Exist Polynomial Time Algorithms Research Articles

Counting kernels in directed graphs with arbitrary orientations

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem are all obtained by restricting both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular interval graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear-time solvability on the subclass of threshold graphs.

Impact of soft ride time constraints on the complexity of scheduling in Dial-A-Ride Problems

The Dial-a-Ride problem can contain various constraints for pickup-delivery requests, such as time windows and ride time constraints. Given a tour as a sequence of pickups and deliveries, there exist polynomial time algorithms to find a schedule respecting these constraints (provided that there exists one). However, if no such schedule exists, it is natural to ask for a schedule minimising constraint violations. We model a generic fixed-sequence scheduling problem, which we call Min Pickup-Delivery Scheduling, that allows violation of time window constraints (by late visits) and ride time constraints, both penalised with linear penalty functions. Although our model only considers these two types of time constraints, we show that it can express other common time constraints such as earliness and waiting times. We prove that Min Pickup-Delivery Scheduling is APX-hard, even in the very restrictive case where only the ride time constraints can be violated and the time windows have the same length. Conversely, when only the time window constraints can be violated (by late visits), we present a polynomial time algorithm to solve the problem. Then we focus on instances with some specific structural properties and present two polynomial-time algorithms: one for the case where all the ride time constraints are bounded by a constant, and a second one for the case where all the pickups precede all the deliveries in the sequence of stops.

A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges

We consider three variants of the problem of finding a maximum weight restricted 2-matching in a subcubic graph G. (A 2-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on the variant a restricted 2-matching means a 2-matching that is either triangle-free or square-free or both triangle- and square-free. Since computing a maximum weight square-free 2-matching in a subcubic graph is NP-hard, in the second and third variant we additionally assume that the edge-weights are vertex-induced on each square. While there exist polynomial time algorithms for the first two types of 2-matchings, they are quite complicated or use advanced methodology. For each of the three problems we present a simple reduction to the computation of a maximum weight b-matching. The reduction is conducted with the aid of half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints. For a subset of triangles and/or squares of G, we replace each edge of such a triangle/square with two half-edges. Two half-edges of one edge e of weight w(e) may get different weights, not necessarily equal to 12w(e). In the metric setting when the edge weights satisfy the triangle inequality, this has a geometric interpretation connected to how an incircle partitions the edges of a triangle. Our algorithms are additionally faster than those known before. The running time of each of them is O(n2log⁡n), where n denotes the number of vertices in the graph.

Covering edges in networks

AbstractIn this paper we consider the covering problem on a network G = (V, E) with edge demands. The task is to cover a subset J ⊆ E of the edges with a minimum number of facilities within a predefined coverage radius. We focus on both the nodal and the absolute version of this problem. In the latter, facilities may be placed everywhere in the network. While there already exist polynomial time algorithms to solve the problem on trees, we establish a finite dominating set (i.e., a finite subset of points provably containing an optimal solution) for the absolute version in general graphs. Complexity and approximability results are given and a greedy strategy is proved to be a (1 + ln(|J|))‐approximate algorithm. Finally, the different approaches are compared in a computational study.

A parallel branch-and-bound algorithm to compute a tighter tardiness bound for preemptive global EDF

In this paper we present a parallel exact algorithm to compute an upper bound to tardiness of preemptive global EDF (G-EDF) schedulers, named harmonic bound, which has been proved to be up to 30% tighter than previously proposed bounds. Tightness is a crucial property of tardiness bounds: a too loose bound may cause a feasible soft real-time application to be mistakenly deemed unfeasible. Unfortunately, no polynomial-time algorithm is known to date to compute the harmonic bound. Although there is no proof of hardness of any sort either, the complex formula of the bound apparently provides no hints to devise algorithms with sub-exponential worst-case cost. In this paper we address this issue by proposing a parallel, exact, branch-and-bound algorithm to compute the harmonic bound, called harm-BB, which proves to be extremely fast in a large number of experiments. More specifically, we compare its execution times with those of existing polynomial-time algorithms for other known tardiness bounds on 630,000 random task sets. harm-BB outperforms, or is comparable to, the competitor algorithms in all scenarios but the ones with the highest number of processors (7 and 8) and tasks ( $$\sim $$ 50). In the latter scenarios harm-BB is indeed slower than the other algorithms; yet, it was still feasible, as it takes only about 2.8 s to compute the bound on a commodity dual-core CPU. Even better, we show that harm-BB has a high parallel efficiency, thus its execution time may be largely cut down on highly-parallel platforms.

