DOCSLIB.ORG

A Some NP-Complete Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

A Some NP-Complete Problems To ask the hard question is simple. But what does it mean? What are we going to do? W. H. Auden In this appendix we present a brief list of NP-complete problems; we restrict ourselves to problems which either were mentioned before or are closely related to subjects treated in the book. A much more extensive list can be found in Garey and Johnson [GaJo79]. Chinese postman (cf. Section 14.5) Let G =(V,A,E) be a mixed graph, where A is the set of directed edges and E the set of undirected edges of G. Moreover, let w be a nonnegative length function on A ∪ E,andc be a positive number. Does there exist a cycle of length at most c in G which contains each edge at least once and which uses the edges in A according to their given orientation? This problem was shown to be NP-complete by Papadimitriou [Pap76], even when G is a planar graph with maximal degree 3 and w(e) = 1 for all edges e. However, it is polynomial for graphs and digraphs; that is, if either A = ∅ or E = ∅. See Theorem 14.5.4 and Exercise 14.5.6. Chromatic index (cf. Section 9.3) Let G be a graph. Is it possible to color the edges of G with k colors, that is, does χ(G) ≤ k hold? Holyer [Hol81] proved that this problem is NP-complete for each k ≥ 3; this holds even for the special case where k =3andG is 3-regular. Chromatic number (cf. Section 9.1) Let G be a graph. Is it possible to color the vertices of G with k colors, that is, does χ(G) ≤ k hold? Karp [Kar72] proved that this problem is NP-complete for each k ≥ 3. Even the special case where k =3andG is a planar graph with maximal 502 A Some NP-Complete Problems degree 4 remains NP-complete; see [GaJS76]. Assuming P =NP,thereisnot even a polynomial approximative algorithm which always needs fewer than 2χ(G) colors; see [GaJo76]. For perfect graphs, the chromatic number can be computed in polynomial time; see [GrLS93]. Clique (cf. Exercise 2.8.3) Let G =(V,E) be a graph and c ≤|V | a positive integer. Does G contain a clique consisting of c vertices? This problem is NP-complete by a result of Karp [Kar72]; thus determin- ing the clique number ω(G) is an NP-hard problem. Also, the related question of whether G contains a clique with at least r|V | vertices, where 0 <r<1, is NP-complete for fixed r.AssumingP= NP, there is not even a polynomial ε-approximative algorithm for determining a maximal clique; see [ArSa02]. However, the problem can be solved in polynomial time for perfect (in partic- ular, for bipartite) graphs; see [GrLS93]. Diameter (cf. Section 3.9) Let G =(V,E) be a connected graph, and let c ≤|V | be a positive integer. Recall that the diameter of G can be determined efficiently; see Section 3.9. However, the following two related problems are NP-complete; see [ChTh78] and [GaJo79]. (1) Does there exist a strongly connected orientation H of G with diameter at most c? Note that Robbins’ theorem allows us to check efficiently whether a strongly connected orientation exists (and to find such an orientation, if possible); see Section 1.6. (2) Let C be a given set of at most |E| nonnegative integers. Does there exist a mapping w : E → C such that G has weighted diameter at most c, that is, such that any two vertices u, v have distance d(u, v) ≤ c in the network (G, w)? This problem remains NP-complete even for C = {0, 1}. Discrete metric realization (cf. Section 3.2) Let D =(dxy)beann × n matrix with integer entries representing distances in a finite metric space . Is there a network (G, w) of total length ≤ k realizing D? Winkler [Win88] proved that this problem – and also the the analogous real problem – is NP-complete. Disjoint paths (cf. Section 7.1) Let G =(V,E) be a graph, s and t be two vertices of G,andk and c be two A Some NP-Complete Problems 503 positive integers. Does G contain k vertex disjoint paths of length at most c from s to t? Itai, Pearl and Shiloach [ItPS82] proved that this problem is NP-complete for each fixed k ≥ 5 (whereas it is polynomial for fixed k ≤ 4). Similar results hold for edge disjoint paths from s to t, and for the analogous problems where each path should contain precisely c edges. In contrast, the maximal number of (edge or vertex) disjoint paths from s to t can be determined efficiently using network flow methods if no restrictions are added; see Section 7.1. Graph