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During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers.

There has been a relatively long sequence of papers on space in resolution and resolution-based proof systems, and it is probably fair to say that resolution is reasonably well understood from this point of view. For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in [Alekhnovich et al.'02], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any $k$-CNF formula in constant space.

In this paper, we prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al.'02]:

1. We prove an $\Omega(n)$ space lower bound in PC for the canonical $3$-CNF version of the pigeonhole principle formulas $\mathsf{PHP}_n^m$ with $m$ pigeons and $n$ holes, and show that this is tight.
2. For PCR, we prove an $\Omega(n)$ space lower bound for a bitwise encoding of the functional pigeonhole principle with $m$ pigeons and $n$ holes. These formulas have width $O(\log n)$, and so this is an exponential improvement over [Alekhnovich et al.'02] measured in the width of the formulas.
3. We then present another encoding of a version of the pigeonhole principle that has constant width, and prove an $\Omega(n)$ space lower bound in PCR for these formulas as well.
4. Finally, we prove that any $k$-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into a $3$-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [FOCS, 2007]. In that framework the parameterized version of any proof system is not fpt-bounded for some technical reasons, but we remark that this question becomes much more interesting if we restrict ourselves to those parameterized contradictions $(F,k)$ in which $F$ itself is a contradiction. We call such parameterized contradictions strong, and with one important exception (vertex cover) all interesting contradictions we are aware of are strong. It follows from the gap complexity theorem of Dantchev, Martin and Szeider [FOCS, 2007] that tree-like Parameterized Resolution is not fpt-bounded w.r.t. strong parameterized contradictions.

The main result of this paper significantly improves upon this by showing that even the parameterized version of bounded-depth Frege is not fpt-bounded w.r.t. strong contradictions. More precisely, we prove that the pigeonhole principle requires proofs of size $n^{\Omega(k)}$ in bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev, Martin and Szeider [FOCS, 2007].

In the opposite direction, we interpret a well-known $\mathsf{FPT}$ algorithm for vertex cover as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.

We initiate the study of the proof complexity of propositional encoding of (weak cases of) concrete independence results. We study the proof complexity of Paris-Harrington's Large Ramsey Theorem for bicolorings of graphs.

We prove a non-trivial conditional lower bound in Resolution and a quasi-polynomial upper bound in bounded-depth Frege. The lower bound is conditional on a hardness assumption for a Weak (quasi-polynomial) Pigeonhole Principle in $\mathsf{Res}(2)$. The proof technique of the lower bound extends the idea of using a combinatorial principle to blow-up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced Ramsey principle for triangles is established. This is obtained by applying some highly non-trivial constructions due to Erdös and Mills.

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a $k$-clique requires $n^{\Omega(k)}$ steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in Beyersdorff et al. (ECCC TR10-198) of understanding the Resolution complexity of this family of formulas.

Given a set $S$ of points (stations) located in the $d$-dim. Euclidean space and a root $b \in S$, the $h$-$\mathsf{hops Convergecast}$ problem asks to find for a minimal energy-cost range assignment which allows to perform the converge-cast primitive (i.e. node accumulation) towards $b$ in at most $h$ hops. For this problem no polynomial time algorithm is known even for $h = 2$.

The main goal of this work is the design of an efficient distributed heuristic (i.e. protocol) and the analysis (both theoretical and experimental) of its expected solution cost. In particular, we introduce an efficient parameterized randomized protocol for $h$-$\mathsf{hops Convergecast}$ and we analyze it on random instances created by placing $n$ points uniformly at random in a $d$-cube of side length $L$. We prove that for $h = 2$, its expected approximation ratio is bounded by some constant factor. Finally, for $h = 3,..., 8$, we provide a wide experimental study showing that our protocol has very good performances when compared with previously introduced (centralized) heuristics.

The Minimum-Energy Broadcast problem is to assign a transmission range to every station of an ad hoc wireless networks so that (i) a given source station is allowed to perform broadcast operations and (ii) the overall energy consumption of the range assignment is minimized.

We prove a nearly tight asymptotical bound on the optimal cost for the Minimum-Energy Broadcast problem on square grids. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum-Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about $\pi$ and it yields $\Omega(\sqrt{n})$ source hops, where $n$ is the number of stations.

As a by product, we get nearly tight bounds for the Minimum-Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius.

Finally, we emphasize that our upper bounds are obtained via polynomial time constructions.

The $d$-Dim $h$-hops MST problem is defined as follows: Given a set $S$ of points in the $d$- dimensional Euclidean space and $s \in S$, find a minimum-cost spanning tree for $S$ rooted at $s$ with height at most $h$. We investigate the problem for any constants $h$ and $d > 0$. We prove the first non trivial lower bound on the solution cost for almost all Euclidean instances (i.e. the lower-bound holds with hight probability). Then we introduce an easy-to-implement, very fast divide et impera heuristic and we prove that its solution cost matches the lower bound.

In this note we show that the asymmetric Prover-Delayer game developed in Beyersdorff, Galesi, Lauria [2010a] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form $2^{\Omega(n\log n)}$ for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki [1999] and Dantchev and Riis [2001].

