XP Algorithm Research Articles

The Complexity of Object Association in Multiple Object Tracking

Object association, i.e., the identification of which observations correspond to the same object, is a central task for the area of multiple object tracking. Two prominent models capturing this task have been introduced in the literature: the Lifted Multicut model and the more recent Lifted Paths model. Here, we carry out a detailed complexity-theoretic study of the problems arising from these two models that is aimed at complementing previous empirical work on object association. We obtain a comprehensive complexity map for both models that takes into account natural restrictions to instances such as possible bounds on the number of frames, number of tracked objects and branching degree, as well as less explicit structural restrictions such as having bounded treewidth. Our results include new fixed-parameter and XP algorithms for the problems as well as hardness proofs which altogether indicate that the Lifted Paths problem exhibits a more favorable complexity behavior than Lifted Multicut.

Integer programming in parameterized complexity: Five miniatures

Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k)poly(n). We focus on: •Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.•Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.•Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, Max-q-Cut, and certain other coloring problems by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, indefinite quadratic programs in fixed dimension, parametric integer programs in fixed dimension, and 2-stage stochastic integer programs.

Individual-Based Stability in Hedonic Diversity Games

In hedonic diversity games (HDGs), recently introduced by Bredereck, Elkind, and Igarashi (2019), each agent belongs to one of two classes (men and women, vegetarians and meat-eaters, junior and senior researchers), and agents' preferences over coalitions are determined by the fraction of agents from their class in each coalition. Bredereck et al. show that while an HDG may fail to have a Nash stable (NS) or a core stable (CS) outcome, every HDG in which all agents have single-peaked preferences admits an individually stable (IS) outcome, which can be computed in polynomial time. In this work, we extend and strengthen these results in several ways. First, we establish that the problem of deciding if an HDG has an NS outcome is NP-complete, but admits an XP algorithm with respect to the size of the smaller class. Second, we show that, in fact, all HDGs admit IS outcomes that can be computed in polynomial time; our algorithm for finding such outcomes is considerably simpler than that of Bredereck et al. We also consider two ways of generalizing the model of Bredereck et al. to k ≥ 2 classes. We complement our theoretical results by empirical analysis, comparing the IS outcomes found by our algorithm, the algorithm of Bredereck et al. and a natural better-response dynamics.

An Optimal XP Algorithm for Hamiltonian Cycle on Graphs of Bounded Clique-Width

In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time $$n^{\mathcal {O}(k)}$$. This improves the naive algorithm that runs in time $$n^{\mathcal {O}(k^2)}$$ by Espelage et al. (Graph-theoretic concepts in computer science, vol 2204. Springer, Berlin, 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $$n^{o(k)}$$ (Fomin et al. in SIAM J Comput 43:1541–1563, 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges.

Adapting The Directed Grid Theorem into an FPT Algorithm

Originally proved in 1986 by Robertson and Seymour, the Grid Theorem is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem isn directed graphs was conjectured by Johnson et al. in 2001, and proved recently by Kawarabayashi and Kreutzer in 2015. Namely, they showed that there is a function f(k) such that every directed graph of directed tree-width at least f(k) contains a cylindrical grid of size k as a butterfly minor. Moreover, they claim that their proof can be turned into an XP algorithm, with parameter k, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this article, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer and we improve the XP algorithm into an FPT algorithm.The first step of the proof is an XP algorithm by Johnson et al. in 2001 that decides whether a directed graph D has directed tree-width at most 3k − 2 or admits a haven of order k. It is worth mentioning that a skecth of an FPT algorithm for this problem appears in Chapter 9 of the book ”Classes of Directed Graphs”, from 2018, with an approximation factor of 5k + 2. Our first contribution is to adapt the proof from Johnson et al. to find either an arboreal decomposition of width at most 3k − 2 or a haven of order k in a directed graph D in FPT time, by making use of important separators. We then follow the roadmap of the proof by Kawarabayashi and Kreutzer by locally improving the complexity at some steps, in particular concerning the problem of finding hitting sets for brambles of large order.

