NP-completeness Proofs Research Articles

Temporally connected components

We discuss a variety of extensions of connected components in temporal graphs, focusing on extensions using connectivity over time through temporal paths (or journeys). Starting with components induced by temporal sources or sinks, we build up to components induced by multiple sources or sinks, and eventually components where all vertices are sources and sinks, i.e. temporally connected components. The cases of bounded components (i.e. defined on time windows), and open or closed components, are also considered. Our contributions mainly include structural results on the number of components, and algorithmic and complexity results of corresponding decision problems. Several new NP-completeness proofs are provided while exploring the boundaries between easy and difficult problems.

On the computational complexity of the strong geodetic recognition problem

A vertex set $S$ of a graph $G$ is a strong geodetic set if it is possible to choose exactly one shortest path for each pair of vertices in $S$, and cover all the remaining vertices of the graph with these paths. In the \textsc{Strong Geodetic} problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be \NP-complete for general graphs. In this work we introduce the \textsc{Strong Geodetic Recognition} problem (SGR), which consists in determining whether a given vertex set~$S \subseteq V(G)$ is strong geodetic. We demonstrate that this version is \NP-complete. We investigate and compare the computational complexity of both decision problems restricted to some graph classes, deriving polynomial-time algorithms, \NP-completeness proofs, and initial parameterized complexity results, including an answer to an open question in the literature for the complexity of SG for chordal graphs.

Pairwise Preferences in the Stable Marriage Problem

We study the classical, two-sided stable marriage problem under pairwise preferences. In the most general setting, agents are allowed to express their preferences as comparisons of any two of their edges, and they also have the right to declare a draw or even withdraw from such a comparison. This freedom is then gradually restricted as we specify six stages of orderedness in the preferences, ending with the classical case of strictly ordered lists. We study all cases occurring when combining the three known notions of stability—weak, strong, and super-stability—under the assumption that each side of the bipartite market obtains one of the six degrees of orderedness. By designing three polynomial algorithms and two NP-completeness proofs, we determine the complexity of all cases not yet known and thus give an exact boundary in terms of preference structure between tractable and intractable cases.

Total Domination, Connected Vertex Cover and Steiner Tree with Conflicts

Total dominating set, connected vertex cover and Steiner tree are well-known graph problems. Despite the fact that they are NP-complete to optimize, it is easy (even trivial) to find solutions, regardless of their size. In this paper, we study a variant of these problems by adding conflicts, that are pairs of vertices that cannot be both in a solution. This new constraint leads to situations where it is NP-complete to decide if there exists a solution avoiding conflicts. This paper proposes NP-completeness proofs of the existence of a solution for different restricted classes of graphs and conflicts, improving recent results. We also propose polynomial time constructions in several restricted cases and we introduce a new parameter, the stretch, to capture the locality of the conflicts.

Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset \(E'\) of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of \(E'\). We say that \(E'\) is a perfect edge dominating set of G, if every edge not in \(E'\) is dominated by exactly one edge of \(E'\). The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most \(d \ge 3\) and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a \(P_5\)-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a \(P_5\).

Oriented coloring in planar, bipartite, bounded degree 3 acyclic oriented graphs

It has been a challenging problem to determine the smallest graph class where a problem is proved to be hard. In the literature, this has been pointed out to be very important in order to establish the real nature of a combinatorial problem.An orientedk-coloring of an oriented graph G→=(V,E→) is a partition of V into k subsets such that there are no two adjacent vertices belonging to the same subset, and all the arcs between a pair of subsets have the same orientation. The decision problem k−oriented chromatic number (ocnk) consists of an oriented graph G→ and an integer k>0, plus the question if there exists an oriented k-coloring of G→. By its strong appeal, many papers have presented NP-completeness proofs for ocnk. It was not known the complexity status of ocnk when the input graph G→ satisfies that the underlying graph G has maximum degree 3.In this paper we prove that ocn4 is NP-complete for an acyclic oriented graph G→ such that G is at same time: connected, planar, bipartite, and with maximum degree 3.Our result defines a P versus NP-complete dichotomy with respect to the maximum degree Δ(G): ocnk is polynomial if Δ(G)<3 and NP-complete if Δ(G)≥3, since it is known that ocn3 is in P, and that ocnk is in P when the underlying graph has Δ(G)≤2.

Finding a subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems.

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: given an oriented graph (digraph) G, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether G must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs.

Consistency of Sequence-Based Gene Clusters

In comparative genomics, differences or similarities of gene orders are determined to predict functional relations of genes or phylogenetic relations of genomes. For this purpose, various combinatorial models can be used to specify gene clusters--groups of genes that are co-located in a set of genomes. Several approaches have been proposed to reconstruct putative ancestral gene clusters based on the gene order of contemporary species. One prevalent and natural reconstruction criterion is consistency: For a set of reconstructed gene clusters, there should exist a gene order that comprises all given clusters. For permutation-based gene cluster models, efficient methods exist to verify this condition. In this article, we discuss the consistency problem for different gene cluster models on sequences with restricted gene multiplicities. Our results range from linear-time algorithms for the simple model of adjacencies to NP-completeness proofs for more complex models like common intervals.

Complexity dichotomy on partial grid recognition

Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even trivial. A natural step, outstanding thus far, was to provide a broad classification of graphs that make for polynomial or NP-complete instances. We provide such a classification based on the set of allowed vertex degrees in the input graphs, yielding a full dichotomy on the complexity of the problem. As byproducts, the previous NP-completeness result for binary trees was strengthened to strictly binary trees, and the three-dimensional version of the problem was for the first time proven to be NP-complete. Our results were made possible by introducing the concepts of consistent orientations and robust gadgets, and by showing how the former allows NP-completeness proofs by local replacement even in the absence of the latter.

Finding the Most Relevant Fragments in Networks

We study a point pattern detection problem on networks, motivated by applications in geographical analysis, such as crime hotspot detection. Given a network N (a connected graph with non-negative edge lengths) together with a set of sites, which lie on the edges or vertices of N, we look for a connected subnetwork F of N of small total length that contains many sites. The edges of F can form parts of the edges of N. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path or a tree. We give polynomial-time algorithms, NP-hardness and NP-completeness proofs, approximation algorithms, and also fixed-parameter tractable algorithms.

Detecting induced subgraphs

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B , we study the decision problem Π B whose instance is a graph G and question is “Does G contain a realisation of B as an induced subgraph?”. For several B ’s, the complexity of Π B is known and here we give the complexity for several more. Our NP-completeness proofs for Π B ’s rely on the NP-completeness proof of the following problem. Let S be a set of graphs and d be an integer. Let Γ S d be the problem whose instance is ( G , x , y ) where G is a graph whose maximum degree is at most d , with no induced subgraph in S and x , y ∈ V ( G ) are two non-adjacent vertices of degree 2. The question is “Does G contain an induced cycle passing through x , y ?”. Among several results, we prove that Γ 0̸ 3 is NP-complete. We give a simple criterion on a connected graph H to decide whether Γ { H } + ∞ is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.

NP-completeness results for edge modification problems

The aim of edge modification problems is to change the edge set of a given graph as little as possible in order to satisfy a certain property. Edge modification problems in graphs have a lot of applications in different areas, and many polynomial-time algorithms and NP-completeness proofs for this kind of problems are known. In this work we prove new NP-completeness results for these problems in some graph classes, such as interval, circular-arc, permutation, circle, bridged, weakly chordal and clique-Helly graphs.

