NP-hardness Results Research Articles

Fast Optimal Genome Tiling with Applications to Microarray Design and Homology Search

In this paper, we consider several variations of the following basic tiling problem: given a sequence of real numbers with two size-bound parameters, we want to find a set of tiles of maximum total weight such that each tiles satisfies the size bounds. A solution to this problem is important to a number of computational biology applications such as selecting genomic DNA fragments for PCR-based amplicon microarrays and performing homology searches with long sequence queries. Our goal is to design efficient algorithms with linear or near-linear time and space in the normal range of parameter values for these problems. For this purpose, we first discuss the solution to a basic online interval maximum problem via a sliding-window approach and show how to use this solution in a nontrivial manner for many of the tiling problems introduced. We also discuss NP-hardness results and approximation algorithms for generalizing our basic tiling problem to higher dimensions. Finally, computational results from applying our tiling algorithms to genomic sequences of five model eukaryotes are reported.

Fast Optimal Genome Tiling with Applications to Microarray Design and Homology Search

In this paper, we consider several variations of the following basic tiling problem: given a sequence of real numbers with two size-bound parameters, we want to find a set of tiles of maximum total weight such that each tiles satisfies the size bounds. A solution to this problem is important to a number of computational biology applications such as selecting genomic DNA fragments for PCR-based amplicon microarrays and performing homology searches with long sequence queries. Our goal is to design efficient algorithms with linear or near-linear time and space in the normal range of parameter values for these problems. For this purpose, we first discuss the solution to a basic online interval maximum problem via a sliding-window approach and show how to use this solution in a nontrivial manner for many of the tiling problems introduced.We also discuss NP-hardness results and approximation algorithms for generalizing our basic tiling problem to higher dimensions. Finally, computational results from applying our tiling algorithms to genomic sequences of five model eukaryotes are reported.

Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses

We prove hardness results for approximating set splitting problems and also instances of satisfiability problems which have no “mixed” clauses, i.e., every clause has either all its literals unnegated or all of them negated. Results of Hastad imply tight hardness results for set splitting when all sets have size exactly $k \ge 4$ elements and also for non-mixed satisfiability problems with exactly $k$ literals in each clause for $k \ge 4$. We consider the case $k=3$. For the MAX E3-SET SPLITTING, problem in which all sets have size exactly 3, we prove an NP-hardness result for approximating within any factor better than ${\frac{19}{20}}$. This result holds even for satisfiable instances of MAX E3-SET SPLITTING, and is based on a PCP construction due to Hastad. For “non-mixed MAX 3SAT,” we give a PCP construction which is a slight variant of the one given by Hastad for MAX 3SAT, and use it to prove the NP-hardness of approximating within a factor better than ${\frac{11}{12}}$.

Total balancedness condition for Steiner tree games

In Steiner tree game associated with a graph G=( V, E), players consist of a subset N⊆ V of nodes. The characteristic function value of a subset S⊆ N of the players is the minimum weight of a Steiner tree that spans S. We show that it is NP-hard to determine whether a Steiner tree game is totally balanced, i.e., cores for all its subgames are non-empty. In addition, the NP-hardness result is also proven for deciding whether the core is non-empty, or whether an imputation is a member of the core.

The complexity of the characterization of networks supporting shortest-path interval routing

Interval Routing is a routing method that was proposed in order to reduce the size of the routing tables by using intervals and was extensively studied and implemented. Some variants of the original method were also defined and studied. The question of characterizing networks which support optimal (i.e., shortest path) Interval Routing has been thoroughly investigated for each of the variants and under different models, with only partial answers, both positive and negative, given so far. In this paper, we study the characterization problem under the most basic model (the one unit cost), and with the most restrictive memory requirements (one interval per edge). We prove that this problem is NP-hard (even for the restricted class of graphs of diameter at most 3). Our result holds for all variants of Interval Routing. It significantly extends some related NP-hardness result, and implies that, unless P = NP , partial characterization results of some classes of networks which support shortest path Interval Routing, cannot be pushed further to lead to efficient characterizations for these classes.

Some comments on sequencing with controllable processing times

We discuss sequencing problems on a single machine with controllable job processing times. For the maximum job cost criterion, we present several polynomial time results. For the total weighted job completion time criterion, we present an NP-hardness result. Our results settle several open questions in this area.

Locating New Stops in a Railway Network

Given a railway network together with information on the population and their use of the railway infrastructure, we are considering the effects of introducing new train stops in the existing railway network. One effect concerns the accessibility of the railway infrastructure to the population, measured in how far people live from their nearest train stop. The second effect we study is the change in travel time for the railway customers that is induced by new train stops. Based on these two models, we introduce two combinatorial optimization problems and give NP-hardness results for them. We suggest an algorithmic approach for the model based on travel time and present a real-world application with its first experimental results.

The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant

We show that approximating the shortest vector problem (in any $\ell_p$ norm) to within any constant factor less than $\sqrt[p]2$ is hard for NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai's constructive variant of Sauer's lemma that greatly simplifies Ajtai's original proof.

Computational complexity of real structured singular value in l/sub p/ setting

This paper studies a generalized real structured singular value (/spl mu/) problem where uncertain parameters are bounded by an l/sub p/ norm. Two results are presented. The first one shows that this generalized /spl mu/ problem is NP-hard for any given rational number p/spl isin/[1, /spl infin/]. The NP-hardness holds as long as le, the size of the largest repeated block, exceeds one. This result generalizes the known NP-hardness result for the conventional /spl mu/ problem (with p=/spl infin/). The second result, which strengthens the first one, considers the approximability problem of the generalized /spl mu/. We show that the problem of obtaining an estimate for the generalized /spl mu/ with some guaranteed bounds on the relative error remains to be NP-hard, regardless how large this bound is.

