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Dominik Scheder

Publications and source records attributed to Dominik Scheder.

17 recordsLinked to original sources
arXiv2026

Solving Stackelberg Vertex Cover on trees using split and join

The Stackelberg Vertex Cover problem is a bilevel optimization problem with two players on a graph $G = (F \cup P, E)$ where each vertex from $F$ has a weight and the first player selects a price for each vertex in $P$. Afterwards, the second player finds a minimum vertex cover $X$ and the first player receives the set price for each vertex from $X \cap P$. The goal is to maximize the revenue of the first player. This problem was recently shown to be NP-complete for bipartite graphs while being solvable in linear time on paths. We present three new algorithms for solving Stackelberg Vertex Cover on certain kinds of trees: (1) a pseudo-polynomial algorithm working on general trees when all weights are integer, i.e., it is FPT with the maximum weight as a parameter; (2) a strongly polynomial algorithm for trees having the property that the least common ancestor of any two vertices from $P$ is again in $P$ (this case includes paths); and (3) an FPT-algorithm for trees, where the parameter is the maximum number $P$-vertices $v_i$ that an $F$-vertex $u$ can reach while using no other $P$-vertices. These algorithms are based on a lemma that allows us to split instances at a vertex $u$ into multiple sub-instances, which follows from LP duality and integrality of the vertex cover LP on bipartite graphs. The lemma requires that the minimum vertex covers of the sub-instances agree on $u$ (either all include $u$ or all don't). For this we introduce the concept of commitments. Finally, we show that the Stackelberg Vertex Cover problem with commitments is weakly NP-complete.

cs.DS↗
arXiv2025

PLS-complete problems with lexicographic cost functions: Max-$k$-SAT and Abelian Permutation Orbit Minimization

How hard is it to find a local optimum? If we are given a graph and want to find a locally maximal cut--meaning that the number of edges in the cut can't be improved by moving a single vertex from one side to the other--then just iterating improving steps finds a local maximum in $ |E|$ steps. If, on the other hand, the edges are weighted, this problem becomes hard for the class PLS (Polynomial Local Search). We are interested in optimization problems with lexicographic costs. For Max-Cut this would mean that the edges $e_1,\dots, e_m$ have costs $c(e_i) = 2^i$. For such a cost function finding a global Max-Cut is easy. In contrast, we show that it is PLS-complete to find an assignment for a 4-CNF formula that is locally maximal (when the clauses have lexicographic weights); and also for a 3-CNF when we allow switching two variables at a time. We use these results to answer a question in Scheder and Tantow, who showed that finding a lexicographic local minimum of a string $s \in \{0,1\}^n$ under the action of a list of given permutations $\pi_1, \dots, \pi_k \in S_{n}$ is PLS-complete. They ask whether the problem stays PLS-complete when the $\pi_1,\dots,\pi_k$ commute, i.e., generate an Abelian subgroup $G$ of $S_n$. We show that it does, and in fact stays PLS-complete even (1) when every element in $G$ has order two or (2) when $G$ is cyclic. Additionally, we use it to further investigate the complexity of computing pure $\alpha$-Nash equilibria in congestion games. Using lexicographic 4-SAT, we obtain a simple proof of the PLS-completeness originally shown by Skopalik and V\"ocking that can be extended to exponential and polynomial delay functions with positive coefficients. The number of strategies per player and players per resource is bounded. However, the degree of the polynomials is not bounded by a constant.

cs.CC↗
arXiv2025

PLS-completeness of string permutations

Bitstrings can be permuted via permutations and compared via the lexicographic order. In this paper we study the complexity of finding a minimum of a bitstring via given permutations. As a global optima is known to be NP-complete, we study the local optima via the class PLS and show hardness for PLS. Additionally, we show that even for one permutation the global optimization is NP-complete and give a formula that has these permutation as symmetries. This answers an open question inspired from Kolodziejczyk and Thapen and stated at the SAT and interactions seminar in Dagstuhl.

cs.CC↗
arXiv2022

PPSZ is better than you think

PPSZ, for long time the fastest known algorithm for $k$-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to $0$ or $1$, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being implied by a small set of clauses). We show that PPSZ performs exponentially better than previously known, for all $k \geq 3$. For Unique-$3$-SAT we bound its running time by $O(1.306973^{n})$, which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick, which runs in time $O(1.306995^n)$. Before that, the best known upper bound for Unique-$3$-SAT was $O(1.3070319^n)$. All improvements are achieved without changing the original PPSZ. The core idea is to pretend that PPSZ does not process the variables in uniformly random order, but according to a carefully designed distribution. We write "pretend" since this can be done without any actual change to the algorithm.

