonical form for trees in linear time use specialized canonical forms for trees and no canonical forms defined for all graphs. For a graph $$G=(V,E)$$ the maximal canonical form is obtained by relabelling the vertices with $$1,\ldots ,|V|$$ in a way that the binary number with $$|V|^2$$ bits that is the result of concatenating the rows of the adjacency matrix is maximal. This maximal canonical form is not only defined for all graphs but even plays a special role among the canonical forms for graphs due to some nesting properties allowing orderly algorithms. We give an $$O(|V|^2)$$ algorithm to compute the maximal canonical form of a tree.

Complexity and inapproximability results for the Power Edge Set problem

We consider the single channel PMU placement problem called the Power Edge Set problem. In this variant of the PMU placement problem, (single channel) PMUs are placed on the edges of an electrical network. Such a PMU measures the current along the edge on which it is placed and the voltage at its two endpoints. The objective is to find the minimum placement of PMUs in the network that ensures its full observability, namely measurement of all the voltages and currents. We prove that PES is NP-hard to approximate within a factor (1.12)- $$\epsilon $$ , for any $$\epsilon > 0$$ . On the positive side we prove that PES problem is solvable in polynomial time for trees and grids.

Safe sets in graphs: Graph classes and structural parameters

A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V \setminus S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity of the problem. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between tree-depth and vertex cover number.

Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of [Formula: see text], in which any destination point can be reached from any starting point by always moving in the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygonal regions with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles and even for trees, but can be solved optimally for trees in polynomial time, if we allow only certain types of GRR contacts. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

A Dynamic Programming Algorithm for the Ridersharing Problem Restricted with Unique Destination and Zero Detour on Trees

We deal with the problem of sharing vehicles by individuals with similar itineraries which is to find the minimum number of drivers, each of which has a vehicle capacity and a detour to realize all trips. Recently, Gu et al. showed that the problem is NP-hard even for star graphs restricted with unique destination, and gave a polynomial-time algorithm to solve the problem for paths restricted with unique destination and zero detour. In this paper we will give a dynamic programming algorithm to solve the problem in polynomial time for trees restricted with unique destination and zero detour. In our best knowledge it is a first polynomial-time algorithm for trees.

The scaling of the minimum sum of edge lengths in uniformly random trees

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.

RASCAL: A Randomized Approach for Coevolutionary Analysis.

A popular method for coevolutionary inference is cophylogenetic reconstruction where the branch length of the phylogenies have been previously derived. This approach, unlike the more generalized reconstruction techniques that are NP-Hard, can reconcile the shared evolutionary history of a pair of phylogenetic trees in polynomial time. This approach, while proven to be highly successful, requires a high polynomial running time. This is quickly becoming a limiting factor of this approach due to the continual increase in size of coevolutionary data sets. One existing method that combats this issue proposes a trade-off of accuracy for an asymptotic time complexity reduction. This technique in almost 70% of cases converges on Pareto optimal solutions in linear time. We build on this prior work by proposing an alternate linear time algorithm (RASCAL) that offers a significant accuracy increase, with RASCAL converging on Pareto optimal solutions in 85% of cases and unlike prior methods can ensure, with high probability, that all optimal solutions can be recovered, provided sufficient replicates are performed.

Two-Step Optimization Approach for the Design of Multiplierless Linear-Phase FIR Filters

Deterministic tree search algorithms for the design of multiplierless linear phase finite impulse response filters are generally time consuming. Many researches therefore focus on how to restrict the number of discrete values assigned to each coefficient during a tree search. In this paper, a two-step tree search algorithm is proposed. In the first step, a polynomial-time tree search algorithm where each coefficient is fixed to a single one discrete value is introduced. Since the synthesis of large coefficients is dominant in the hardware cost over small coefficients, in the second step optimization, the small coefficients obtained in the first step is kept unaltered and the large coefficients are further divided into several groups and the coefficients are optimized group by group alternatingly. Such a two-step search strategy maximally utilizes the limited computational resources and can achieve lower hardware cost design in a shorter design time, compared with existing algorithms.

Injective Colorings with Arithmetic Constraints

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring $$c:V(G)\rightarrow \mathbb {Z}$$c:V(G)?Z of a graph $$G$$G is an additive coloring if for every $$uv, vw$$uv,vw in $$E(G)$$E(G), $$c(u)+c(w)\ne 2c(v)$$c(u)+c(w)?2c(v). The smallest integer $$k$$k such that an injective (resp. additive) coloring of a given graph $$G$$G exists with $$k$$k colors (resp. colors in $$\{1,\ldots ,k\}$${1,?,k}) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by $$\chi _i(G)$$?i(G) and $$\chi '_a(G)$$?a?(G), respectively. In the first part of this work, we present several upper bounds for the additive chromatic index. On the one hand, we prove a super linear upper bound in terms of the injective chromatic number for arbitrary graphs, as well as a linear upper bound for bipartite graphs and trees. Complete graphs are extremal graphs for the super linear bound, while complete balanced bipartite graphs are extremal graphs for the linear bound. On the other hand, we prove a quadratic upper bound in terms of the maximum degree. In the second part, we study the computational complexity of computing $$\chi '_a(G)$$?a?(G). We prove that it can be computed in polynomial time for trees. We also prove that for bounded treewidth graphs, to decide whether $$\chi '_a(G)\le k$$?a?(G)≤k, for a fixed $$k$$k, can be done in polynomial time. On the other hand, we show that for cubic graphs it is NP-complete to decide whether $$\chi '_a(G)\le 4$$?a?(G)≤4. We also prove that for every $$\epsilon >0$$∈>0 there is a polynomial time approximation algorithm with approximation factor $$n^{1/3+\epsilon }$$n1/3+∈ for $$\chi '_a(G)$$?a?(G), when restricted to split graphs. However, unless $$\mathsf P =\mathsf NP $$P=NP, for every $$\epsilon >0$$∈>0 there is no polynomial time approximation algorithm with approximation factor $$n^{1/3-\epsilon }$$n1/3-∈ for $$\chi '_a(G)$$?a?(G), even when restricted to split graphs.

