Computing sparse permanents faster

Servedio, Rocco A.; Wan, Andrew

doi:10.1016/j.ipl.2005.06.007

Cited by 22 publications

(28 citation statements)

References 3 publications

(10 reference statements)

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“…When bipartite graphs are concerned, the classical algorithm of Ryser [12] has been improved for graphs of bounded average degree first by Servedio and Wan [13] and, very recently, by Izumi and Wadayama [9]. Our last result is the following theorem.…”

Section: Introductionmentioning

confidence: 89%

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Faster exponential-time algorithms in graphs of bounded average degree

Cygan

Pilipczuk

2015

Information and Computation

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We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O ⋆ (2 (1−ε d )n ) time 1 and exponential space for a constant ε d depending only on d. Thus, we generalize the recent results of Björklund et al. [TALG 2012] on graphs of bounded degree.Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O ⋆ (2 n/2 ) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Björklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O ⋆ (2 (1−ε 2d )n/2 ) time and exponential space, where ε 2d is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d.Moreover we obtain a simple algorithm that counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most [3], and breaking these time complexity barriers seems like a very challenging task.From a broader perspective, improving upon a trivial brute-force or a simple dynamic programming algorithm is one of the main goals the field of exponential-time algorithms. Although the last few years brought a number of positive results in that direction, most notably the O ⋆ (1.66 n ) randomized algorithm for finding a Hamiltonian cycle in an undirected graph [2], it is conjectured (the so-called Strong Exponential Time Hypothesis [8]) that the problem of satisfying a general CNF-SAT formulae does not admit any exponentially better algorithm than the trivial brute-force one. A number of lower bounds were proven using this assumption [6,10,11].In 2008 Björklund et al. [5] observed that the classical dynamic programming algorithm for TSP can be trimmed to running time O ⋆ (2 (1−ε∆)n ) in graphs of maximum degree ∆. The cost of this improvement is the use of exponential space, as we can no longer easily translate the dynamic programming algorithm into an inclusion-exclusion formula. The ideas from [5] were also applied to the Fast Subset Convolution algorithm, yielding a similar improvements for the problem of computing the chromatic number in graphs of bounded * Partially supported by NCN grant N206567140 and Foundation for Polish Science.

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Section: Introductionmentioning

confidence: 89%

“…Our last result is the following theorem. Hence, we improve the running time of [9,13] in terms of the dependency on d. We would like to emphasise that our proof of Theorem 1.3 is elementary and does not need the advanced techniques of coding theory used in [9].…”

Section: Introductionmentioning

confidence: 98%

Faster exponential-time algorithms in graphs of bounded average degree

Cygan

Pilipczuk

2015

Information and Computation

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“…G is also known as the permanent of the 0-1 adjacency matrix of G (see Servedio and Wan [13] for the permanent of a matrix).…”

Section: Anonymity Metricmentioning

confidence: 99%

Data Caching for Enhancing Anonymity

Bagai

Tang

2011

2011 IEEE International Conference on Advanced Information Networking and Applications

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Abstract-The benefits of caching for reducing access time to frequently needed data, in order to improve system performance, are already well-known. In this paper, a proposal for employing data caching for increasing the level of anonymity provided by an anonymity system is presented. This technique is especially effective for user sessions containing bidirectional communication, such as anonymous web browsing. A framework is first constructed for capturing the effect of attacks on anonymity systems that have the ability to serve some incoming user requests from their cache. A system-wide metric is then presented for measuring the anonymity provided by such systems. It is shown that the anonymity level of such systems rises with the amount of data caching performed by them. This behavior is illustrated in an example threshold mix network.

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“…Hence, we improve the running time of [13,12] in terms of the dependency on d. We would like to emphasise that our proof of Theorem 1 is elementary and does not need the advanced techniques of coding theory used in [13].…”

Section: Introductionmentioning

confidence: 98%

“…In the case of the problem of counting perfect matchings in bipartite graphs, the classic algorithm of Ryser [4] the best known improvement in general graphs is an algorithm running in expected time O (2 (1−O(n 2/3 log n))·n/2 ) due to Bax and Franklin [11]. If one assumes bounded average degree, faster algorithms have been given by Servedio and Wan [12] and, very recently, by Izumi and Wadayama [13]. In Section 2 we continue this line of research and show the following.…”

Section: Introductionmentioning

confidence: 99%

Faster Exponential-Time Algorithms in Graphs of Bounded Average Degree

Cygan

Pilipczuk

2013

Automata, Languages, and Programming

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We present a number of exponential-time algorithms for problems in sparse matrices and graphs of bounded average degree. First, we obtain a simple algorithm that computes a permanent of an n×n matrix over an arbitrary commutative ring with at most dn non-zero entries using O (2 (1−1/(3.55d))n ) time and ring operations 1 , improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].Second, we present a simple algorithm for counting perfect matchings in an n-vertex graph in O (2 n/2 ) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Björklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations.Third, we show a combinatorial lemma that bounds the number of "Hamiltonian-like" structures in a graph of bounded average degree. Using this result, we show that

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Computing sparse permanents faster

Cited by 22 publications

References 3 publications

Faster exponential-time algorithms in graphs of bounded average degree

Faster exponential-time algorithms in graphs of bounded average degree

Data Caching for Enhancing Anonymity

Faster Exponential-Time Algorithms in Graphs of Bounded Average Degree

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