partitioning (cf. Section 9.2) Let G =(V,E) be a graph and c a positive integer. The question of whether G can be partitioned into at most c subgraphs of a given type is NP-complete for many classes of subgraphs: for triangles and, more generally, for subgraphs with a given isomorphism type, for Hamiltonian subgraphs, for forests, for cliques, and for matchings. We refer the reader to [GaJo79, §A1.1] and the references given there. In particular, determining the clique partition number θ(G)isanNP- hard problem in general. For perfect graphs, this problem can be solved in polynomial time; see [GrLS93]. Hamiltonian cycle (cf. Sections 1.4 and 2.8) Let G =(V,E) be a graph. Does G contain a Hamiltonian cycle? Karp [Kar72] proved that this problem is NP-complete; it remains NP- complete even if we know a Hamiltonian path of G [PaSt77]; see Theorem 15.7.3. The special cases for bipartite graphs and for planar, 3-connected, 3- regular graphs are still NP-complete; see [Kri75] and [GaJT76]. The analogous problem for directed Hamiltonian cycles in digraphs likewise is NP-complete [Kar72]; see Exercise 2.7.6. Hamiltonian path (cf. Exercise 2.7.7) Does the graph G =(V,E) contain a Hamiltonian path? This problem and the analogous directed problem are NP-complete, even if the start and end vertices of the Hamiltonian path are fixed; see [GaJo79]. Independent set (cf. Exercise 2.8.3) Let G =(V,E) be a graph and c ≤|V | a positive integer. Does G contain an independent set with c elements? Note that the independent sets of G are precisely the cliques of the complementary graph G. This problem is therefore NP-complete in general, but polynomial for perfect graphs (see Clique and Vertex cover). 504 A Some NP-Complete Problems The independent set problem remains NP-complete when restricted to 3-regular planar graphs; see [GaJS76]. Induced subgraph Let G =(V,E) be a graph and c a positive integer. The problem of whether G contains an induced subgraph on c vertices that belongs to a prescribed class of graphs is often NP-complete: for cliques and independent sets (see Clique and Independent set), and also for planar subgraphs, for bipartite subgraphs, for subforests, etc. We refer to [GaJo79, §A1.2] and the references given there. Integer linear programming (cf. Section 14.3) Let A be an m × n integer matrix, c ∈ Zn and b ∈ Zm integer vectors, and d an integer. Does there exist an integer vector x ∈ Zn satisfying x ≥ 0, AxT ≤ bT ,andcxT ≥ d? This problem is NP-complete by a result of Karp [Kar72], whereas the corresponding linear program (where x may have rational entries) can be solved in polynomial time by the work of Khachyan [Kha79]. Longest cycle Let N =(G, w) be a network with a nonnegative length function w, where G is a graph, and let c be a positive integer. Does N contain a cycle of length at least c? This problem is NP-complete even when all edges have length 1; see Ex- ercise 2.7.8. Similar results hold for the analogous directed problem. Longest path (cf. Sections 2.7 and 3.1) Let s and t be two vertices in a network N =(G, w) on a graph G, where w is a nonnegative length function, and let c be a positive integer. Does there exist a path from s to t of length at least c? This problem is NP-complete even when all edges have length 1; see Exercise 2.7.8. Similar results hold for the analogous directed problem. Matroid intersection (cf. Section 5.4) Let (E,Si)(i =1, 2, 3) be three matroids on the same set E, and let c be a positive integer. Does E have a subset U of cardinality c which is an indepen- dent set for all three matroids? This problem is NP-complete; see Theorem 5.4.13. Note that the corre- sponding problem for the intersection of two matroids is solvable in polynomial time (even in the weighted case); see [Law75, Law76], [Edm79], [Cun86], and [Whi87]. A Some NP-Complete Problems 505 Max cut (cf. Chapter 6) Let G =(V,E) be a graph with a nonnegative capacity function c, and let b be a positive integer. Does there exist a cut (S, T )ofG with capacity c(S, T ) ≥ b? This problem is NP-complete by a result of Karp [Kar72]; this holds even in the special case where G has maximal degree 3 and c(e) = 1 for all edges e;see [Yan78]. Thus determining a cut of maximal capacity is an NP-hard problem, whereas the analogous problem for cuts of minimal capacity is easy.
Recommended publications
  • On Computing Longest Paths in Small Graph Classes

