4 research outputs found
Complexity and Approximability of Parameterized MAX-CSPs
International audienceWe study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance.We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a (1−ϵ)-approximation in time f(k,ϵ)⋅poly(n). 3. There is no FPTAS unless FPT=W[1].For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results
Parameterized Compilation Lower Bounds for Restricted CNF-formulas
We show unconditional parameterized lower bounds in the area of knowledge
compilation, more specifically on the size of circuits in decomposable negation
normal form (DNNF) that encode CNF-formulas restricted by several graph width
measures. In particular, we show that
- there are CNF formulas of size and modular incidence treewidth
whose smallest DNNF-encoding has size , and
- there are CNF formulas of size and incidence neighborhood diversity
whose smallest DNNF-encoding has size .
These results complement recent upper bounds for compiling CNF into DNNF and
strengthen---quantitatively and qualitatively---known conditional low\-er
bounds for cliquewidth. Moreover, they show that, unlike for many graph
problems, the parameters considered here behave significantly differently from
treewidth
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect