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Extensor-Coding
[article]

2018
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arXiv
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pre-print

We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ±ε. Our algorithm runs in time ε^-2 4^k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem

arXiv:1804.09448v1
fatcat:vr5sbwdwpbbsrgxjgppwlz3him
## more »

... ngest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic 2^k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.##
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Note on "The Complexity of Counting Surjective Homomorphisms and Compactions"
[article]

2017
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arXiv
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pre-print

Our proof is an application of a powerful framework of Lovász (2012), and it is analogous to proofs of Curticapean,

arXiv:1710.01712v1
fatcat:u4p5ixsypfeercdzgvfiabvvbu
*Dell*, and Marx (STOC 2017) who studied the "dual" problem in which the pattern graph ...##
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Lovász Meets Weisfeiler and Leman
[article]

2018
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arXiv
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pre-print

Thirty years of graph matching in pattern recognition. 10 Radu Curticapean,

arXiv:1802.08876v2
fatcat:clduredu25azpfhzroddazj6ha
*Holger**Dell*, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. ...##
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Counting Answers to Existential Questions
[article]

2019
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arXiv
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pre-print

Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem's parameterized and data complexity, where the query is considered to be small or fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating

arXiv:1902.04960v2
fatcat:sm53qhw6dzanleeutgyztfxqdu
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... y: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis as well as a #W[2]-hardness result. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; this essentially establishes #A[2]-completeness. Using ideas stemming from Lov\'asz, we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.##
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Fine-grained dichotomies for the Tutte plane and Boolean #CSP
[article]

2016
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arXiv
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pre-print

*Dell*, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis ...

*Dell*et al. [8] proved for all hard points, except for points with y = 1, that an exp o(n/ log 3 n) -time algorithm for T x,y on simple graphs would violate #ETH. ...

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Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths
[article]

2021
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arXiv
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pre-print

self-contained proof that counting k-matchings modulo odd integers q is Mod_q-W[1]-complete and prove that counting k-paths modulo 2 is Parity-W[1]-complete, answering an open question by Björklund,

arXiv:2107.00629v1
fatcat:6ch2zonbobgjdjrrkxdr4cz2je
*Dell*...##
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Complexity and Approximability of the Cover Polynomial

2011
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Computational Complexity
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The cover polynomial and its geometric version introduced by Chung & Graham and D'Antona & Munarini, respectively, are two-variate graph polynomials for directed graphs. They count the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, they can be thought of as interpolations between determinant and permanent, and are proposed as directed analogues of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is #Phard to evaluate at all

doi:10.1007/s00037-011-0018-0
fatcat:nugx3hfs2zdhlomz53pmits2qq
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... to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomials: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial and its geometric version is #P-hard under polynomial-time Turing reductions, while only three points in the cover polynomial and two points in the geometric cover polynomial are easy. We also study the complexity of approximately evaluating the geometric cover polynomial. Under the reasonable complexity assumptions RP = NP and RFP = #P, we give a succinct characterization of a large class of points at which approximating the geometric cover polynomial within any polynomial factor is not possible.##
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Finding Detours is Fixed-parameter Tractable
[article]

2016
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arXiv
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pre-print

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time

arXiv:1607.07737v1
fatcat:xg3fkjj6nfff3c4p464uh2sk74
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... unning time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.##
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Is Valiant–Vazirani's isolation probability improvable?

2013
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Computational Complexity
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The Valiant-Vazirani Isolation Lemma [TCS, vol. 47, pp. 85-93, 1986] provides an efficient procedure for isolating a satisfying assignment of a given satisfiable circuit: Given a Boolean circuit C on n input variables, the procedure outputs a new circuit C on the same n input variables such that (i) every satisfying assignment of C also satisfies C, and (ii) if C is satisfiable, then C has exactly one satisfying assignment. In particular, if C is unsatisfiable, then (i) implies that C is

doi:10.1007/s00037-013-0059-7
fatcat:5f7ljbwnhvfatpomcvxpw6luym
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... es that C is unsatisfiable. The Valiant-Vazirani procedure is randomized, and when C is satisfiable it produces a uniquely satisfiable circuit C with probability Ω(1/n). Is it possible to have an efficient deterministic witnessisolating procedure? Or, at least, is it possible to improve the success probability of a randomized procedure to a large constant? We argue that the answer is likely 'No'. More precisely, we prove that there exists a non-uniform randomized polynomial-time witness-isolating procedure with success probability bigger than 2/3 if and only if NP ⊆ P/poly. Thus, an improved witness-isolating procedure would imply the collapse of the polynomial-time hierarchy. We establish similar results for other variants of witness isolation, such as reductions that remove all but an odd number of satisfying assignments of a satisfiable circuit. We also consider a blackbox setting of witness isolation that generalizes the setting of the Valiant-Vazirani Isolation Lemma, and give an upper bound of O(1/n) on the success probability for a natural class of randomized witness-isolating procedures.##
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Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

2018
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Algorithmica
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*Dell*, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12] , in combination with the results of Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential ...

