50 research outputs found
Fully Proportional Representation as Resource Allocation: Approximability Results
We model Monroe's and Chamberlin and Courant's multiwinner voting systems as
a certain resource allocation problem. We show that for many restricted
variants of this problem, under standard complexity-theoretic assumptions,
there are no constant-factor approximation algorithms. Yet, we also show cases
where good approximation algorithms exist (briefly put, these variants
correspond to optimizing total voter satisfaction under Borda scores, within
Monroe's and Chamberlin and Courant's voting systems).Comment: 26 pages, 1 figur
A Systematic Review of Approximability Results for Traveling Salesman Problems leveraging the TSP-T3CO Definition Scheme
The traveling salesman (or salesperson) problem, short TSP, is a problem of
strong interest to many researchers from mathematics, economics, and computer
science. Manifold TSP variants occur in nearly every scientific field and
application domain: engineering, physics, biology, life sciences, and
manufacturing just to name a few. Several thousand papers are published on
theoretical research or application-oriented results each year. This paper
provides the first systematic survey on the best currently known
approximability and inapproximability results for well-known TSP variants such
as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP,
Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour
Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent
TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The
foundation of our survey is the definition scheme T3CO, which we propose as a
uniform, easy-to-use and extensible means for the formal and precise definition
of TSP variants. Applying T3CO to formally define the variant studied by a
paper reveals subtle differences within the same named variant and also brings
out the differences between the variants more clearly. We achieve the first
comprehensive, concise, and compact representation of approximability results
by using T3CO definitions. This makes it easier to understand the
approximability landscape and the assumptions under which certain results hold.
Open gaps become more evident and results can be compared more easily
Compromise Solutions for Robust Combinatorial Optimization with Variable-Sized Uncertainty
In classic robust optimization, it is assumed that a set of possible parameter realizations, the uncertainty set, is modeled in a previous step and part of the input. As recent work has shown, finding the most suitable uncertainty set is in itself already a difficult task. We consider robust problems where the uncertainty set is not completely defined. Only the shape is known, but not its size. Such a setting is known as variable-sized uncertainty. In this work we present an approach how to find a single robust solution, that performs well on average over all possible uncertainty set sizes. We demonstrate that this approach can be solved efficiently for min-max robust optimization, but is more involved in the case of min-max regret, where positive and negative complexity results for the selection problem, the minimum spanning tree problem, and the shortest path problem are provided. We introduce an iterative solution procedure, and evaluate its performance in an experimental comparison
Gap Amplification for Reconfiguration Problems
In this paper, we demonstrate gap amplification for reconfiguration problems.
In particular, we prove an explicit factor of PSPACE-hardness of approximation
for three popular reconfiguration problems only assuming the Reconfiguration
Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result
is that under RIH, Maxmin Binary CSP Reconfiguration is PSPACE-hard to
approximate within a factor of . Moreover, the same result holds even
if the constraint graph is restricted to -expander for arbitrarily
small . The crux of its proof is an alteration of the gap
amplification technique due to Dinur (J. ACM, 2007), which amplifies the
vs. gap for arbitrarily small up to the vs.
gap. As an application of the main result, we demonstrate that
Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are
PSPACE-hard to approximate within a factor of under RIH. Our proof is
based on a gap-preserving reduction from Label Cover to Set Cover due to Lund
and Yannakakis (J. ACM, 1994). However, unlike Lund--Yannakakis' reduction, the
expander mixing lemma is essential to use. We highlight that all results hold
unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the
first explicit inapproximability results for reconfiguration problems without
resorting to the parallel repetition theorem. We finally complement the main
result by showing that it is NP-hard to approximate Maxmin Binary CSP
Reconfiguration within a factor better than .Comment: 41 pages, to appear in Proc. 35th Annu. ACM-SIAM Symp. Discrete
Algorithms (SODA), 202
Computational Results for Extensive-Form Adversarial Team Games
We provide, to the best of our knowledge, the first computational study of
extensive-form adversarial team games. These games are sequential, zero-sum
games in which a team of players, sharing the same utility function, faces an
adversary. We define three different scenarios according to the communication
capabilities of the team. In the first, the teammates can communicate and
correlate their actions both before and during the play. In the second, they
can only communicate before the play. In the third, no communication is
possible at all. We define the most suitable solution concepts, and we study
the inefficiency caused by partial or null communication, showing that the
inefficiency can be arbitrarily large in the size of the game tree.
Furthermore, we study the computational complexity of the equilibrium-finding
problem in the three scenarios mentioned above, and we provide, for each of the
three scenarios, an exact algorithm. Finally, we empirically evaluate the
scalability of the algorithms in random games and the inefficiency caused by
partial or null communication
Backdoor Sets for CSP
A backdoor set of a CSP instance is a set of variables whose instantiation moves the instance into a fixed class of tractable instances (an island of tractability). An interesting algorithmic task is to find a small backdoor set efficiently: once it is found we can solve the instance by solving a number of tractable instances. Parameterized complexity provides an adequate framework for studying and solving this algorithmic task, where the size of the backdoor set provides a natural parameter. In this survey we present some recent parameterized complexity results on CSP backdoor sets, focusing on backdoor sets into islands of tractability that are defined in terms of constraint languages
Team-maxmin equilibrium: Efficiency bounds and algorithms
The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources-corresponding to the team members-that cannot communicate. It is known that such equilibrium always exists and it is unique (except degenerate cases) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in) efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances