This paper presents several ways to compute lower and upper bounds for MaxSAT based on calling a complete SAT solver. Preliminary results indicate that (i) the bounds are of high quality, (ii) the bounds can boost the search of MaxSAT solvers on some benchmarks, and (iii) the upper bounds computed by a Stochastic Local Search procedure (SLS) can be substantially improved when its search is init...

The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧,∨,¬. Suppose L ∈ {0, 1}∗ is a language. Let Ln = L∩{0, 1}. We say that L is computed by a family of circuits C1, C2, . . . if on an input x = (x1, . . . , xn), Cn(x) is 1 when x ∈ Ln and is 0 when x / ∈ Ln. For a circuit C, we define size(C) to be the number of edges in the graph representing C, and dept...

Dobkin and Lipton introduced the connected components argument to prove lower bounds in the linear decision tree model for membership problems, for example the element uniqueness problem. In this paper we apply the same idea to obtain lower bound statements for a variety of problems, each having the avor of element uniqueness. In fact one of these problems is a parametric version of element uni...

The information inequality has been shown to be an effective tool for providing lower bounds for the minimax risk. Bounds based on the chisquare distance can sometimes offer a considerable improvement especially when applied iteratively. This paper compares these two methods in four examples including the bounded normal mean problem as well as obervations from a Poisson distribution.

Preprocessing (data reduction or kernelization) to reduce instance size is one of the most commonly deployed heuristics in the implementation practice to tackle computationally hard problems. However, a systematic theoretical study of them remained elusive so far. One of the reasons for this is that if an input to an NP -hard problem can be processed in polynomial time to an equivalent one of s...

We prove a lower bound for swapping the order of Arthur and Merlin in two-round MerlinArthur games using black-box techniques. Namely, we show that any AM-game requires time Ω(t) to black-box simulate MA-games running in time t. Thus, the known simulations of MA by AM with quadratic overhead, dating back to Babai’s original paper on Arthur-Merlin games, are tight within this setting. The black-...

Now that we have a good handle on the performance of ERM and its variants, it is time to ask whether we can do better. For example, consider binary classification: we observe n i.i.d. training samples from an unknown joint distribution P on X× {0,1}, where X is some feature space, and for a fixed class F of candidate classifiers f :X→ {0,1} we let f̂n be the ERM solution f̂n = argmin f ∈F 1 n n ∑...

An axis-parallel b-dimensional box is a Cartesian product R1×R2×· · ·×Rb where Ri is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lo...

In this paper we present lower bounds for compact routing schemes. We give (1) networks on n vertices which for any interval routing scheme, (n) routers of the network require (n) intervals on some outgoing link and (2) for each d 3, networks of maximal degree d which for any interval routing scheme, (n) routers each require (n= logn) intervals on some outgoing link. Our results give the best k...