Complexity column

Randomized online graph coloring

Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The condition which we follow in graph coloring is that the incident vertices/edges have not the same color. There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and others are online algorithm. Where offline means the graph is known in advance and the online means that the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is added to the graph and the main issue is that we want to minimize the number of colors. We cannot change the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors online in multiple rounds.

We present a genetic-based approach to solve SAT problem and NP-complete problems. The main idea of the approach presented here is to exploit the fact that, although all NP-complete problems are equally difficult in a general computational sense, some have much better genetic representations than others, leading to much more successful use of genetic-based algorithm on some NP-complete problems than on others. Since any NP-complete problem can be mapped into any other one in polynomial time by a transformation, the approach described here consists of identifying and finding a canonical or generic NP-complete problem on which genetic algorithm work well, and solving other NP-complete problems indirectly by translating them onto the canonical problem. We presented some initial results where we have the Boolean Satisfiability Problem (SAT) as a canonical problem, and results on Hamiltonian Circuit problem which represent a family of NP-complete problems, it can be solved efficiently by mapping them first onto SAT problems.KeywordsSATGenetic AlgorithmNP-complete ProblemHamiltonian Circuit

Throughout the academic year, educational institutions use an official and structured document called an academic calendar to communicate and arrange important dates and events. It functions as a thorough timetable that delineates the exact dates of terms, semesters, and sessions that are dedicated to educational activities. The dates of the semesters and terms, deadlines for registration, class schedules, exam periods, holidays and breaks, dates of graduation, special events and academic activities, administrative deadlines etc. are all commonly included in the academic calendar. A crucial component of educational institutions is the examination procedure, which is scheduled with dates and hours decided upon before the academic year starts. An educational institution's course schedule and exam schedule, which require a combination of resources including teachers, subjects, students, and classrooms, are the most significant components of the academic calendar. The academic institute creates the course calendar and exam schedule with the fewest possible resources in order to avoid conflicts by meeting different priority and essential requirements. Graph coloring issues, which are NP complete problems, can be used to solve this scheduling challenge. In this study, under certain limitations, Exam scheduling was created using graph coloring.

Several implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

Complexity-separating graph classes for vertex, edge and total colouring

In this article, we consider the interplay of generalized quantifiers and built-in relations over finite structures, in particular, in the range of logics capturing the circuit complexity classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) . It is well known that for capturing \(\mathrm{AC^{0}}\) first-order logic has to be equipped with order and, e.g., predicates for addition and multiplication, whereas for \(\mathrm{TC^{0}}\) generalized quantifiers such as majority quantifiers are necessary. The sharp division between the classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) can be explained by the fact that \(\mathrm{AC^{0}}\) is not closed under restricting \(\mathrm{AC^{0}}\) -computable queries into simple subsequences of the input, whereas \(\mathrm{TC^{0}}\) is closed under such relativization as its queries can be expressed in terms of first-order formulas using universe-independent generalized quantifiers and order as the only built-in relation. In the terminology of abstract logics, the above means that logics capturing \(\mathrm{AC^{0}}\) do not have the relativization property, and hence, they are not regular logics unlike the logics capturing \(\mathrm{TC^{0}}\) . This weakness of \(\mathrm{AC^{0}}\) has been also elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington et al.) was that if a language \( L \) has a neutral letter, then \( L \) can be defined in \(\operatorname{FO}_{\mathcal{A}}\) , first-order logic with the collection of all numerical built-in relations \(\mathcal{A}\) , if and only if \( L \) can be already defined in \(\operatorname{FO}_{\leq}\) . Our approach is two-fold. First, we study universe-independent cardinality quantifiers \(\operatorname{\mathsf{Q}}\) defined by a parameter set \(S\subseteq\mathbb{N}\) and formulate a combinatorial criterion for \( S \) implying that all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) can be defined in \(\operatorname{FO}_{\leq}(\operatorname{\mathsf{Q}})\) . For instance, this criterion is satisfied if \( S \) is the range of some polynomial with positive integer coefficients of degree at least two. Second, by adapting the key properties of abstract logics to accommodate built-in relations, we define the regular interior \(\operatorname{\mathcal{R}-int}(\mathcal{L})\) (the largest regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}^{*}\subseteq\mathcal{L}\) ) and regular closure \(\operatorname{\mathcal{R}-cl}(\mathcal{L})\) (the least regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}\subseteq\mathcal{L}^{*}\) ), of a logic \(\mathcal{L}\) with built-in relations, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of \(\mathcal{L}\) . By extending the results of Barrington et al., we further show that if \(\mathcal{B}=\{+\}\) , or \(\mathcal{B}\) contains only unary relations besides \(\leq\) , then \(\operatorname{\mathcal{R}-int}(\operatorname{FO}_{\mathcal{B}})\equiv \operatorname{FO}_{\leq}\) . In contrast, our results from the first part of the article imply that if \(\mathcal{B}\) contains \(\leq\) and the range of a polynomial of degree at least two, then \(\operatorname{\mathcal{R}-cl}(\operatorname{FO}_{\mathcal{B}})\) includes all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) .