There are methods to turn short refutations in Polynomial Calculus ($\mathsf{Pc}$) and Polynomial Calculus with Resolution ($\mathsf{Pcr}$) into refutations of low degree. Bonet and Galesi [Bonet and Galesi 1999, 2003] asked if such size-degree trade-offs for $\mathsf{Pc}$ [Clegg et al. 1996; Impagliazzo et al. 1999] and $\mathsf{Pcr}$ [Alekhnovich et al. 2004] are optimal.

We answer this question by showing a polynomial encoding of Graph Ordering Principle on $m$ variables which requires $\mathsf{Pc}$ and $\mathsf{Pcr}$ refutations of degree $\Omega(\sqrt{m})$. Trade-offs optimality follows from our result and from the short refutations of Graph Ordering Principle in [Bonet and Galesi 1999, 2001].

We then introduce the algebraic proof system $\mathsf{Pcr}_{k}$ which combines together Polynomial Calculus ($\mathsf{Pc}$) and $k$-DNF Resolution ($\mathsf{Res}_{k}$). We show a size hierarchy theorem for $\mathsf{Pcr}_{k}$: $\mathsf{Pcr}_{k}$ is exponentially separated from $\mathsf{Pcr}_{k+1}$. This follows from the previous degree lower bound and from techniques developed for $\mathsf{Res}_{k}$.

Finally we show that random formulas in conjunctive normal form ($3$-CNF) are hard to refute in $\mathsf{Pcr}_{k}$.

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless $W[P]$-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Computing, 38(4), 2008).

The Minimum-Energy Broadcast problem is to assign a transmission range to every station of an ad hoc wireless networks so that (i) a given source station is allowed to perform broadcast operations and (ii) the overall energy consumption of the range assignment is minimized.

We prove a nearly tight asymptotical bound on the optimal cost for the Minimum-Energy Broadcast problem on square grids. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum-Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about $\pi$ and it yields $\Omega(\sqrt{n})$ source hops, where $n$ is the number of stations.

As a by product, we get nearly tight bounds for the Minimum-Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius.

Finally, we emphasize that our upper bounds are obtained via polynomial time constructions.

The $d$-Dim $h$-hops MST problem is defined as follows: Given a set $S$ of points in the $d$- dimensional Euclidean space and $s \in S$, find a minimum-cost spanning tree for $S$ rooted at $s$ with height at most $h$. We investigate the problem for any constants $h$ and $d > 0$. We prove the first non trivial lower bound on the solution cost for almost all Euclidean instances (i.e. the lower-bound holds with hight probability). Then we introduce an easy-to-implement, very fast divide et impera heuristic and we prove that its solution cost matches the lower bound.

In this note we show that any $k$-CNF which can be refuted by a quasi-polynomial $\mathsf{Res}^{*}(\mathsf{polylog})$ refutation has a “narrow” refutation in $\mathsf{Res}$ (i.e., of poly-logarithmic width). Notice that while $\mathsf{Res}^{*}(\mathsf{polylog})$ is a complete proof system, this is not the case for $\mathsf{Res}$ if we ask for a narrow refutation. In particular is not even possible to express all CNFs with narrow clauses. But even for constant width CNF the former system is complete and the latter is not (see for example Bonet, Galesi 2001). We are going to show that the formulas “left out” are the ones which require large $\mathsf{Res}^{*}(\mathsf{polylog})$ refutations. We also show the converse implication: a narrow Resolution refutation can be simulated by a short $\mathsf{Res}^{*}(\mathsf{polylog})$ refutation.

We give upper bounds for a positional game — in the sense of Beck — based on the Paris-Harrington principle for bi-colorings of graphs and uniform hypergraphs of arbitrary dimension. The bounds show a striking difference with respect to the bounds of the combinatorial principle itself. Our results confirm a phenomenon already observed by Beck and others: the upper bounds for the game version of a combinatorial principle are drastically smaller than the upper bounds for the principle itself. In the case of Paris-Harrington games the difference is qualitatively very striking. For example, the bounds for the game on $3$-uniform hypergraphs are a fixed stack of exponentials while the bounds on the corresponding combinatorial principle are known to be Ackermannian! For higher dimensions, the combinatorial Paris-Harrington numbers are known to be cofinal in the Schwichtenberg-Wainer Hiearchy of fast-growing functions up to $\varepsilon_0$, while we show that the game Paris-Harrington numbers are fixed stacks of exponentials.

Given a propositional formula $\phi$ we are interested in studying how difficult is to prove that $\phi$ is a tautology. Multivariate polynomials can be manipulated in a specific way to obtain such a proof. We study the power of this algebraic manipulation. In particular its running time is deeply related with the degree of the polynomials involved. We prove that this method behaves very badly in some cases, by showing that there are tautologies with very short proofs but which require high degree polynomials.

This does not exclude that another method of polynomial manipulation could lead to an efficient algorithm. In this case we are able to prove that if such efficient algorithm exists, it could be used to solve problems believed to be very hard.

Algebraic proof systems discussed so far are limited. We define a slightly stronger system $\mathsf{Pcr}_{k}$. We give examples of tautologies that are easy for this system, but very hard for systems known in literature. Nevertheless we show that a random tautology is still very hard to prove for $\mathsf{Pcr}_{k}$.