The importance of the P versus NP question

The P versus NP problem is to determine whether every language accepted by some nondeterministic Turing machine in polynomial time is also accepted by some deterministic Turing machine in polynomial time. Unquestionably this problem has caught the interest of the mathematical community. For example, it is the first of seven million-dollar “Millennium Prize Problems” listed by the Clay Mathematics Institute [www.claymath.org]. The Riemann Hypothesis and Poincare Conjecture, both mathematical classics, are farther down the list. On the other hand, Fields Medalist Steve Smale lists P versus NP as problem number three, after Riemann and Poincare, in “Mathematical Problems for the Next Century” [Smale 1998]. But P versus NP is also a problem of central interest in computer science. It was posed thirty years ago [Cook 1971; Levin 1973] as a problem concerned with the fundamental limits of feasible computation. Although this question is front and center in complexity theory, NP-completeness proofs have become pervasive in many other areas of computer science, including artificial intelligence, databases, programming languages, and computer networks (see Garey and Johnson [1979] for 300 early examples). If the question is resolved, what would be the consequences? Consider first a proof of P=NP. It is possible that the proof is nonconstructive, in the sense that it does not yield an algorithm for any NP-complete problem. Or it might give an impractical algorithm, for example, running in time n100. In either of these cases, the proof would probably have few practical consequences other than to disappoint complexity theorists. However, experience has shown that when natural problems are proved to be in P, a feasible algorithm can be found. There are potential counterexamples to this assertion; most famously, the deep results of Robertson and Seymour [1993–1995], who prove that every minor closed family of graphs can be recognized in time O(n3), but their algorithm has such huge constants it is not practical. But practical algorithms are known for some specific minor-closed families (such as planar graphs), and possibly could be found for other examples if sufficient effort is expended. If P=NP is proved by exhibiting a truly feasible algorithm for an NP-complete problem such as SATISFIABILITY (deciding whether a collection of propositional clauses has a satisfying assignment), the practical consequences would be stunning. First, most of the hundreds of problems shown to be NP-complete can be efficiently reduced to SATISFIABILITY, so many of the optimization problems important to industry could be solved. Second, mathematics would be transformed, because computers could find a formal proof of any theorem which has a proof of reasonable length. This is because formal proofs (say in Zermelo–Fraenkel set theory) are

Precedence Constrained Scheduling: A Case in P

‘Unit execution time’ precedence constrained scheduling (UET) is an NP-complete problem with very few special cases known to be solvable in P-time. In this article we present a practically useful case of UET solvable in P-time: we show that if the task graph is given in levels that are ‘locally’ of in-degree two and of ‘width’ more than 1.55 times the number of processors (plus 1), then an optimal schedule can be found in P-time. Task graphs which represent algebraic computations fall ordinarily in this category. Our algorithm is based on a limited look-ahead technique which allows us to use it in an on-line fashion. In the appendix we give two short NP-completeness proofs which suggest that both ‘width’ and ‘degree’ restrictions are needed to get a polynomially solvable subcase.

VLSI Routing in Polynomial Time

There are plenty of NP-complete problems in VLSI design, like channel routing or switchbox routing in the 2-layer Manhattan model. However, there are quite a few polynomially solvable problems as well, some of them (like the single row routing in the 2-layer Manhattan model) date back to the fifties. Here we survey some old and some more recent results (both polynomially solvable subcases and NP-completeness proofs) in a systematic way, taking into account two hierarchies, namely that of geometry (what to route) and technology (how to route) at the same time. Due to lack of space, this extended abstract does not contain references; a more detailed version is available from the author.

Edge-packing planar graphs by cyclic graphs

Maximum G edge-packing is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H. This paper considers the cases where G and H are planar and G is cyclic. Recent work on the general problem is surveyed, inadequacies and limitations in these results are identified, and NP-completeness proofs for key cases are presented.

Graph Sandwich Problems

The graph sandwich problem for property Π is defined as follows: Given two graphs G 1 = ( V, E 1) and G 2 = ( V, E 2) such that E 1 ⊆ E 2, is there a graph G = ( V, E) such that E 1 ⊆ E ⊆ E 2 which satisfies property Π? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NP-completeness of others. We describe polynomial time algorithms for threshold graphs, split graphs, and cographs. For the sandwich problem for threshold graphs, the only case in which a previous algorithm existed, we obtain a faster algorithm. NP-completeness proofs are given for comparability graphs, permutation graphs, and several other families. For Eulerian graphs; one Version of the problem is polynomial and another is NP-complete.

Decomposability of partially defined Boolean functions

The problem of recognizing decomposability of partially defined Boolean functions is considered. The results include polynomial time algorithms for certain important types of decomposition, as well as NP-completeness proofs for more complex structures.

Some results concerning the complexity of restricted colorings of graphs

We consider the complexity of restricted colorings of a graph in which each vertex (or edge) receives one color from a list of permissible colors associated with that vertex (edge). Since the problem is strongly NP-complete, we assume various restrictions imposed on the number and form of permissible colors and the structure of a graph. In this way we obtain some evidence for comparing the complexity of the restricted vertex coloring problem versus that of edge coloring and arrive at a number of results about special cases that are either positive (polynomial solvability) or negative (NP-completeness proofs).