In this paper, we present a novel computing model, called probe machine (PM). Unlike the turing machine (TM), PM is a fully parallel computing model in the sense that it can simultaneously process multiple pairs of data, rather than sequentially process every pair of linearly adjacent data. We establish the mathematical model of PM as a nine-tuple consisting of data library, probe library, data controller, probe controller, probe operation, computing platform, detector, true solution storage, and residue collector. We analyze the computation capability of the PM model, and in particular, we show that TM is a special case of PM. We revisit two NP-complete problems, i.e., the graph coloring and Hamilton cycle problems, and devise two algorithms on basis of the established PM model, which can enumerate all solutions to each of these problems by only one probe operation. Furthermore, we show that PM can be implemented by leveraging the nano-DNA probe technologies. The computational power of an electronic computer based on TM is known far more than that of the human brain. A question naturally arises: will a future computer based on PM outperform the human brain in more ways beyond the computational power?

We have proposed the mapping of graph coloring problems into swarm dynamics. Empirical evidence that flock steering behaviors augmented with the notion of hostility (enmity and friendliness) are enough to perform efficiently the task of coloring the nodes of graphs even in the case 3-coloration hard graph topologies. We discuss here what are the minimal cognitive capabilities that allow the emergent behavior of swarms to solve such NP-complete problem without mediating an explicit knowledge representation.

For some weighted NP-complete problems, checking whether a proposed solution is optimal is a nontrivial task. Such is the case for the traveling salesman problem, or the spin-glass problem in three dimensions. In this Letter, we consider the weighted tripartite matching problem, a well known NP-complete problem. We write mean-field finite temperature equations for this model and derive their zero temperature limit. We show that any solution of the zero temperature equations provides an exact absolute ground state of the system. As a consequence, we propose a criterion which can be checked in polynomial time, and such that given a putative optimal solution, if the criterion is satisfied, then the solution is indeed optimal. This criterion is generalized to a class of variants of the multiple traveling salesmen problems.

The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, its formalization is attributed to Stephen Cook and Leonid Levin. Despite extensive research, a definitive answer remains elusive. Closely tied to this is the concept of NP-completeness. If a single NP-complete problem could be solved efficiently, it would imply that all problems in NP can be solved efficiently, proving that P equals NP. Garey and Johnson defined K-CLOSURE such that for any edge $(u, v)$ in the directed graph, either node $u$ is in the set $V'$ or node $v$ is not in $V'$. This implies that either both nodes are in $V'$ or both are not in $V'$. Our previous work in IPI Letters presented a polynomial-time algorithm for K-CLOSURE. While no errors have been identified in this work, many believe that Garey and Johnson's original definition was incorrect, and their citation of Queyranne was a misunderstanding. Many argue that the empty set serves as a simple counterexample to Garey and Johnson's definition of K-CLOSURE. This paper proposes that K-CLOSURE is actually an NP-complete problem, which would imply that P equals NP.

Approximation algorithms for Hamming clustering problems

We consider the problem of designing a near-optimal linear decision tree to classify two given point sets B and W in $\Re^n$. A linear decision tree defines a polyhedral subdivision of space; it is a classifier if no leaf region contains points from both sets. We show hardness results for computing such a classifier with approximately optimal depth or size in polynomial time. In particular, we show that unless NP = ZPP, the depth of a classifier cannot be approximated within any constant factor, and that the total number of nodes cannot be approximated within any fixed polynomial. Our proof uses a simple connection between this problem and graph coloring and uses the result of Feige and Kilian on the inapproximability of the chromatic number. We also study the problem of designing a classifier with a single inequality that involves as few variables as possible and point out certain aspects of the difficulty of this problem.

Joint replenishment problems constitute an important class of models in inventory management. They exhibit aspects of possible coordination among multiple products to save costs. Their computational complexity had been open even if there are just two products that need to be synced. In “Integer factorization: Why two-item joint replenishment is hard,” Schulz and Telha present a simple framework based on integer factorization to establish the computational hardness of two variants of the joint replenishment problem with two items. Whereas difficult to solve in practice and not believed to be solvable in polynomial time, integer factorization is not as difficult as NP-complete problems. The authors show that a similar technique can be used to show even the NP-completeness of one variant of the joint replenishment problem (again with just two items).

With the increasing number of Web Services with similar or identical functionality, the non-functional properties of a Web Service will become more and more important. Hence, a choice needs to be made to determine which services are to participate in a given composite service. In general, multi-QoS constrained Web Services composition, with or without optimization, is a NP-complete problem on computational complexity that cannot be exactly solved in polynomial time. A lot of heuristics and approximation algorithms with polynomial- and pseudo-polynomial-time complexities have been designed to deal with this problem. However, these approaches suffer from excessive computational complexities that cannot be used for service composition in runtime. In this paper, we propose a efficient approach for multi-QoS constrained Web Services selection. Firstly, a user preference model was proposed to collect the user's preference. And then, a correlation model of candidate services are established in order to reduce the search space. Based on these two model, a heuristic algorithm is then proposed to find a feasible solution for multi-QoS constrained Web Services selection with high performance and high precision. The experimental results show that the proposed approach can achieve the expecting goal.