cs.DS↗
arXiv2021

Impatient PPSZ -- a Faster algorithm for CSP

PPSZ is the fastest known algorithm for (d,k)-CSP problems, for most values of d and k. It goes through the variables in random order and sets each variable randomly to one of the d colors, excluding those colors that can be ruled out by looking at few constraints at a time. We propose and analyze a modification of PPSZ: whenever all but 2 colors can be ruled out for some variable, immediately set that variable randomly to one of the remaining colors. We show that our new "impatient PPSZ" outperforms PPSZ exponentially for all k and all d >= 3 on formulas with a unique satisfying assignment.

cs.DS↗
arXiv2016

Tighter Hard Instances for PPSZ

We construct uniquely satisfiable $k$-CNF formulas that are hard for the algorithm PPSZ. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most $(2 + ε) / k$; the saving of an algorithm on an input formula with $n$ variables is the largest $γ$ such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least $2^{ - (1 - γ) n}$. Since PPSZ (both weak and strong) is known to have savings of at least $\frac{π^2 + o(1)}{6k}$, this is optimal up to the constant factor. In particular, for $k=3$, our upper bound is $2^{0.333\dots n}$, which is fairly close to the lower bound $2^{0.386\dots n}$ of Hertli [SIAM J. Comput.'14]. We also construct instances based on linear systems over $\mathbb{F}_2$ for which strong PPSZ has savings of at most $O\left(\frac{\log(k)}{k}\right)$. This is only a $\log(k)$ factor away from the optimal bound. Our constructions improve previous savings upper bound of $O\left(\frac{\log^2(k)}{k}\right)$ due to Chen et al. [SODA'13].

cs.CC↗
arXiv2012

A New Bound for 3-Satisfiable MaxSat and its Algorithmic Application

Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 2/3 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment $τ$ will satisfy at least $2m/3+ m_U/3+ρn'$ clauses, where m is the number of clauses of F, m_U is the number of clauses of F containing a variable from U, n' is the total number of variables in clauses not containing a variable in U, and ρis a positive absolute constant. Both U and $τ$ can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MAXSAT-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2m/3 + k clauses, where k is the parameter.

cs.DM↗
arXiv2012

In Defense of Bureaucracy in the Metric Facility Location Problem

Our work is devoted to the metric facility location problem and addresses the selfish behavior of the players. It contributes to the line of work initiated by Procaccia and Tennenholtz [EC09] on approximate mechanism design without money. We explore and argue for an intuitive and simple rule of complexity O(nk),a so-called proportionality mechanism. The mechanism works in k consecutive rounds,each time choosing a random player at whose position to place the next facility; each time the probabilities of players to be picked are distributed proportionally to their distances to the current set of the facilities. Lu et al. [EC10] showed that the proportionality rule is incentive compatible for k=1,2, but fails to be so for k>2. We tweak the model slightly such that for any k, the proportionality mechanism becomes incentive compatible. In the new model we allow the government to be bureaucratic, i.e., to have the power of to force each player to choose from only a specific set of available facilities. In the proportionality mechanism, we force every player that receives a facility at his reported location to connect to exactly that facility. We extend the proportionality mechanism to a more general setting with a private network of facilities already present in the metric space and show that it is truthful as well. We further show that for any fixed k, the proportionality rule achieves in expectation a constant approximation guarantee to the optimal solution; namely at most a ratio of 4k. On the other hand, we show a lower bound of ln k(1+o(1)), and we suspect the truth to be closer to this lower bound. Thus, our work is the first among those on incentive compatible facility location that treats effectively (with a constant factor of approximation) the general case of an arbitrary number of facilities.