SOKOBAN and other motion planning problems

We consider a natural family of motion planning problems with movable obstacles and obtain hardness results for them. Some members of the family are shown to be PSPACE-complete thus improving and extending (and also simplifying) a previous NP-hardness result of Wilfong. The family considered includes a motion planning problem which forms the basis of a popular computer game called SOKOBAN. The decision problem corresponding to SOKOBAN is shown to be NP-hard. The motion planning problems considered are related to the “warehouseman's problem” considered by Hopcroft, Schwartz and Sharir, and to geometric versions of the motion planning problem on graphs considered by Papadimitriou, Raghavan, Sudan and Tamaki.

Machine scheduling with availability constraints

We will give a survey on results related to scheduling problems where machines are not continuously available for processing. We will deal with single and multi machine problems and analyze their complexity. We survey NP-hardness results, polynomial optimization and approximation algorithms. We also distinguish between on-line and off-line formulations of the problems. Results are concerned with criteria on completion times and due dates.

On A Scheduling Problem of Time Deteriorating Jobs

We consider a scheduling problem with a single machine and a set of jobs which have to be processed sequentially. While waiting for processing, jobs may deteriorate, causing the processing requirement of each job to grow after a fixed waiting timet0. We prove that the problem of minimizing the makespan—completion time for all jobs—is NP-hard. Next we consider the problem for a natural special case where the job requirement grows linearly at a job-specific rate aftert0. We develop a fully polynomial time approximation scheme for the problem in this case. We also give further NP-hardness results, and a polynomial time algorithm for the case where the job-specific rate is proportional to the initial processing requirement of each job.

New results on drawing angle graphs

An angle graph is a graph with a fixed cyclic order of the edges around each vertex and an angle specified for every pair of consecutive edges incident on a vertex. We study the problem of constructing a drawing of an angle graph that preserves its angles, and present several new results. • • We disprove the conjectures of Vijayan (1986) about the relationship between the planarity of an angle graph and the planarity of its biconnected components. • • We show that testing an angle graph for planarity is NP-hard. • • Using our NP-hardness result, we show that given a triconnected planar graph and an angle α, the problem of determining whether it admits a planar straight-line drawing in which the angle between any two consecutive edges incident on the same vertex is at least α, is NP-hard. • • A multilayered angle graph is one whose edges are assigned to a given set of layers. A multilayered angle graph is multiplanar if it admits a drawing in which edges assigned to the same layer do not cross. We prove that testing a multilayered angle graph for multiplanarity is NP-hard even if we restrict the angles to be multiples of 90° and the number of layers to two. • • We give a simple linear time algorithm for testing whether a series-parallel angle graph admits a (nonplanar) drawing that preserves its angles.

Robust proofs of NP-hardness for protein folding: general lattices and energy potentials.

This paper addresses the robustness of intractability arguments for simplified models of protein folding that use lattices to discretize the space of conformations that a protein can assume. We present two generalized NP-hardness results. The first concerns the intractability of protein folding independent of the lattice used to define the discrete protein-folding model. We consider a previously studied model and prove that for any reasonable lattice the protein-structure prediction problem is NP-hard. The second hardness result concerns the intractability of protein folding for a class of energy formulas that contains a broad range of mean force potentials whose form is similar to commonly used pair potentials (e.g., the Lennard-Jones potential). We prove that protein-structure prediction is NP-hard for any energy formula in this class. These are the first robust intractability results that identify sources of computational complexity of protein-structure prediction that transcend particular problem formulations.

On rectangle intersection and overlap graphs

Let R be a family of isooriented rectangles in the plane. A graph G=(V, E) is called a rectangle intersection (respectively, overlap) graph for R if there is a one-to-one correspondence between V and R such that two vertices in V are adjacent to each other if and only if the corresponding rectangles in R intersect (respectively, overlap) each other. We first prove that the maximum independent set problem is NP-hard even for both cubic planar rectangle intersection and cubic planar rectangle overlap graphs. We then show the NP-completeness of the vertex coloring problem with three colors for both planar rectangle intersection and planar rectangle overlap graphs even when the degree of every vertex is limited to four. These NP-hardness results are obtained for the tightest degree constraint cases. >

Minimizing the number of late jobs on unrelated machines

We consider the problem of preemptively scheduling a set of independent jobs with release times on an arbitrary number of unrelated machines so as to minimize the number of late jobs. We show that the problem is NP-hard in the strong sense, giving the first strong NP-hardness result for this problem.

Some no-wait shops scheduling problems: Complexity aspect

We investigate the computational complexity of no-wait shops scheduling problems. The problem of finding optimal finish time schedules is shown to be NP-hard for flowshops with two machine centres where each machine centre has one or more parallel machines for processing tasks. The complexity results are also reported for no-wait shops scheduling when all nonzero tasks have unit or identical processing time requirement. A polynomial time algorithm for 3-machine flowshops is proposed for optimal finish time schedules. Finding optimal finish time schedules in 2-machine jobshops in NP-hard. Also we establish NP-hard results for 3-machine jobshops for both optimal finish time and mean flow time schedules. Some results are deduced with the present work and with those reported earlier.