*Dell*et al. [8] proved for all hard points, except for points with y = 1, that an exp o(n/ log 3 n) -time algorithm for T x,y on simple graphs would violate #ETH. ...

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Is Valiant-Vazirani's Isolation Probability Improvable?

2012
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2012 IEEE 27th Conference on Computational Complexity
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The Valiant-Vazirani Isolation Lemma [TCS, vol. 47, pp. 85-93, 1986] provides an efficient procedure for isolating a satisfying assignment of a given satisfiable circuit: Given a Boolean circuit C on n input variables, the procedure outputs a new circuit C on the same n input variables such that (i) every satisfying assignment of C also satisfies C, and (ii) if C is satisfiable, then C has exactly one satisfying assignment. In particular, if C is unsatisfiable, then (i) implies that C is

doi:10.1109/ccc.2012.22
dblp:conf/coco/DellKMW12
fatcat:kzanaystp5h4jmyvl6kyf2nklu
## more »

... es that C is unsatisfiable. The Valiant-Vazirani procedure is randomized, and when C is satisfiable it produces a uniquely satisfiable circuit C with probability Ω(1/n). Is it possible to have an efficient deterministic witnessisolating procedure? Or, at least, is it possible to improve the success probability of a randomized procedure to a large constant? We argue that the answer is likely 'No'. More precisely, we prove that there exists a non-uniform randomized polynomial-time witness-isolating procedure with success probability bigger than 2/3 if and only if NP ⊆ P/poly. Thus, an improved witness-isolating procedure would imply the collapse of the polynomial-time hierarchy. We establish similar results for other variants of witness isolation, such as reductions that remove all but an odd number of satisfying assignments of a satisfiable circuit. We also consider a blackbox setting of witness isolation that generalizes the setting of the Valiant-Vazirani Isolation Lemma, and give an upper bound of O(1/n) on the success probability for a natural class of randomized witness-isolating procedures.##
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AND-compression of NP-complete Problems: Streamlined Proof and Minor Observations
[chapter]

2014
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Lecture Notes in Computer Science
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Our framework for ruling out such compressions is almost identical to Lemma 1 of

doi:10.1007/978-3-319-13524-3_16
fatcat:dimjvdvr6rfhnkmeb7uqymaeqy
*Dell*and Marx (2012) and*Dell*and van Melkebeek (2014) , and to Definition 2.2 of Hermelin and Wu (2012) . ... proof, its overall structure, is more modular and more similar to arguments used previously by Ko (1983) , Fortnow and Santhanam (2011), and van Melkebeek (2014) for compression-type procedures and*Dell*...##
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Exponential Time Complexity of the Permanent and the Tutte Polynomial
[chapter]

2010
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Lecture Notes in Computer Science
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We show conditional lower bounds for well-studied #P-hard problems: • The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. • The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). • The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case

doi:10.1007/978-3-642-14165-2_37
fatcat:cd5klsagcfatta5f4hqltalzn4
## more »

... x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n/ poly log n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.##
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Kernelization of Packing Problems
[chapter]

2012
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Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
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[JCSS 2009], Fortnow and Santhanam [JCSS 2011],

doi:10.1137/1.9781611973099.6
dblp:conf/soda/DellM12
fatcat:nbclqme3dzdvffolelibcbrtke
*Dell*and Van Melkebeek [JACM 2014] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexity-theoretic ... As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by*Dell*and Van Melkebeek [JACM 2014] with shorter elementary arguments ... ., [Tho10, Guo09, More recently,*Dell*and Van Melkebeek [DvM10] rened the complexity results of [FS08, BDFH09] to prove conditional lower bounds also for problems that do admit polynomial kernels ...##
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Counting edge-injective homomorphisms and matchings on restricted graph classes
[article]

2018
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arXiv
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pre-print

We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #W[1]-hard on graphs G that are line graphs or bipartite graphs with degree 2 on one side. In our proofs, we use that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k length-2 paths into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of

arXiv:1702.05447v2
fatcat:sy24us5odjdp5awy4ex4lyjut4
## more »

... distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes H of pattern graphs, we complement this result: If the graphs in H have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from H is #W[1]-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.
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