cs.GT↗
arXiv2011

Improving PPSZ for 3-SAT using Critical Variables

A critical variable of a satisfiable CNF formula is a variable that has the same value in all satisfying assignments. Using a simple case distinction on the fraction of critical variables of a CNF formula, we improve the running time for 3-SAT from O(1.32216^n) by Rolf [2006] to O(1.32153^n). Using a different approach, Iwama et al. [2010] very recently achieved a running time of O(1.32113^n). Our method nicely combines with theirs, yielding the currently fastest known algorithm with running time O(1.32065^n). We also improve the bound for 4-SAT from O(1.47390^n) [Iwama, Tamaki 2004] to O(1.46928^n), where O(1.46981^n) can be obtained using the methods of [Iwama, Tamaki 2004] and [Rolf 2006].

cs.DS↗
arXiv2010

Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least a^a^...^a clauses, where a is approximately 2 and the height of this tower is roughly k.

cs.DM↗
arXiv2010

Unsatisfiable CNF Formulas need many Conflicts

A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. A CNF formula without any conflicts is satisfiable. The Lovasz Local Lemma implies that a k-CNF formula is satisfiable if each clause conflicts with at most 2^k/e-1 clauses. It does not, however, give any good bound on how many conflicts an unsatisfiable formula has globally. We show here that every unsatisfiable k-CNF formula requires 2.69^k conflicts and there exist unsatisfiable k-CNF formulas with 3.51^k conflicts.

cs.DM↗
arXiv2010

A Full Derandomization of Schoening's k-SAT Algorithm

Schoening in 1999 presented a simple randomized algorithm for k-SAT with running time O(a^n * poly(n)) for a = 2(k-1)/k. We give a deterministic version of this algorithm running in time O((a+epsilon)^n * poly(n)), where epsilon > 0 can be made arbitrarily small.

cs.DS↗
arXiv2010

Using CSP To Improve Deterministic 3-SAT

We show how one can use certain deterministic algorithms for higher-value constraint satisfaction problems (CSPs) to speed up deterministic local search for 3-SAT. This way, we improve the deterministic worst-case running time for 3-SAT to O(1.439^n).

cs.DS↗
arXiv2010

Using a Skewed Hamming Distance to Speed Up Deterministic Local Search

Schoening presents a simple randomized algorithm for (d,k)-CSP problems with running time (d(k-1)/k)^n poly(n). Here, d is the number of colors, k is the size of the constraints, and n is the number of variables. A derandomized version of this, given by Dantsin et al., achieves a running time of (dk/(k+1))^n poly(n), inferior to Schoening's. We come up with a simple modification of the deterministic algorithm, achieving a running time of (d(k-1)/k * k^d/(k^d-1))^n \poly(n). Though not completely eleminating the gap, this comes very close to the randomized bound for all but very small values of d. Our main idea is to define a graph structure on the set of d colors to speed up local search.

cs.CC↗
arXiv2008

Satisfiability of Almost Disjoint CNF Formulas

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4 4^k. More generally, a (k,d)-CSP is a constraint satisfaction problem in conjunctive normal form where each variable can take on one of d values, and each constraint contains k variables and forbids exacty one of the d^k possible assignments to these variables. Call a (k,d)-CSP l-disjoint if no two distinct constraints have l or more variables in common. Let m_l(k,d) denote the largest integer m such that any l-disjoint (k,d)-CSP with at most m constraints is satisfiable. We show that 1/k (d^k/(ed^(l-1)k))^(1+1/(l-1))<= m_l(k,d) < c (k^2/l ln(d) d^k)^(1+1/(l-1)). for some constant c. This means for constant l, upper and lower bound differ only in a polynomial factor in d and k.

cs.DM↗
arXiv2007

Unsatisfiable Linear k-CNFs Exist, for every k

We call a CNF formula linear if any two clauses have at most one variable in common. Let Linear k-SAT be the problem of deciding whether a given linear k-CNF formula is satisfiable. Here, a k-CNF formula is a CNF formula in which every clause has size exactly k. It was known that for k >= 3, Linear k-SAT is NP-complete if and only if an unsatisfiable linear k-CNF formula exists, and that they do exist for k >= 4. We prove that unsatisfiable linear k-CNF formulas exist for every k. Let f(k) be the minimum number of clauses in an unsatisfiable linear k-CNF formula. We show that f(k) is Omega(k2^k) and O(4^k*k^4), i.e., minimum size unsatisfiable linear k-CNF formulas are significantly larger than minimum size unsatisfiable k-CNF formulas. Finally, we prove that, surprisingly, linear k-CNF formulas do not allow for a larger fraction of clauses to be satisfied than general k-CNF formulas.

cs.DM↗