Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page

Notes on Boolean Read-k and Multilinear Circuits

A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant of f, the polynomial contains at least one monomial with the same set of variables, each appearing with degree at most k. Every monotone circuit is a read-k circuit for some k. We show that already read-1 (OR,AND) circuits are not weaker than monotone arithmetic constant-free (+,x) circuits computing multilinear polynomials, are not weaker than non-monotone multilinear (OR,AND,NOT) circuits computing monotone Boolean functions, and have the same power as tropical (min,+) circuits solving combinatorial minimization problems. Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied Mathematic

Graphs and Circuits: Some Further Remarks

We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl\u27ak, R"odl and Savick\u27y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits

Yet harder knapsack problems

AbstractAlready 30 years ago, Chvátal has shown that some instances of the zero-one knapsack problem cannot be solved in polynomial time using a particular type of branch-and-bound algorithms based on relaxations of linear programs together with some rudimentary cutting-plane arguments as bounding rules. We extend this result by proving an exponential lower bound in a more general class of branch-and-bound and dynamic programming algorithms which are allowed to use memoization and arbitrarily powerful bound rules to detect and remove subproblems leading to no optimal solution

Lower Bounds for DeMorgan Circuits of Bounded Negation Width

We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width

Very Large Cliques are Easy to Detect

It is known that, for every constant $kgeq 3$, the presence of a $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex graph cannot be detected by a monotone boolean circuit using fewer than $Omega((n/log n)^k)$ gates. We show that, for every constant $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be detected by a monotone circuit using only $O(n^2log n)$ gates. Moreover, if we allow unbounded fanin, then $O(log n)$ gates are enough