On NP-Intermediate, Isomorphism Problems, and Polynomial Hierarchy
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On Treewidth and Graph Minors
On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx. -
Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
A Logical Approach to Isomorphism Testing and Constraint Satisfaction
A logical approach to Isomorphism Testing and Constraint Satisfaction Oleg Verbitsky Humboldt University of Berlin, Germany ESSLLI 2016, 1519 August 1/21 Part 8: k with : FO# k ≥ 3 Applications to Isomorphism Testing 2/21 Outline 1 Warm-up 2 Weisfeiler-Leman algorithm 3 Bounds for particular classes of graphs 4 References 3/21 Outline 1 Warm-up 2 Weisfeiler-Leman algorithm 3 Bounds for particular classes of graphs 4 References 4/21 Closure properties of denable graphs Denition G denotes the complement graph of G: V (G) = V (G) and uv 2 E(G) () uv2 = E(G). Exercise Prove that, if is denable in k , then is also denable G FO# G in k . FO# 5/21 Closure properties of denable graphs Denition G1 + G2 denotes the vertex-disjoint union of graphs G1 and G2. Exercise Let k ≥ 3. Suppose that connected graphs G1;:::;Gm are denable in k . Prove that is also denable FO# G1 + G2 + ::: + Gm in k . FO# Hint As an example, let k = 3 and suppose that G = 5A + 4B and H = 4A + 5B for some connected A and B such that W#(A; B) ≤ 3. In the rst round, Spoiler makes a counting move in G by marking all vertices in all A-components. Duplicator is forced to mark at least one vertex v in a B-component of H. Spoiler pebbles v, and Duplicator can only pebble some vertex u in an A-component of G. From now on, Spoiler plays in the components that contain u and v using his winning strategy for the game on A and B. -
NP-Completeness: Reductions Tue, Nov 21, 2017
CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
Monte-Carlo Algorithms in Graph Isomorphism Testing
Monte-Carlo algorithms in graph isomorphism testing L´aszl´oBabai∗ E¨otv¨osUniversity, Budapest, Hungary Universit´ede Montr´eal,Canada Abstract Abstract. We present an O(V 4 log V ) coin flipping algorithm to test vertex-colored graphs with bounded color multiplicities for color-preserving isomorphism. We are also able to generate uniformly distributed ran- dom automorphisms of such graphs. A more general result finds gener- ators for the intersection of cylindric subgroups of a direct product of groups in O(n7/2 log n) time, where n is the length of the input string. This result will be applied in another paper to find a polynomial time coin flipping algorithm to test isomorphism of graphs with bounded eigenvalue multiplicities. The most general result says that if AutX is accessible by a chain G0 ≥ · · · ≥ Gk = AutX of quickly recognizable groups such that the indices |Gi−1 : Gi| are small (but unknown), the order of |G0| is known and there is a fast way of generating uniformly distributed random members of G0 then a set of generators of AutX can be found by a fast algorithm. Applications of the main result im- prove the complexity of isomorphism testing for graphs with bounded valences to exp(n1/2+o(1)) and for distributive lattices to O(n6 log log n). ∗Apart from possible typos, this is a verbatim transcript of the author’s 1979 technical report, “Monte Carlo algorithms in graph isomorphism testing” (Universit´ede Montr´eal, D.M.S. No. 79-10), with two footnotes added to correct a typo and a glaring omission in the original. -
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-Completeness of the Local Hamiltonian Problem
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-completeness of the Local Hamiltonian problem Scribe: Jenish C. Mehta The Complexity Class BQP The complexity class BQP is the quantum analog of the class BPP. It consists of all languages that can be decided in quantum polynomial time. More formally, Definition 1. A language L 2 BQP if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n = poly(jxj) qubits, where the circuit is considered a sequence of unitary operators each on 2 qubits, i.e Cx = UTUT−1...U1 where each 2 2 Ui 2 L C ⊗ C , such that: 2 i. Completeness: x 2 L ) Pr(Cx accepts j0ni) ≥ 3 1 ii. Soundness: x 62 L ) Pr(Cx accepts j0ni) ≤ 3 We say that the circuit “Cx accepts jyi” if the first output qubit measured in Cxjyi is 0. More j0i specifically, letting P1 = j0ih0j1 be the projection of the first qubit on state j0i, j0i 2 Pr(Cx accepts jyi) =k (P1 ⊗ In−1)Cxjyi k2 The Complexity Class QMA The complexity class QMA (or BQNP, as Kitaev originally named it) is the quantum analog of the class NP. More formally, Definition 2. A language L 2 QMA if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n + q = poly(jxj) qubits, such that: 2q i. Completeness: x 2 L ) 9jyi 2 C , kjyik2 = 1, such that Pr(Cx accepts j0ni ⊗ 2 jyi) ≥ 3 2q 1 ii. -
Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021
Treewidth versus clique number. I. Graph classes with a forbidden structure∗† Cl´ement Dallard1 Martin Milaniˇc1 Kenny Storgelˇ 2 1 FAMNIT and IAM, University of Primorska, Koper, Slovenia 2 Faculty of Information Studies, Novo mesto, Slovenia [email protected] [email protected] [email protected] Treewidth is an important graph invariant, relevant for both structural and algo- rithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call (tw,ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, in- duced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw,ω)-bounded. Our results yield an infinite family of χ-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is (tw,ω)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Breˇsar, Hartinger, Kos, and Milaniˇc from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milaniˇc, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of (tw,ω)- boundedness related to list k-coloring and clique problems. In addition, we propose a question about the complexity of the Maximum Weight Independent Set prob- lem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor. -
NP-Completeness Part II
NP-Completeness Part II Outline for Today ● Recap from Last Time ● What is NP-completeness again, anyway? ● 3SAT ● A simple, canonical NP-complete problem. ● Independent Sets ● Discovering a new NP-complete problem. ● Gadget-Based Reductions ● A common technique in NP reductions. ● 3-Colorability ● A more elaborate NP-completeness reduction. ● Recent News ● Things happened! What were they? Polynomial-Time Reductions ● If A ≤P B and B ∈ P, then A ∈ P. ● If A ≤P B and B ∈ NP, then A ∈ NP. P NP NP-Hardness ● A language L is called NP-hard if for every A ∈ NP, we have A ≤P L. ● A language in L is called NP-complete if L is NP-hard and L ∈ NP. ● The class NPC is the set of NP-complete problems. NP NP-Hard NPC P The Tantalizing Truth Theorem: If any NP-complete language is in P, then P = NP. Proof: Suppose that L is NP-complete and L ∈ P. Now consider any arbitrary NP problem A. Since L is NP-complete, we know that A ≤p L. Since L ∈ P and A ≤p L, we see that A ∈ P. Since our choice of A was arbitrary, this means that NP ⊆ P, so P = NP. ■ P = NP The Tantalizing Truth Theorem: If any NP-complete language is not in P, then P ≠ NP. Proof: Suppose that L is an NP-complete language not in P. Since L is NP-complete, we know that L ∈ NP. Therefore, we know that L ∈ NP and L ∉ P, so P ≠ NP. ■ NP NPC P How do we even know NP-complete problems exist in the first place? Satisfiability ● A propositional logic formula φ is called satisfiable if there is some assignment to its variables that makes it evaluate to true. -
The Polynomial Hierarchy and Alternations
Chapter 5 The Polynomial Hierarchy and Alternations “..synthesizing circuits is exceedingly difficulty. It is even more difficult to show that a circuit found in this way is the most economical one to realize a function. The difficulty springs from the large number of essentially different networks available.” Claude Shannon 1949 This chapter discusses the polynomial hierarchy, a generalization of P, NP and coNP that tends to crop up in many complexity theoretic inves- tigations (including several chapters of this book). We will provide three equivalent definitions for the polynomial hierarchy, using quantified pred- icates, alternating Turing machines, and oracle TMs (a fourth definition, using uniform families of circuits, will be given in Chapter 6). We also use the hierarchy to show that solving the SAT problem requires either linear space or super-linear time. p p 5.1 The classes Σ2 and Π2 To understand the need for going beyond nondeterminism, let’s recall an NP problem, INDSET, for which we do have a short certificate of membership: INDSET = {hG, ki : graph G has an independent set of size ≥ k} . Consider a slight modification to the above problem, namely, determin- ing the largest independent set in a graph (phrased as a decision problem): EXACT INDSET = {hG, ki : the largest independent set in G has size exactly k} . Web draft 2006-09-28 18:09 95 Complexity Theory: A Modern Approach. c 2006 Sanjeev Arora and Boaz Barak. DRAFTReferences and attributions are still incomplete. P P 96 5.1. THE CLASSES Σ2 AND Π2 Now there seems to be no short certificate for membership: hG, ki ∈ EXACT INDSET iff there exists an independent set of size k in G and every other independent set has size at most k.