On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω > log ( n ) added to an Erdős-Rényi graph G ∼ G ( n ,1/2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω (√ n ). By contrast, information theoretically, one can recover planted cliques so long as ω > log ( n ). In this work, we continue the investigation of algorithms from the Sum of Squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [2] and Deshpande and Montanari [25]. Our main result is that degree four SoS does not recover the planted clique unless ω > √ n / polylog n , improving on the bound ω > n 1/3 due to Reference [25]. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of References [2, 25, 27].

The joint movement of pebbles in solving the ( $$N^2-1$$ N 2 - 1 )-puzzle suboptimally and its applications in rule-based cooperative path-finding

The present paper deals with the problem of solving the ($$n^2 - 1$$n2-1)-puzzle and cooperative path-finding (CPF) problems sub-optimally by rule-based algorithms. To solve the puzzle, we need to rearrange $$n^2 - 1$$n2-1 pebbles in the $$n \times n$$n×n-sized square grid using one vacant position to achieve the goal configuration. An improvement to the existing polynomial-time algorithm is proposed and experimentally analyzed. The improved algorithm represents an attempt to move pebbles in a more efficient way compared to the original algorithm by grouping them into so-called snakes and moving them together as part of a snake formation. An experimental evaluation has shown that the snakeenhanced algorithm produces solutions which are 8---9 % shorter than the solutions generated by the original algorithm. Snake-like movement has also been integrated into the rule-based algorithms used in solving CPF problems sub-optimally, which is a closely related task. The task in CPF consists in moving a group of abstract robots on an undirected graph to specific vertices. The robots can move to unoccupied neighboring vertices; no more than one robot can be placed in each vertex. The ($$n^2 - 1$$n2-1)-puzzle is a special case of CPF where the underlying graph is a 4-connected grid and only one vertex is vacant. Two major rule-based algorithms for solving CPF problems were included in our study--BIBOX and PUSH-and-SWAP (PUSH-and-ROTATE). The use of snakes in the BIBOX algorithm led to consistent efficiency gains of around 30 % for the ($$n^2 - 1$$n2-1)-puzzle and up to 50 % in for CPF problems on biconnected graphs with various ear decompositions and multiple vacant vertices. For the PUSH-and-SWAP algorithm, the efficiency gain achieved from the use of snakes was around 5---8 %. However, the efficiency gain was unstable and hardly predictable for PUSH-and-SWAP.

On Bharathi–Kempe–Salek conjecture for influence maximization on arborescence

Bharathi et al. (WINE, pp 306---311, 2007) conjectured that the influence maximization problem is NP-hard for arborescence directed into a root. In this note, we show that this conjecture is not true for deterministic diffusion model and linear threshold (LT) model, that is, there exist polynomial-time algorithms for the influence maximization problem in those two models on arborescence directed into a root. This means that if the conjecture in the independent cascade (IC) model is true, then it would give an interesting difference between the IC model and the LT model.

