Effective Polynomial-time Research Articles

Diminishable parameterized problems and strict polynomial kernelization

Kernelization – a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems – plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-s ize kernels under the assumption of NP⊈coNP/poly. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [CiE 2009, Theory Comput. Syst. 2011], we underline the applicability of their framework by showing that a variety of fixed-parameter tractable problems, including graph problems and Turing machine computation problems, does not admit strict polynomial kernels under the assumption of P≠NP, an assumption being weaker than the assumption of NP⊈coNP/poly. Finally, we study an adaption of the framework to a relaxation of the notion of strict kernels, where in the latter one is not allowed to increase the parameter of the reduced instance by more than a constant times the input parameter.

Towards energy-efficient task scheduling on smartphones in mobile crowd sensing systems

In a mobile crowd sensing system, a smartphone undertakes many different sensing tasks that demand data from various sensors. In this paper, we consider the problem of scheduling different sensing tasks assigned to a smartphone with the objective of minimizing sensing energy consumption while ensuring Quality of SenSing (QoSS). First, we consider a simple case in which each sensing task only requests data from a single sensor. We formally define the corresponding problem as the Minimum Energy Single-sensor task Scheduling (MESS) problem and present a polynomial-time optimal algorithm to solve it. Furthermore, we address a more general case in which some sensing tasks request multiple sensors to report their measurements simultaneously. We present an Integer Linear Programming (ILP) formulation as well as two effective polynomial-time heuristic algorithms, for the corresponding Minimum Energy Multi-sensor task Scheduling (MEMS) problem. Extensive simulation results show that the proposed algorithms achieve significant energy savings, compared to a widely-used baseline approach; moreover, the proposed heuristic algorithms produce close-to-optimal solutions.

Partial Solvers for Parity Games: Effective Polynomial-Time Composition

Partial methods play an important role in formal methods and beyond. Recently such methods were developed for parity games, where polynomial-time partial solvers decide the winners of a subset of nodes. We investigate here how effective polynomial-time partial solvers can be by studying interactions of partial solvers based on generic composition patterns that preserve polynomial-time computability. We show that use of such composition patterns discovers new partial solvers - including those that merge node sets that have the same but unknown winner - by studying games that composed partial solvers can neither solve nor simplify. We experimentally validate that this data-driven approach to refinement leads to polynomial-time partial solvers that can solve all standard benchmarks of structured games. For one of these polynomial-time partial solvers not even a sole random game from a few billion random games of varying configuration was found that it won't solve completely.

Interval scheduling and colorful independent sets

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural "compactness" parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are: 1. We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is NP-hard otherwise. 2. We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization). 3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter "maximum number of live jobs" to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter k and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less than five minutes.

Average parameterization and partial kernelization for computing medians

We propose an effective polynomial-time preprocessing strategy for intractable median problems. Developing a new methodological framework, we show that if the input objects of generally intractable problems exhibit a sufficiently high degree of similarity between each other on average, then there are efficient exact solving algorithms. In other words, we show that the median problems Swap Median Permutation, Consensus Clustering, Kemeny Score, and Kemeny Tie Score all are fixed-parameter tractable with respect to the parameter “average distance between input objects”. To this end, we develop the novel concept of “partial kernelization” and, furthermore, identify polynomial-time solvable special cases for the considered problems.

Certain classical groups are not well-defined

In this article we show that the isomorphism type of certain semilinear classical groups may depend on the choice of form matrix, as well as the dimension and the field size. When there is more than one isomorphism type, we count them and present effective polynomial-time algorithms to determine whether two such groups are isomorphic.

Link Scheduling With Power Control for Throughput Enhancement in Multihop Wireless Networks

Joint scheduling and power control schemes have previously been proposed to reduce power dissipation in wireless ad hoc networks. However, instead of power consumption, throughput is a more important performance concern for some emerging multihop wireless networks, such as wireless mesh networks. This paper examines joint link scheduling and power control with the objective of throughput improvement. The MAximum THroughput link Scheduling with Power Control (MATH-SPC) problem is first formulated and then a mixed integer linear programming (MILP) formulation is presented to provide optimal solutions. However, simply maximizing the throughput may lead to a severe bias on bandwidth allocation among links. To achieve a good tradeoff between throughput and fairness, a new parameter called the demand satisfaction factor (DSF) to characterize the fairness of bandwidth allocation and formulate the MAximum Throughput fAir link Scheduling with Power Control (MATA-SPC) problem is defined. An MILP formulation and an effective polynomial-time heuristic algorithm, namely, the serial linear programming rounding (SLPR) heuristic, to solve the MATA-SPC problem are also presented. Numerical results show that bandwidth can be fairly allocated among all links/flows by solving the MILP formulation or by using the heuristic algorithm at the cost of a minor reduction of network throughput. In addition, extensions to end-to-end throughput and fairness and multiradio wireless multihop networks are discussed

Optimal reliability design in an electrical distribution system via a polynomial-time algorithm

In this paper, the optimal design of reliability indices in an electrical distribution system and their impact to planning are studied. By formulating the cost due to interrupted KVA-hour, initial interruption cost, and modification cost in terms of component reliability indices, an optimization problem is derived while simultaneously considering multiple load points and feeder capacity limits, resulting in a nonlinear programming problem. To solve this nonlinear programming problem, an effective polynomial-time algorithm is presented. The numerical test shows that not only significant savings can be achieved, but also information on identifying value-saving feeder/components can be revealed as well. This is a valuable tool to distribution planning/operations.

An effective polynomial-time heuristic for the minimum-cardinality IIS set-covering problem

The state-of-the-art methods for analyzing infeasible linear programs concentrate on isolating Irreducible Infeasible Sets (IISs) of constraints. However, when there are numerous infeasibilities in the model, it is also useful to identify a minimum-cardinality set of constraints which, if removed from the LP, renders it feasible. This set of constraintscovers the IISs. This paper presents a heuristic algorithm for finding a small-cardinality set of constraints which covers the IISs in an infeasible LP. Empirical tests show that it finds a true minimum-cardinality cover over all of the examples in a test set, though this performance cannot be guaranteed in general.

Algorithms to solve some classes of network minimax problems and their applications

We consider some classes of network minimax problems that generalize combinatorial versions of shortest and longest path problems in networks. Effective polynomial-time algorithms are proposed for solving these problems.