4 research outputs found
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
A tight lower bound for counting Hamiltonian cycles via matrix rank
For even k ϵ N, the matchings connectivity matrix Mk is a binary matrix indexed by perfect matchings on k vertices; the entry at (M;M) is 1 i M [ M0 forms a single cycle. Cygan et al. (STOC 2013) showed that the rank of Mk over Z2 is ( p ϵ k ) and used this to give an O ((2 + p 2)pw) time algorithm for counting Hamiltonian cycles modulo ϵ on graphs of pathwidth pw, carrying over to the decision problem via witness isolation. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix withinMk, which enabled a \pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar \pattern propagation" when only a black-box lower bound on the asymptotic rank of Mk is given; no stronger structural insights such as the existence of large permutation submatrices in Mk are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes p) parameterized by pathwidth. To apply this technique, we prove that the rank of Mk over the rationals is 4k=poly(k), using the representation theory of the symmetric group and various insights from algebraic combinatorics. We also show that the rank of Mk over Zp is (1:57k) for any prime p 6= 2. Combining our rank bounds with the new pattern propagation technique, we show that Hamiltonian cycles cannot be counted in time O∗((6 -ϵ)pw) for any ϵ > 0 unless SETH fails. This bound is tight due to a O∗(6pw) time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes p = 2 in time O∗(3:57pw), indicating that the modulus can affect the complexity in intricate ways
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which pairs of perfect matchings on vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of over is and used this to give an time algorithm for counting Hamiltonian cycles modulo on graphs of pathwidth . The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within , which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of is given; no stronger structural insights such as the existence of large permutation submatrices in are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes ) parameterized by pathwidth. To apply this technique, we prove that the rank of over the rationals is . We also show that the rank of over is for any prime and even for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time for any unless SETH fails. This bound is tight due to a time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes in time , indicating that the modulus can affect the complexity in intricate ways
Matchings and Representation Theory
In this thesis we investigate the algebraic properties of matchings via representation theory. We identify three scenarios in different areas of combinatorial mathematics where the algebraic structure of matchings gives keen insight into the combinatorial problem at hand. In particular, we prove tight conditional lower bounds on the computational complexity of counting Hamiltonian cycles, resolve an asymptotic version of a conjecture of Godsil and Meagher in Erdos-Ko-Rado combinatorics, and shed light on the algebraic structure of symmetric semidefinite relaxations of the perfect matching proble