A Tight Lower Bound for Planar Steiner Orientation

In the Steiner Orientation problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs mathscr {T}. The question is whether we can orient the undirected edges in a way such that there is a directed sleadsto t path for each terminal pair (s,t)in mathscr {T}. Arkin and Hassin [DAM’02] showed that the Steiner Orientation problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2. From the viewpoint of exact algorithms, Cygan et al. [ESA’12, SIDMA’13] designed an XP algorithm running in n^{O(k)} time for all kge 1. Pilipczuk and Wahlström [SODA’16, TOCT’18] showed that the Steiner Orientation problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS’01] the Steiner Orientation problem does not admit an f(k)cdot n^{o(k/log k)} algorithm for any computable function f. In this paper, we give a short and easy proof that the n^{O(k)} algorithm of Cygan et al. is asymptotically optimal, even if the input graph is planar. Formally, we show that the Planar Steiner Orientation problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)cdot n^{o(k)} time for any computable function f. Moreover, under a stronger hypothesis called Gap-ETH of Dinur [ECCC’16] and Manurangsi and Raghavendra [ICALP’17], we are able to show that there is no constant vartheta >0 such that Planar Steiner Orientation admits an (frac{19}{20}+vartheta )-approximation in FPT time, i.e., no f(k)cdot n^{o(k)} time algorithm can distinguish between the case when all k pairs are satisfiable versus the case when less than k cdot (frac{19}{20}+vartheta ) pairs are satisfiable. To the best of our knowledge, this is the first FPT inapproximability result on planar graphs.

Linear representation of transversal matroids and gammoids parameterized by rank

Given a bipartite graph G=(U⊎V,E), a linear representation of the transversal matroid associated with G on the ground set U, can be constructed in randomized polynomial time. In fact one can get a linear representation deterministically in time 2O(m2n), where m=|U| and n=|V|, by looping through all the choices made in the randomized algorithm. Other important matroids for which one can obtain linear representation deterministically in time similar to the one for transversal matroids include gammoids and strict gammoids. Strict gammoids are duals of transversal matroids and gammoids are restrictions of strict gammoids. We give faster deterministic algorithms to construct linear representations of transversal matroids, gammoids and strict gammoids. All our algorithms run in time (mr)mO(1), where m is the cardinality of the ground set and r is the rank of the matroid. In the language of parameterized complexity, we give an XP algorithm for finding linear representations of transversal matroids, gammoids and strict gammoids parameterized by the rank of the given matroid.

<i>In silico</i> docking studies of RP2 (X-Linked retinitis pigmentosa) protein using anthocyanins as potential inhibitors

Mutations occurring in X-Linked retinitis pigmentosa (RP2) gene causes retinal damage, ultimately leading to vision loss. Owing to the complications of the disease, there is no effective treatment till date. In the present study, we have refined the crystal structure of RP2 using protein preparation wizard of Glide software. Active site of the protein was predicted with SiteMap program. Five anthocyanin compounds from Syzygium cumini fruit peel were docked into the active site of RP2. Molecular docking calculations were performed by employing Glide XP algorithm. Cyanidin 3,5 diglucoside with lowest Gscore (-12.62 kcal/mol) was rescored subsequently with Prime-MMGBSA binding free energy calculations. Binding affinity (-93.62 kcal/mol) of the inhibitor suggests its potential use in the treatment of retinitis pigmentosa.

A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width

In this paper we develop new algorithmic machinery for solving hard problems on graphs of bounded rank-width and on digraphs of bounded bi-rank-width in polynomial (XP, to be precise) time. These include, particularly, graph coloring and chromatic polynomial problems, the Hamiltonian path and c-min-leaf outbranching, the directed cut, and more generally MSOL-partitioning problems on digraphs. Our focus on a formally clean and unified approach for the considered algorithmic problems is in contrast with many previous published XP algorithms running on graphs of bounded clique-width, which mostly used ad hoc techniques and ideas. The new contributions include faster algorithms for computing the chromatic number and the chromatic polynomial on graphs of bounded rank-width, and new algorithms for solving the defective coloring, the min-leaf outbranching, and the directed cut problems.