Perfect correspondences between dot-depth and polynomial-time hierarchies

We introduce the polynomial-time tree reducibility (ptt-reducibility) for formal languages. Our main result establishes a one–one correspondence between this reducibility and inclusions between complexity classes. More precisely, for languages B and C it holds that B ptt-reduces to C if and only if the unbalanced leaf-language class of B is robustly contained in the unbalanced leaf-language class of C. Formerly, such correspondence was only known for balanced leaf-language classes. Moreover, we show that restricted to regular languages, the levels 0, 1/2, 1, and 3/2 of the dot-depth hierarchy are closed under ptt-reducibility. Our results also have applications in complexity theory: We obtain the first gap theorem of leaf-language definability above the Boolean closure of NP.

Parameterized algorithms for load coloring problem

One way to state the Load Coloring Problem (LCP) is as follows. Let G=(V,E) be graph and let f:V→{red,blue} be a 2-coloring. An edge e∈E is called red (blue) if both end-vertices of e are red (blue). For a 2-coloring f, let rf′ and bf′ be the number of red and blue edges and let μf(G)=min{rf′,bf′}. Let μ(G) be the maximum of μf(G) over all 2-colorings.We introduce the parameterized problem k-LCP of deciding whether μ(G)⩾k, where k is the parameter. We prove that this problem admits a kernel with at most 7k. Ahuja et al. (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width t can be found in time O⁎(2t). We also show that either G is a Yes-instance of k-LCP or the treewidth of G is at most 2k. Thus, k-LCP can be solved in time O⁎(4k).

Generalized rainbow connectivity of graphs

Let C={c1,c2,…,ck} be a set of k colors, and let ℓ→=(ℓ1,ℓ2,…,ℓk) be a k-tuple of nonnegative integers ℓ1,ℓ2,…,ℓk. For a graph G=(V,E), let f:E→C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is ℓ→-rainbow connected if every two vertices of G have a path P connecting them such that the number of edges on P that are colored with cj is at most ℓj for each index j∈{1,2,…,k}. Given a k-tuple ℓ→ and an edge-colored graph, we study the problem of determining whether the edge-colored graph is ℓ→-rainbow connected. In this paper, we first study the computational complexity of the problem with regard to certain graph classes: the problem is NP-complete even for cacti, while is solvable in polynomial time for trees. We then give an FPT algorithm for general graphs when parameterized by both k and ℓmax=max⁡{ℓj|1⩽j⩽k}.

Shock Graph and Shape matching

The aim of this paper to develop a theory for generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. The shocks are organized into a directed, acyclic shock graph and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar which permits a reduction of a shock graph to a unique rooted shock tree. A novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time.

Multi-Session Function Computation and Multicasting in Undirected Graphs

In the function computation problem, certain nodes of an undirected graph have access to independent data, while some other nodes of the graph require certain functions of the data; this model, motivated by sensor networks and cloud computing, is the focus of this paper. We study the maximum rates at which function computation is possible on a capacitated graph; the capacities on the edges of the graph impose constraints on the communication rate. We consider a simple class of computation strategies based on Steiner-tree packing (so-called computation trees), which does not involve block coding and has minimal delay. With a single terminal requiring function computation, computation trees are known to be optimal when the underlying graph is itself a directed tree, but have arbitrarily poor performance in general directed graphs. Our main result is that computation trees are near optimal for a wide class of function computation requirements even at multiple terminals in undirected graphs. The key technical contribution involves connecting approximation algorithms for Steiner cuts in undirected graphs to the function computation problem. Furthermore, we show that existing algorithms for Steiner tree packings allow us to compute approximately optimal packings of computation trees in polynomial time. We also show a close connection between the function computation problem and a communication problem involving multiple multicasts.

TreeCmp: Comparison of Trees in Polynomial Time

When a phylogenetic reconstruction does not result in one tree but in several, tree metrics permit finding out how far the reconstructed trees are from one another. They also permit to assess the accuracy of a reconstruction if a true tree is known. TreeCmp implements eight metrics that can be calculated in polynomial time for arbitrary (not only bifurcating) trees: four for unrooted (Matching Split metric, which we have recently proposed, Robinson-Foulds, Path Difference, Quartet) and four for rooted trees (Matching Cluster, Robinson-Foulds cluster, Nodal Splitted and Triple). TreeCmp is the first implementation of Matching Split/Cluster metrics and the first efficient and convenient implementation of Nodal Splitted. It allows to compare relatively large trees. We provide an example of the application of TreeCmp to compare the accuracy of ten approaches to phylogenetic reconstruction with trees up to 5000 external nodes, using a measure of accuracy based on normalized similarity between trees.

Minimizing maximum risk for fair network connection with interval data

In this paper we introduce a minimax model for network connection problems with interval parameters. We consider how to connect given nodes in a network with a path or a spanning tree under a given budget, where each link is associated with an interval and can be established at a cost of any value in the interval. The quality of an individual link (or the risk of link failure, etc.) depends on its construction cost and associated interval. To achieve fairness of the network connection, our model aims at the minimization of the maximum risk over all links used. We propose two algorithms that find optimal paths and spanning trees in polynomial time, respectively. The polynomial solvability indicates salient difference between our minimax model and the model of robust deviation criterion for network connection with interval data, which gives rise to NP-hard optimization problems.