    On Computing Longest Paths in Small Graph Classes

    On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not.
  • Algebraically Grid-Like Graphs Have Large Tree-Width

    Algebraically Grid-Like Graphs Have Large Tree-Width

    Algebraically grid-like graphs have large tree-width Daniel Weißauer Department of Mathematics University of Hamburg Abstract By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of ’grid-likeness’ of a graph. The grid contains a long cycle on the perimeter, which is the F2-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let k,p be integers, γ a real number and G a graph. Suppose that G contains a cycle of length at least 2γpk which is the F2-sum of cycles of length at most p and whose metric is distorted by a factor of at most γ. Then G has tree-width at least k. 1 Introduction For a positive integer n, the (n × n)-grid is the graph Gn whose vertices are all pairs (i, j) with 1 ≤ i, j ≤ n, where two points are adjacent when they are at Euclidean distance 1. The cycle Cn, which bounds the outer face in the natural drawing of Gn in the plane, has length 4(n−1) and is the F2-sum of the rectangles bounding the inner faces. This is by itself not a distinctive feature of graphs with large tree-width: The situation is similar for the n-wheel Wn, arXiv:1802.05158v1 [math.CO] 14 Feb 2018 the graph consisting of a cycle Dn of length n and a vertex x∈ / Dn which is adjacent to every vertex of Dn.
  • Approximating the Maximum Clique Minor and Some Subgraph Homeomorphism Problems

    Approximating the Maximum Clique Minor and Some Subgraph Homeomorphism Problems

    Approximating the maximum clique minor and some subgraph homeomorphism problems Noga Alon1, Andrzej Lingas2, and Martin Wahlen2 1 Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv. [email protected] 2 Department of Computer Science, Lund University, 22100 Lund. [email protected], [email protected], Fax +46 46 13 10 21 Abstract. We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and √ Thomason supplies an O( n) bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollob´asand Thomason √ and of Koml´osand Szemer´ediprovides an O( n) bound on the ap- proximation factor for the maximum homeomorphic clique achievable in γ polynomial time. On the other hand, we show an Ω(n1/2−O(1/(log n) )) O(1) lower bound (for some constant γ, unless N P ⊆ ZPTIME(2(log n) )) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound.
  • Short Cycles

    Short Cycles

    Short Cycles Minimum Cycle Bases of Graphs from Chemistry and Biochemistry Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium an der Fakultat¨ fur¨ Naturwissenschaften und Mathematik der Universitat¨ Wien Vorgelegt von Petra Manuela Gleiss im September 2001 An dieser Stelle m¨ochte ich mich herzlich bei all jenen bedanken, die zum Entstehen der vorliegenden Arbeit beigetragen haben. Allen voran Peter Stadler, der mich durch seine wissenschaftliche Leitung, sein ub¨ er- w¨altigendes Wissen und seine Geduld unterstutzte,¨ sowie Josef Leydold, ohne den ich so manch tieferen Einblick in die Mathematik nicht gewonnen h¨atte. Ivo Hofacker, dermich oftmals aus den unendlichen Weiten des \Computer Universums" rettete. Meinem Bruder Jurgen¨ Gleiss, fur¨ die Einfuhrung¨ und Hilfstellungen bei meinen Kampf mit C++. Daniela Dorigoni, die die Daten der atmosph¨arischen Netzwerke in den Computer eingeben hat. Allen Kolleginnen und Kollegen vom Institut, fur¨ die Hilfsbereitschaft. Meine Eltern Erika und Franz Gleiss, die mir durch ihre Unterstutzung¨ ein Studium erm¨oglichten. Meiner Oma Maria Fischer, fur¨ den immerw¨ahrenden Glauben an mich. Meinen Schwiegereltern Irmtraud und Gun¨ ther Scharner, fur¨ die oftmalige Betreuung meiner Kinder. Zum Schluss Roland Scharner, Florian und Sarah Gleiss, meinen drei Liebsten, die mich immer wieder aufbauten und in die reale Welt zuruc¨ kfuhrten.¨ Ich wurde teilweise vom osterreic¨ hischem Fonds zur F¨orderung der Wissenschaftlichen Forschung, Proj.No. P14094-MAT finanziell unterstuzt.¨ Zusammenfassung In der Biochemie werden Kreis-Basen nicht nur bei der Betrachtung kleiner einfacher organischer Molekule,¨ sondern auch bei Struktur Untersuchungen hoch komplexer Biomolekule,¨ sowie zur Veranschaulichung chemische Reaktionsnetzwerke herange- zogen. Die kleinste kanonische Menge von Kreisen zur Beschreibung der zyklischen Struk- tur eines ungerichteten Graphen ist die Menge der relevanten Kreis (Vereingungs- menge aller minimaler Kreis-Basen).
  • Downloaded the Biochemical Pathways Information from the Saccharomyces Genome Database (SGD) [29], Which Contains for Each Enzyme of S

    Downloaded the Biochemical Pathways Information from the Saccharomyces Genome Database (SGD) [29], Which Contains for Each Enzyme of S

    Algorithms 2015, 8, 810-831; doi:10.3390/a8040810 OPEN ACCESS algorithms ISSN 1999-4893 www.mdpi.com/journal/algorithms Article Finding Supported Paths in Heterogeneous Networks y Guillaume Fertin 1;*, Christian Komusiewicz 2, Hafedh Mohamed-Babou 1 and Irena Rusu 1 1 LINA, UMR CNRS 6241, Université de Nantes, Nantes 44322, France; E-Mails: [email protected] (H.M.-B.); [email protected] (I.R.) 2 Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Berlin D-10587, Germany; E-Mail: [email protected] y This paper is an extended version of our paper published in the Proceedings of the 11th International Symposium on Experimental Algorithms (SEA 2012), Bordeaux, France, 7–9 June 2012, originally entitled “Algorithms for Subnetwork Mining in Heterogeneous Networks”. * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +33-2-5112-5824. Academic Editor: Giuseppe Lancia Received: 17 June 2015 / Accepted: 29 September 2015 / Published: 9 October 2015 Abstract: Subnetwork mining is an essential issue in the analysis of biological, social and communication networks. Recent applications require the simultaneous mining of several networks on the same or a similar vertex set. That is, one searches for subnetworks fulfilling different properties in each input network. We study the case that the input consists of a directed graph D and an undirected graph G on the same vertex set, and the sought pattern is a path P in D whose vertex set induces a connected subgraph of G. In this context, three concrete problems arise, depending on whether the existence of P is questioned or whether the length of P is to be optimized: in that case, one can search for a longest path or (maybe less intuitively) a shortest one.
  • On Digraph Width Measures in Parameterized Algorithmics (Extended Abstract)

    On Digraph Width Measures in Parameterized Algorithmics (Extended Abstract)

    On Digraph Width Measures in Parameterized Algorithmics (extended abstract) Robert Ganian1, Petr Hlinˇen´y1, Joachim Kneis2, Alexander Langer2, Jan Obdrˇz´alek1, and Peter Rossmanith2 1 Faculty of Informatics, Masaryk University, Brno, Czech Republic {xganian1,hlineny,obdrzalek}@fi.muni.cz 2 Theoretical Computer Science, RWTH Aachen University, Germany {kneis,langer,rossmani}@cs.rwth-aachen.de Abstract. In contrast to undirected width measures (such as tree- width or clique-width), which have provided many important algo- rithmic applications, analogous measures for digraphs such as DAG- width or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer–Ordyniak, Dankelmann–Gutin–Kim, or Lampis– Kaouri–Mitsou, have given some evidence for this. We support this di- rection by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAG- depth. On the positive side, we also note that taking Kant´e’s directed generalization of rank-width as a parameter makes many problems fixed parameter tractable. 1 Introduction The very successful concept of graph tree-width was introduced in the context of the Graph Minors project by Robertson and Seymour [RS86,RS91], and it turned out to be very useful for efficiently solving graph problems. Tree-width is a property of undirected graphs. In this paper we will be interested in directed graphs or digraphs. Naturally, a width measure specifically tailored to digraphs with all the nice properties of tree-width would be tremendously useful. The properties of such a measure should include at least the following: i) The width measure is small on many interesting instances.
  • Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

    Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

    Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree Karthekeyan Chandrasekaran∗ Elena Grigorescuy Gabriel Istratez Shubhang Kulkarniy Young-San Liny Minshen Zhuy November 23, 2020 Abstract A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence as well as maximum-sized binary tree. We show the following results: 1. The longest heapable subsequence problem can be solved in kO(log k)n time, where k is the number of distinct values in the input sequence. 2. We introduce the alphabet size as a new parameter in the study of computational prob- lems in permutation DAGs. Our result on longest heapable subsequence implies that the maximum-sized binary tree problem in a given permutation DAG is fixed-parameter tractable when parameterized by the alphabet size. 3. We show that the alphabet size with respect to a fixed topological ordering can be com- puted in polynomial time, admits a min-max relation, and has a polyhedral description. 4. We design a fixed-parameter algorithm with run-time wO(w)n for the maximum-sized bi- nary tree problem in undirected graphs when parameterized by treewidth w.
  • Section 2.6. Even Subgraphs

    Section 2.6. Even Subgraphs

    2.6. Even Subgraphs 1 Section 2.6. Even Subgraphs Note. Recall from Section 2.4, “Decompositions and Coverings,” that an even graph is a graph in which every vertex has even degree. In this section we consider the symmetric difference of even subgraphs of a given graph and we see how even subgraphs interact with edge cuts. We also define three vector spaces associated with a graph, the edge space, the cycle space, and the bond space. Definition. For a given graph G, an even subgraph of G is a spanning subgraph of G which is even. (Sometimes we may use the term “even subgraph of G” to indicate just the edge set of the subgraph.) Note. Theorem 2.10 and Proposition 2.13 of the previous section allow us to address the symmetric difference of two even subgraphs of a given graph, as follows. Corollary 2.16. The symmetric difference of two even subgraphs is an even subgraph. Note. Bondy and Murty comment that “. we show in Chapters 4 [“Trees”] and 21 [“Integer Flows and Coverings”], [that] the even subgraphs of a graph play an important structural role.” See page 64. In the context of even subgraphs, the term “cycle” may be used to mean the edge set of a cycle, and the term “disjoint cycles” may mean edge-disjoint cycles. With this convention, Veblen’s Theorem (Theorem 2.7), which states that a graph admits a cycle decomposition if and only if it is even, can be restated as follows. 2.6. Even Subgraphs 2 Theorem 2.17.
  • Planarity and Duality of Finite and Infinite Graphs

    Planarity and Duality of Finite and Infinite Graphs

    JOURNAL OF COMBINATORIAL THEORY, Series B 29, 244-271 (1980) Planarity and Duality of Finite and Infinite Graphs CARSTEN THOMASSEN Matematisk Institut, Universitets Parken, 8000 Aarhus C, Denmark Communicated by the Editors Received September 17, 1979 We present a short proof of the following theorems simultaneously: Kuratowski’s theorem, Fary’s theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski’s theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whit- ney, MacLane, Tutte, and Fournier (in the case of Whitney’s theorem and MacLane’s theorem this has already been done by Tutte). In connection with Tutte’s planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte’s condition in terms of overlap graphs is equivalent to Kuratowski’s condition, we characterize completely the infinite graphs satisfying MacLane’s condition and we prove that the 3- connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte’s criterion involving non-separating cy- cles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.
  • Some NP-Complete Problems

