Lower Bounds for Matroid Optimization Problems with a Linear Constraint

TL;DR

This work studies matroid optimization with a linear constraint (MOL) and proves that none of the non-trivial MOL problems admit a Fully PTAS, resolving the status for BM, CMB, KCM, and related problems. The authors introduce $\Pi$-matroids to encode hidden properties and derive unconditional hardness for oracle-EMB, then obtain reductions from EMB to all non-trivial MOL problems. They further extend hardness to the standard computational model via SAT-matroids and $f$-decoded encodings, under $\text{P} \neq \text{NP}$, establishing NP-hardness for these problems in the encoded setting. The results delineate a tight boundary for approximation schemes on general matroids and motivate targeted study of Fully PTAS possibilities on restricted matroid classes, with $\Pi$-matroids offering a versatile tool for future lower-bound proofs.

Abstract

We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid independent set, or a matroid basis constraint, (ii) additional linear constraint. A notable member in this family is budgeted matroid independent set (BM), which can be viewed as classic $0/1$-knapsack with a matroid constraint. While special cases of BM, such as knapsack with cardinality constraint and multiple-choice knapsack, admit a fully polynomial-time approximation scheme (Fully PTAS), the best known result for BM on a general matroid is an Efficient PTAS. Prior to this work, the existence of a Fully PTAS for BM, and more generally, for any problem in the family of MOL problems, has been open. In this paper, we answer this question negatively by showing that none of the (non-trivial) problems in this family admits a Fully PTAS. This resolves the complexity status of several well studied problems. Our main result is obtained by showing first that exact weight matroid basis (EMB) does not admit a pseudo-polynomial time algorithm. This distinguishes EMB from the special cases of $k$-subset sum and EMB on a linear matroid, which are solvable in pseudo-polynomial time. We then obtain unconditional hardness results for the family of MOL problems in the oracle model (even if randomization is allowed), and show that the same results hold when the matroids are encoded as part of the input, assuming $P \neq NP$. For the hardness proof of EMB, we introduce the $Π$-matroid family. This intricate subclass of matroids, which exploits the interaction between a weight function and the matroid constraint, may find use in tackling other matroid optimization problems.

Figures (3)

• Figure 1: The independent sets of the $\Pi$-matroid $M_{n,k,\alpha}(\Pi)$, with parameters $n = 4$, $k = 2$, and $\alpha = 5$. The secret family $\Pi$ contains all independent sets in the graph $G$, where $\{2,3\}$ is the only independent set in $G$ with $k$ elements.
• Figure 2: An illustration of the proof of \ref{['thm:emb']}. The figure presents the sequences of queries to the membership oracles by the algorithm on the instances $I$ and $I_S$ for a string of bits $b$, such that $S \notin Q(b)$. The label "yes" ("no") indicates that the queried set is (not) independent in the matroid. The only query that distinguishes between $I$ and $I_S$ is on the set $S$, which is not queried; thus, the algorithm returns the same output for $I$ and $I_S$.
• Figure 3: An example of a partition matroid$(E,{\mathcal{I}})$, which can be efficiently encoded. The ground set is $E = \{u_1,u_2,u_3,v_1,v_2,w_1,w_2,w_3\}$, partitioned into three sets: $U,V,W$. The independent sets are all subsets of $E$ containing at most one element from $U,V$, and $W$; that is, ${\mathcal{I}} = \{S \subseteq E~|~\forall X \in \{U,V,W\}: |S \cap X| \leq 1\}$. A simple efficient encoding of $(E,{\mathcal{I}})$ is $I = (E,U,V,W)$. Membership can be decided efficiently given $I$, by checking the feasibility of a given set $S$ w.r.t. $U,V$ and $W$.

Theorems & Definitions (40)

• Theorem 1.0
• Definition 1.1
• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Theorem 2.4
• proof
• Claim 2.5