55 research outputs found
The Minrank of Random Graphs
The minrank of a graph is the minimum rank of a matrix that can be
obtained from the adjacency matrix of by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
for all regimes of . In particular, for any constant ,
we show that with high probability,
where is chosen from . This bound gives a near quadratic
improvement over the previous best lower bound of (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits
String Matching: Communication, Circuits, and Learning
String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings.
- Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol.
- Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k.
- Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem
On the Quantitative Hardness of CVP
For odd
integers (and ), we show that the Closest Vector Problem
in the norm (\CVP_p) over rank lattices cannot be solved in
2^{(1-\eps) n} time for any constant \eps > 0 unless the Strong Exponential
Time Hypothesis (SETH) fails. We then extend this result to "almost all" values
of , not including the even integers. This comes tantalizingly close
to settling the quantitative time complexity of the important special case of
\CVP_2 (i.e., \CVP in the Euclidean norm), for which a -time
algorithm is known. In particular, our result applies for any
that approaches as .
We also show a similar SETH-hardness result for \SVP_\infty; hardness of
approximating \CVP_p to within some constant factor under the so-called
Gap-ETH assumption; and other quantitative hardness results for \CVP_p and
\CVPP_p for any under different assumptions
Collapsing Superstring Conjecture
In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS.
We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm.
While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS.
We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2
The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications
The orthogonality dimension of a graph over a field is
the smallest integer for which there exists an assignment of a vector with to every vertex , such that whenever and are
adjacent vertices in . The study of the orthogonality dimension of graphs is
motivated by various application in information theory and in theoretical
computer science. The contribution of the present work is two-folded.
First, we prove that there exists a constant such that for every
sufficiently large integer , it is -hard to decide whether the
orthogonality dimension of an input graph over is at most or
at least . At the heart of the proof lies a geometric result, which
might be of independent interest, on a generalization of the orthogonality
dimension parameter for the family of Kneser graphs, analogously to a
long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976).
Second, we study the smallest possible orthogonality dimension over finite
fields of the complement of graphs that do not contain certain fixed subgraphs.
In particular, we provide an explicit construction of triangle-free -vertex
graphs whose complement has orthogonality dimension over the binary field at
most for some constant . Our results involve
constructions from the family of generalized Kneser graphs and they are
motivated by the rigidity approach to circuit lower bounds. We use them to
answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor.
Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle
Conjecture over every finite field.Comment: 19 page
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