    Some NP-Complete Problems

    Appendix A Some NP-Complete Problems To ask the hard question is simple. But what does it mean? What are we going to do? W.H. Auden In this appendix we present a brief list of NP-complete problems; we restrict ourselves to problems which either were mentioned before or are closely re- lated to subjects treated in the book. A much more extensive list can be found in Garey and Johnson [GarJo79]. Chinese postman (cf. Sect. 14.5) Let G =(V,A,E) be a mixed graph, where A is the set of directed edges and E the set of undirected edges of G. Moreover, let w be a nonnegative length function on A ∪ E,andc be a positive number. Does there exist a cycle of length at most c in G which contains each edge at least once and which uses the edges in A according to their given orientation? This problem was shown to be NP-complete by Papadimitriou [Pap76], even when G is a planar graph with maximal degree 3 and w(e) = 1 for all edges e. However, it is polynomial for graphs and digraphs; that is, if either A = ∅ or E = ∅. See Theorem 14.5.4 and Exercise 14.5.6. Chromatic index (cf. Sect. 9.3) Let G be a graph. Is it possible to color the edges of G with k colors, that is, does χ(G) ≤ k hold? Holyer [Hol81] proved that this problem is NP-complete for each k ≥ 3; this holds even for the special case where k =3 and G is 3-regular.
  • Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure

    Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure

    Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure Kalyn R. Lamey Daniel S. Silver Susan G. Williams∗ July 21, 2021 Abstract The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. When G is connected, the subspace C0 of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a cofinite free Zd-action by ±1 ±1 automorphisms, C is dual to a finitely generated module over the polynomial ring F[x1 ; : : : ; xd ] and for it polynomial invariants, the Laplacian polynomials ∆k; k ≥ 0, are defined. Properties of the Laplacian polynomials are discussed. The logarithmic Mahler measure of ∆0 is characterized in terms of the growth of spanning trees. MSC: 05C10, 37B10, 57M25, 82B20 1 Introduction Graphs have been an important part of knot theory investigations since the nineteenth century. In particular, finite plane graphs correspond to alternating links via the medial construction (see section 4). The correspondence became especially fruitful in the mid 1980's when the Jones poly- nomial renewed the interest of many knot theorists in combinatorial methods while at the same time drawing the attention of mathematical physicists. Coloring methods for graphs also have a long history, one that stretches back at least as far as Francis Guthrie's Four Color conjecture of 1852. By contrast, coloring techniques in knot theory are relatively recent, mostly motivated by an observation in the 1963 textbook, Introduction to Knot Theory, by Crowell and Fox [9]. In view of the relationship between finite plane graphs and alternating links, it is not surprising that a corresponding theory of graph coloring exists.
  • Optimal Longest Paths by Dynamic Programming∗

    Optimal Longest Paths by Dynamic Programming∗

    Optimal Longest Paths by Dynamic Programming∗ Tomáš Balyo1, Kai Fieger1, and Christian Schulz1,2 1 Karlsruhe Institute of Technology, Karlsruhe, Germany {tomas.balyo, christian.schulz}@kit.edu, fi[email protected] 2 University of Vienna, Vienna, Austria [email protected] Abstract We propose an optimal algorithm for solving the longest path problem in undirected weighted graphs. By using graph partitioning and dynamic programming, we obtain an algorithm that is significantly faster than other state-of-the-art methods. This enables us to solve instances that have previously been unsolved. 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases Longest Path, Graph Partitioning Digital Object Identifier 10.4230/LIPIcs.SEA.2017. 1 Introduction The longest path problem (LP) is to find a simple path of maximum length between two given vertices of a graph where length is defined as the number of edges or the total weight of the edges in the path. The problem is known to be NP-complete [5] and has several applications such as designing circuit boards [8, 7], project planning [2], information retrieval [15] or patrolling algorithms for multiple robots in graphs [9]. For example, when designing circuit boards where the length difference between wires has to be kept small [8, 7], the longest path problem manifests itself when the length of shorter wires is supposed to be increased. Another example application is project planning/scheduling where the problem can be used to determine the least amount of time that a project could be completed in [2]. We organize the rest of paper as follows.