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61
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipoly ..."
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Cited by 50 (8 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Collective spatial keyword querying
 In SIGMOD
, 2011
"... With the proliferation of geopositioning and geotagging, spatial web objects that possess both a geographical location and a textual description are gaining in prevalence, and spatial keyword queries that exploit both location and textual description are gaining in prominence. However, the queries ..."
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Cited by 47 (10 self)
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With the proliferation of geopositioning and geotagging, spatial web objects that possess both a geographical location and a textual description are gaining in prevalence, and spatial keyword queries that exploit both location and textual description are gaining in prominence. However, the queries studied so far generally focus on finding individual objects that each satisfy a query rather than finding groups of objects where the objects in a group collectively satisfy a query. We define the problem of retrieving a group of spatial web objects such that the group’s keywords cover the query’s keywords and such that objects are nearest to the query location and have the lowest interobject distances. Specifically, we study two variants of this problem, both of which are NPcomplete. We devise exact solutions as well as approximate solutions with provable approximation bounds to the problems. We present empirical studies that offer insight into the efficiency and accuracy of the solutions. 1.
Lower bounds based on the Exponential Time Hypothesis
 Bulletin of the EATCS
, 2011
"... In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponenti ..."
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Cited by 35 (3 self)
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In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponential time hypothesis (SETH). 1
On Problems as Hard as CNFSat
, 2012
"... Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some ba ..."
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Cited by 17 (4 self)
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Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some basic problems, however, there has been no progress over their trivial solution. For others nontrivial solutions have been found, but improving these algorithms further seems to be out of reach. The CNFSat problem is the canonical example of a problem for which the brute force 2 n n O(1) time algorithm remains the best known. The assumption that k–CNFSat requires 2 n time in the worst case when k grows to infinity is known as the strong exponential time hypothesis (SETH) of Impagliazzo and Paturi. In this paper we reveal connections between wellstudied problems, and show that improving over the currently best known algorithms for several of them would violate SETH. Specifically, we show that for every ɛ < 1, an O(2 ɛn) time algorithm for Hitting Set, Set Splitting or NAESat would violate SETH. Here n is the number of elements (or
Saving Space by Algebraization
, 2010
"... The Subset Sum and Knapsack problems are fundamental N Pcomplete problems and the pseudopolynomial time dynamic programming algorithms for them appear in every algorithms textbook. The algorithms require pseudopolynomial time and space. Since we do not expect polynomial time algorithms for Subset ..."
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Cited by 16 (1 self)
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The Subset Sum and Knapsack problems are fundamental N Pcomplete problems and the pseudopolynomial time dynamic programming algorithms for them appear in every algorithms textbook. The algorithms require pseudopolynomial time and space. Since we do not expect polynomial time algorithms for Subset Sum and Knapsack to exist, a very natural question is whether they can be solved in pseudopolynomial time and polynomial space. In this paper we answer this question affirmatively, and give the first pseudopolynomial time, polynomial space algorithms for these problems. Our approach is based on algebraic methods and turns out to be useful for several other problems as well. Then we show how the framework yields polynomial space exact algorithms for the classical Traveling Salesman, Weighted Set Cover and Weighted Steiner Tree problems as well. Our algorithms match the time bound of the best known pseudopolynomial space algorithms for these problems.
Counting Perfect Matchings as Fast as Ryser
, 2012
"... We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5) ..."
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Cited by 10 (1 self)
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We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5)/2) n) ⊂ O(1.619 n) time algorithm by Koivisto, and for polynomial space, the O(1.942 n) time algorithm by Nederlof. Our new algorithm’s runtime matches up to polynomial factors that of Ryser’s 1963 algorithm for bipartite graphs. We present our algorithm in the more general setting of computing the hafnian over an arbitrary ring, analogously to Ryser’s algorithm for permanent computation. We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an n element ground set cannot be solved in O ∗ (2 (1−ɛ1)n) time for any ɛ1> 0 unless there are O ∗ (2 (1−ɛ2)n) time algorithms for computing an n × n 0/1 matrix permanent, for some ɛ2> 0 depending only on ɛ1.
Counting subgraphs via homomorphisms
 In Automata, Languages and Programming: ThirtySixth International Colloquium (ICALP
, 2009
"... We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, ..."
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Cited by 10 (3 self)
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We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser’s formula for counting perfect matchings of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions and the color coding technique, we obtain the following new results: • The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time O(2 n+o(n)); in particular in time O(2 n n 3) for trees and in time 2 n+O( √ n) for planar graphs. • Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally
Exponentialtime approximation of weighted set cover
 Inf. Process. Lett
"... The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for ..."
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The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for problems of that kind. The goal is getting the time complexity still of order O(cn), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes — tradeoffs between approximation factor and the time complexity. We use general transformations from exponentialtime exact algorithms to approximations that are faster but still exponentialtime. For example, we show that for any reduction rate r, one can transform any O∗(cn)time1 algorithm for Set Cover into a (1+ln r)approximation algorithm running in time O∗(cn/r). We believe that results of that kind extend the applicability of exact algorithms for NPhard problems.
TRIMMED MOEBIUS INVERSION AND GRAPHS OF BOUNDED DEGREE
"... We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an nele ..."
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Cited by 8 (2 self)
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We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an nelement universe U and a family F of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from F in time within a polynomial factor (in n) of the number of supersets of the members of F. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to wellstudied combinatorial optimisation problems on graphs of maximum degree ∆. In particular, we show how to compute the Domatic Number in time within a polynomial factor of (2 ∆+1 − 2) n/(∆+1) and the Chromatic Number in time within a polynomial factor of (2 ∆+1 − ∆ − 1) n/(∆+1). For any constant ∆, these bounds are O ` (2 − ɛ) n ´ for ɛ> 0 independent of the number of vertices n.