Finding any nontrivial coarse correlated equilibrium is hard

One of the most appealing aspects of correlated equilibria and coarse correlated equilibria is that natural dynamics quickly arrive at approximations of such equilibria, even in games with many players. In addition, there exist polynomial-time algorithms that compute exact correlated and coarse correlated equilibria. However, in general these dynamics and algorithms do not provide a guarantee on the quality (say, in terms of social welfare) of the resulting equilibrium. In light of these results, a natural question is how good are the correlated and coarse correlated equilibria---in terms natural objectives such as social welfare or Pareto optimality---that can arise from any efficient algorithm or dynamics. We address this question, and establish strong negative results. In particular, we show that in multiplayer games that have a succinct representation, it is NP-hard to compute any coarse correlated equilibrium (or approximate coarse correlated equilibrium) with welfare strictly better than the worst possible. The focus on succinct games ensures that the underlying complexity question is interesting; many multiplayer games of interest are in fact succinct. We show that analogous hardness results hold for correlated equilibria, and persist under the egalitarian objective or Pareto optimality. To complement the hardness results, we develop an algorithmic framework that identifies settings in which we can efficiently compute an approximate correlated equilibrium with near-optimal welfare. We use this framework to develop an efficient algorithm for computing an approximate correlated equilibrium with near-optimal welfare in aggregative games.

Online vector scheduling and generalized load balancing

We give a polynomial time reduction from the vector scheduling problem (VS) to the generalized load balancing problem (GLB). This reduction gives the first non-trivial online algorithm for VS where vectors come in an online fashion. The online algorithm is very simple in that each vector only needs to minimize the Lln(md) norm of the resulting load when it comes, where m is the number of partitions and d is the dimension of vectors. It has an approximation bound of elog(md), which is in O(ln(md)), so it also improves the O(ln2d) bound of the existing polynomial time algorithm for VS. Additionally, the reduction shows that GLB does not have constant approximation algorithms that run in polynomial time unless P=NP.

CONSTRUCTING A MINIMUM PHYLOGENETIC NETWORK FROM A DENSE TRIPLET SET

For a given set L of species and a set T of triplets on L, we seek to construct a phylogenetic network which is consistent with T i.e. which represents all triplets of T. The level of a network is defined as the maximum number of hybrid vertices in its biconnected components. When T is dense, there exist polynomial time algorithms to construct level-0,1 and 2 networks (Aho et al., 1981; Jansson, Nguyen and Sung, 2006; Jansson and Sung, 2006; Iersel et al., 2009). For higher levels, partial answers were obtained in the paper by Iersel and Kelk (2008), with a polynomial time algorithm for simple networks. In this paper, we detail the first complete answer for the general case, solving a problem proposed in Jansson and Sung (2006) and Iersel et al. (2009). For any k fixed, it is possible to construct a level-k network having the minimum number of hybrid vertices and consistent with T, if there is any, in time O(T(k+1)n([4k/3]+1)).

Low-rank exploitation in semidefinite programming for control

Many control-related problems can be cast as semidefinite programs. Even though there exist polynomial time algorithms and excellent publicly available solvers, the time it takes to solve these problems can be excessive. What many of these problems have in common, in particular in control, is that some of the variables enter as matrix-valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this article, we describe how this can be done, and show how our code, called STRUL, can be used in conjunction with the semidefinite programming solver SDPT3. The idea behind the structure exploitation is classical and is implemented in LMI Lab, but we show that when using a modern semidefinite programming framework such as SDPT3, the computational time can be significantly reduced. Finally, we describe how the modelling language YALMIP has been changed in such a way that our code, which can be freely downloaded, can be interfaced using standard YALMIP commands. This greatly simplifies modelling and usage.

Surface- Based Computing Model of Maximum Independent Set Problem

About thirty years ago, the concept of the complexity of the problem was proposed. The most important complex class is P and NP class. Fruitful results of this concept are the existence of the so-called complex class complete problem. If the other issues of this class once solved in polynomial time, then the problem must exist polynomial time algorithms. Therefore, the complete problem is most difficult to solve, but because of their presence, we can choose any of them improved algorithm for a problem, so this kind of problem to get a good solution. DNA computing is a novel method that solving a class of intractable computational problems, in which the computing speeds up exponentially with the problem size. Up to now, many accomplishments have been made to improve its performance and increase its reliability. Maximum Independent Set problem (MIS) is a well-known NP-complete problem. Maximum Clique and Minimum Vertex Covering problem is equivalent to Maximum Independent Set problem. In this paper, we explore solving Maximum Independent Set problem by transforming it into equivalent 0-1 programming problem, and utilizing the surface computing model of that. The proposed method demonstrates universal nature of NP-complete problem.

Approximation Algorithm and Scheme for RNA Maximum Weighted Stacking

Pseudoknotted RNA structure prediction is an important problem in bioinformatics. Existing polynomial time algorithms have no performance guarantee or can handle only limited types of pseudoknots. In this paper for the general problem of pseudoknotted RNA structure prediction, maximum weighted stacking problem is presented based on stacking actions, and its polynomial time approximation algorithm with O(nlogn) time and O(n) space and polynomial time approximation scheme are given. The approximate performance ratio of this approximation algorithm is 3. Compared with existing polynomial time algorithm, they have exact approximation performance and can predict arbitrary pseudoknots.

Fast Algorithm for Pseudoknotted RNA Structure Prediction

Existing polynomial time algorithms can handle only limited types of pseudoknots, or have too high time or space to predict long sequences. In this paper a heuristic algorithm is presented to maximize stems and predict arbitrary pseudoknots with O(n^3) time and O(n) space for a large scale of 5000 bases. Compared with maximum weighted matching algorithm, our algorithm reduce space complexity from O(n^2) to O(n); and the experimental results show that its sensitivity is improved from 80% to 87.8%, and specificity is increased from 53.7% to 75.9%. Compared with genetic algorithm with the accuracy of 83.3% and simulated annealing algorithm with the accuracy of 79.7%, our algorithm increases the predicted accuracy to 87.5%.

On probe permutation graphs

Given a class of graphs G , a graph G is a probe graph of G if its vertices can be partitioned into two sets, P , the probes, and an independent set N , the nonprobes, such that G can be embedded into a graph of G by adding edges between certain vertices of N . If the partition of the vertices into probes and nonprobes is part of the input, then we call the graph a partitioned probe graph of G . In this paper, we provide a recognition algorithm for partitioned probe permutation graphs with time complexity O ( n 2 ) , where n is the number of vertices of the input graph. We show that a probe permutation graph has at most O ( n 4 ) minimal separators. As a consequence, for probe permutation graphs there exist polynomial-time algorithms solving problems like treewidth and minimum fill-in. We also characterize those graphs for which the probe graphs must be weakly chordal.

Exact ground states of large two-dimensional planar Ising spin glasses

Studying spin-glass physics through analyzing their ground-state properties has a long history. Although there exist polynomial-time algorithms for the two-dimensional planar case, where the problem of finding ground states is transformed to a minimum-weight perfect matching problem, the reachable system sizes have been limited both by the needed CPU time and by memory requirements. In this work, we present an algorithm for the calculation of exact ground states for two-dimensional Ising spin glasses with free boundary conditions in at least one direction. The algorithmic foundations of the method date back to the work of Kasteleyn from the 1960s for computing the complete partition function of the Ising model. Using Kasteleyn cities, we calculate exact ground states for huge two-dimensional planar Ising spin-glass lattices (up to 3000;{2} spins) within reasonable time. According to our knowledge, these are the largest sizes currently available. Kasteleyn cities were recently also used by Thomas and Middleton in the context of extended ground states on the torus. Moreover, they show that the method can also be used for computing ground states of planar graphs. Furthermore, we point out that the correctness of heuristically computed ground states can easily be verified. Finally, we evaluate the solution quality of heuristic variants of the L. Bieche approach.

A cutting plane method for solving KYP-SDPs

Semidefinite programs originating from the Kalman–Yakubovich–Popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them. However, the number of variables is often very large making the computational time extremely long. Algorithms more efficient than general purpose solvers are thus needed. To this end structure exploiting algorithms have been proposed, based on the dual formulation. In this paper a cutting plane algorithm is proposed. In a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension.

On the combinatorial structure of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, supplying wireless access to voice/data communication networks for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee. As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.

Edge-coloring of multigraphs

We introduce a monotone invariant π ( G ) on graphs and show that it is an upper bound of the chromatic index of graphs. Moreover, there exist polynomial time algorithms for computing π ( G ) and for coloring edges of a multigraph G by π ( G ) colors. This generalizes the classical edge-coloring theorems of Shannon and Vizing.